Group Actions in Noncommutative Projective Geometry

Faculty of Sciences Department of Mathematics and Computer Science

Group actions in Noncommutative Projective Geometry

Dissertation submitted to obtain the degree of Doctor of Sciences: Mathematics at the University of Antwerp, to be defended by

Kevin De Laet

Supervisor: Antwerpen, 2017 Prof. dr. Lieven Le Bruyn

Faculteit Wetenschappen Departement Wiskunde en Informatica

Groep acties in niet-commutatieve projectieve meetkunde

Proefschrift voorgelegd tot het behalen van de graad van doctor in de wetenschappen: wiskunde aan de Universiteit Antwerpen te verdedigen door

Kevin De Laet

Promotor: Antwerpen, 2017 Prof. dr. Lieven Le Bruyn

Acknowledgements

This thesis could not have been written without the support and guidance of my supervisor, Professor Lieven Le Bruyn, with whom I had uncount- able mathematical discussions over the years and whose courses were the main reason I didn’t quit mathematics in my ﬁrst bachelor year. I would also like to express my gratitude to the other members of the jury:

• Professor Wendy Lowen for her lessons during my bachelor and master years and for being president of the jury, • Professor Michael Artin for being co-founder of noncommutative algebraic geometry, whose results regarding Artin-Schelter regular algebras were the main source of inspiration for this thesis, • Professor Boris Shoykhet for his lessons regarding noncommutative geometry during my master years, • Professor J. Toby Stafford for our mathematical discussions during conferences and seminars. His critical reading of my thesis and his comments improved the final version of this manuscript, • Professor Michel Van den Bergh for being co-founder of noncommutative projective geometry, being my supervisor during my first Phd year and his lessons during my master years, and • Professor Fred Van Oystaeyen for being co-founder of noncommutative algebraic geometry, his lessons during my bachelor and master years and his comments during my preliminary defence, which will inspire my research for the years to come.

In addition, I would also like to thank the following people:

• Professor S. Paul Smith, whose questions led to most of the theorems of Chapter2,

i • my mother Alexia Van Brussel for her unwavering support, • Frederik Caenepeel for sharing an oﬃce with me the last couple of years, • Theo Raedschelders for being proofreader for most of my papers, and • ”the dudes” Micha¨elMetsers, Wout Van Echelpoel and Jochen Van Grootven for making sure I didn’t get too obsessed by my mathematical studies.

ii for my father, Erik De Laet (1961-2012)

Abstract

In this thesis, we consider graded, connected C-algebras, ﬁnitely generated in degree one such that a reductive group G acts on them as gradation preserving algebra automorphisms. The general construction of such algebras is discussed, together with some interesting examples. In addition, some extra results on Sklyanin algebras (a 2-dimensional family of graded algebras parametrized by an elliptic curve E and a point τ ∈ E) of any global dimension n are proved using the action of the 3 Heisenberg group Hn of order n . Of special interest is the case that τ is of order two and that n is odd, in which case the associated Sklyanin algebra is a graded Cliﬀord algebra. This allows us to calculate the center and the PI-degree of these algebras in this particular case. In the last chapter, the study of point modules of quantum polynomial rings is discussed. Quantum polynomial rings of global dimension n form n(n−1) ∗ n a 2 -dimensional family of algebras on which (C ) acts. It is shown that the point modules are parametrized by unions of coordinate subspaces, with the generic point corresponding to the full graph on n points.

Contents

Acknowledgementsi

Nederlandse samenvatting xiii

Introduction xvii Bibliographical comment...... xx

Conventions and notations xxiii

1 Construction of G-algebras1 1.1 G-deformations...... 2

1.1.1 Symmetries on Vk ...... 8 1.2 Character series...... 9 1.3 Twisting...... 10 1.4 Ore extensions...... 12

1.5 Sn+1-deformations of C[V ]...... 13 1.5.1 Classiﬁcation of the simple objects in Proj(A).. 15

2 3-dimensional Sklyanin algebras 21 2.1 Deﬁnitions and basic properties...... 22 2.2 The superpotential and the center...... 24 2.2.1 The superpotential...... 24 2.2.2 The center...... 26

2.3 Birational maps of the projective plane coming from H3- deformations of C[V ]...... 29 2.4 Quotients of non-regular quadratic algebras...... 31

vii 2.4.1 The Hilbert series...... 34

2.4.2 Point modules of A/It ...... 36 2.4.3 The central element...... 38 2.4.4 An analogue of the twisted coordinate ring... 38 2.4.5 Connection with the Cliﬀord algebra...... 42 2.4.6 Roots of unity...... 46 2.4.7 The algebra S([1 : 0 : 0])...... 49 2.4.8 The controlling variety...... 49 2.4.9 The case t = 0, ∞ ...... 52 2.5 Representations of Sklyanin algebras of global dimension 3 at torsion points...... 53 2.5.1 Graded Cayley-Hamilton algebras...... 53

2.5.2 From Proj(A) to trepnA ...... 55 2.5.3 The non-commutative blow-up...... 63

3 Graded Clifford algebras with an action of Hp 69 3.1 Definition...... 69 3.2 Some additional results on graded Clifford algebras... 70 3.3 p = 3...... 72 3.4 p = 5...... 73 3.4.1 The nonregular algebras...... 73 3.4.2 The Koszul property...... 78 3.5 Arbitrary p prime...... 79 3.5.1 The quantum spaces...... 79

4 n-dimensional Sklyanin algebras 85 4.1 Deﬁnition and basic properties...... 85 4.2 Order 2 Sklyanin algebras...... 87 4.2.1 5-dimensional Sklyanin algebras associated to 2- torsion points...... 88

4.2.2 From trepn(A) to Proj(A)...... 92 4.3 Relations for 4-dimensional Sklyanin algebras...... 93

4.3.1 Decomposition of H4-modules...... 94

viii 5 Quantum polynomial rings 99

5.1 Tn+1-deformations...... 99 5.2 Possible conﬁgurations...... 104 5.3 Degeneration graphs...... 108 5.3.1 Quantum P2’s...... 109 5.3.2 Quantum P3’s...... 109 5.3.3 Quantum P4’s...... 110 5.3.4 Quantum P5’s...... 110

Appendices 113

A The ﬁnite Heisenberg groups 115 A.1 Deﬁnition...... 115 A.2 Representation theory...... 115

B Noncommutative algebraic geometry 119 B.1 Introduction...... 119 B.2 Simple objects in qgr(R)...... 120 B.3 Connection with representation theory...... 121 B.4 An example: graded Clifford algebras...... 122 B.4.1 Representations of graded Clifford algebras.... 123 B.4.2 The Proj of graded Clifford algebras...... 123 B.5 Koszul algebras...... 125

C Theta functions 127 C.1 Solution of the functional equations...... 127 C.2 n is odd...... 128

List of Figures

1.1 Point variety of C[x, y][t; δ]...... 19

2.1 The automorphism φ ...... 37

3.1 First conﬁguration...... 76 3.2 Second conﬁguration...... 77

5.1 Degeneration graph for quantum P3’s...... 110 5.2 Degeneration graph for quantum P4’s...... 111

Nederlandse samenvatting

Niet-commutatieve algebra¨ıschemeetkunde is een relatief nieuwe tak van de wiskunde, waarbij men tracht om aan niet-commutatieve algebra’s meetkundige objecten te associëren. Hierbij worden er vaak extra voorwaarden op de beoogde algebra’s gelegd, doordat de meeste algebra¨ısche technieken niet op elke niet-commutatieve algebra van toepassing zijn. Deze voorwaarden kunnen variërenvan ringtheoretische (de PI eigen- shap, het Noethers zijn, domeinen, ...) tot homologe aard (de Koszul eigenschap, de correcte Hilbertreeks hebben, Auslanderregulariteit, ...) De ATV-benadering (Artin-Tate-Van den Bergh) bestudeert de Artin- Schelter (AS) reguliere algebra’s en is ongeveer 25 jaar oud. De algebra’s bestudeerd in deze theorie zijn N-gegradeerde algebra’s die voldoen aan een aantal homologe voorwaarden. De bekendste kwadratische AS- reguliere algebra’s zijn de Sklyanin algebra’s van globale dimensie 3, die het onderwerp waren van mijn masterthesis. Deze doctoraatsthesis is ontstaan uit volgende observatie van mijn masterthesis: gegeven een Sklyanin algebra van globale dimensie drie, dan werkt de Heisenberg groep van orde 27 op dergelijke algebra als graadbe- warende algebra-automorfismen. Dit impliceert onder meer dat voor een 3-dimensionale Sklyanin algebra A de volgende eigenschappen gelden: ∼ •A 1 = V met V een 3-dimensionale simpele representatie van H3 en ∼ •A = T (V )/(R), waarbij R ⊂ V ⊗ V en R = V ∧ V als H3- representatie.

Deze observatie leidde tot de volgende 3 vragen.

1. Kan men de representatietheorie van H3 gebruiken om extra informatie over 3-dimensionale Sklyanin algebra’s te bekomen? Of in een meer algemene context, kan de Hn-actie op de Sklyanin algebra’s van globale dimensie n meer informatie geven over deze algebra’s?

xiii 2. Kan men, vertrekkende vanuit een reductieve groep G en een G- moduul van eindige dimensie V , nieuwe algebra’s bekomen die in- teressant zijn? Bijvoorbeeld: algebra’s met Hilbert reeks (1 − t)−n, algebra’s die eindig zijn over hun centrum, Koszul algebra’s, ...

3. Kunnen de quantum polynoomringen ook in deze theorie gesitueerd worden, met andere woorden bestaat er een reductieve groep G en een eindigdimensionale representatie V zodat de quantum poly- noomalgebra’s uit G en V kunnen gereconstrueerd worden?

Deze 3 vragen worden bestudeerd in deze thesis. Hoofdstuk1 zorgt voor de nodige theorie om vraag 2 te beantwoor- den en toont aan dat (bijna) alles mogelijk is. Het hoofddoel is om gegradeerde algebras A te vinden zodanig dat A =∼ C[V ] als gegradeerd G-moduul, waarbij V een eindigdimensionale G-representatie is. Hiervoor wordt de deﬁnitie van een G-deformatie van graad k voor k ∈ N gegeven en bestuderen we een aantal voorbeelden. Ook een aantal constructies om G-deformaties te vinden worden behandeld, namelijk twisten en Ore extensies. Het volgende hoofdstuk behandelt de eerste vraag, waarbij de Sklyanin algebra’s van globale dimensie drie bestudeerd worden via H3-modulen. Dit hoofdstuk heeft 3 doelen:

• het centraal element van graad 3 dat elke Sklyanin algebra bevat een intrinsieke betekenis geven,

• de representaties van 3-dimensionale Sklyanin algebra’s geassocieerd aan punten van eindige orde behandelen in het kader van positief gegradeerde Cayley-Hamilton algebra’s en als laatste

• de quoti¨entenvan degeneraties van Sklyanin algebra’s onderzoeken zodanig dat als gegradeerd H3-moduul, deze nieuwe algebra’s isomorf zijn aan de polynoomring in 3 variabelen.

In hoofdstuk3 worden gegradeerde Clifford algebra’s van globale priemdi- mensie bestudeerd waarop Hp werkt als automorfismen, waarbij voor p = 5 er een classificatie van de AS-reguliere algebra’s wordt gegeven. p−1 De moduli-ruimte zal voor elk oneven priem p gelijk zijn aan P 2 , waarbij we aantonen dat er exact p + 1 punten zijn die isomorf zijn met de quantum Clifford algebra C−1[x0, ... , xp−1]. Hoofdstuk4 behandelt n-dimensionale Sklyanin algebras met n ≥ 3. We tonen aan dat als n oneven is en het gekozen punt τ ∈ E van orde 2 is, dat deze algebra’s dan Clifford algebra’s zijn. Hiermee kan dan eenvoudig de PI-graad in dit specifiek geval uitgerekend worden. Voor n = 5 bepalen

xiv we ook de point modules en fat point modules met behulp van Appendix B. Het laatste hoofdstuk behandelt de laatste vraag, waarbij de groep G = (C∗)n+1 van belang is. Bijkomend worden ook de mogelijke conﬁguraties van puntmodulen bepaald voor n ≤ 4. De Appendix bestaat uit 3 delen.

• In AppendixA wordt de representatietheorie van Hn, de Heisenberg groep van orde n3 voor elke n ≥ 2 uitgelegd. • In AppendixB wordt de ATV-benadering uitgelegd om gegradeerde algebra’s te bestuderen. Er wordt hierbij vooral nadruk gelegd op de representatietheorie van positief gegradeerde algebra’s, die van belang is in hoofdstuk2 en4. • In AppendixC worden de theta functies van orde n behandeld. Aangezien de Sklyanin algebra’s oorspronkelijk door Odesskii en Feigin in [45] werden gedeﬁnieerd aan de hand van theta functies, is het handig om expliciete formules te hebben voor dergelijke functies.

Deze thesis is slechts een tipje van de sluier. Men kan deze algebra’s bestuderen voor elke eindige groep G en elke eindigdimensionale representatie V van G. Alhoewel in deze thesis vaak de aandacht wordt gelegd op de relaties V ∧ V die gedeformeerd worden, kan men deze constructie ook uitvoeren op andere deelrepresentaties van V ⊗V met dezelfde dimensie. Echter, hoe hoger de dimensie van V , hoe moeilijker het wordt om alle G-deformaties van C[V ] te vinden in het algemeen. Het staat echter vast dat er in dit onderdeel nog veel onderzoek te verrichten is, waarbij voor elke groep G er een connectie zal bestaan met meetkundige objecten waarop G werkt (zoals in het geval van de n-dimensionale Sklyanin algebra’s, waarbij G de Heisenberg groep van orde n3 is en de meetkundige objecten bepaald worden door elliptische curven ingebed in Pn−1).

Introduction

As far as mathematical subjects go, noncommutative algebraic geometry is probably the one that encompasses the most different subtopics. The general consensus seems to be that noncommutative algebraic geometry is about noncommutative algebras fulfilling some properties and associating (classical) commutative structures to these algebras. The question of what these properties have to be is of course open to debate, with different algebraic or homological properties leading to different interesting classes of algebras. This thesis takes the ATV (Artin-Tate-Van den Bergh) approach where one studies connected, positively graded algebras with certain regularity conditions (Artin-Schelter (AS) regularity, Koszulness,...), which are sat- isfied by the commutative polynomial ring in a finite number of variables. The motivating example of these algebras was given by the 3-dimensional Sklyanin algebras, studied in great detail in [3], [4], [6] and many other papers. These algebras are indeed AS-regular and depend on an elliptic curve E and a point τ ∈ E. In my master thesis, a connection was made between these Sklyanin algebras and the Heisenberg group H3 of order 27, a finite 3-group. More specifically, these algebras arose naturally if one takes V to be a simple 3-dimensional representation of H3 and as relations a subspace of V ⊗ V isomorphic to V ∧ V as H3-representation. As a consequence, H3 acts on a 3-dimensional Sklyanin algebra as algebra automorphisms. This observation led to some natural questions.

1. Can one get more information about the 3-dimensional Sklyanin algebras by using the H3-action? More generally, can one study the n-dimensional Sklyanin algebras by the action of Hn for n ≥ 3? 2. Are there other reductive groups G and representations V of G such that one gets interesting algebras using this construction? 3. Where do the quantum algebras (another important class of regular algebras) ﬁt in this picture?

xvii These 3 questions are the topic of this thesis. Chapter1 shows how, given a reductive group G and a finite dimensional representation V of G, the algebras studied here are defined. Some special constructions like twisting and Ore extensions are discussed, together with some examples. Chapter2 is the longest chapter and is a combination of [21] and [22]. It answers the first of the above questions in an affirmative way, with some additional results on the representation variety of Sklyanin algebras associated to torsion points of the elliptic curve. Chapter3 comes from [19] and deals with the second question from above, 3 with G = Hp the Heisenberg group of order p for p prime. The graded Clifford algebras with an action of Hp are discussed, with in dimension 5 a classification of the regular algebras. For general p, it is shown that there are exactly p + 1 algebras which are isomorphic to the quantum algebra C−1[x0, ... , xp−1]. Chapter4 recalls how n-dimensional Sklyanin algebras are constructed in for example [45] by Odesskii and Feigin. Using this definition, it is shown that for order 2 points in odd dimension these algebras are Clifford algebras, which solves the question of the PI-degree for Sklyanin algebras at torsion points in this particular case. In addition, it is explained how a presentation of the 4-dimensional Sklyanin algebras can be found using the representation theory of H4. In the last chapter, the last question is answered. In addition, for dimension n ≤ 4, the possible configurations of point varieties of quantum Pn are calculated. For n = 5, there is a strange phenomenon which makes the classification in higher dimensions next to hopeless. For the readers’ benefit, there are 3 chapters in the Appendix. As the finite Heisenberg groups of order n3 play an important role in this thesis, AppendixA defines these groups for all n ≥ 2 and discusses the representation theory of these groups. AppendixB deals with noncommutative algebraic geometry in the style of ATV. In particular, the motivation behind the study of point modules and fat point modules is explained, with particular emphasis on the connection with representation theory of graded algebras with Hilbert series (1 − t)−n which are finite modules over their center. In AppendixC the theta functions of order n with respect to an elliptic curve E = C/(Z + Zη) are discussed. As the Sklyanin algebras were originally defined by Odesskii and Feigin in [45] using such functions, it seems useful to have explicit descriptions. As the constructed algebras depend on 2 choices (a reductive group G and a representation V ), it is clear that the more complicated G and V , the harder it is to classify all G-deformations of C[V ]. In addition, it is not necessary to deform the relations of C[V ] to get AS-regular algebras,

xviii one can choose any AS-regular algebra (for example C−1[x0, ... , xp−1] as in Chapter3) as the ‘basis’ algebra. The examples in this thesis show how diﬀerent all these algebras behave depending on G and V . I invite the interested reader to apply this construction to his/her favourite group G and a G-representation V and see what kind of algebras are constructed.

xix Bibliographical comment

This thesis is based on the following articles:

• Kevin De Laet, Graded Cliﬀord algebras of prime global dimension with an action of Hp, Communications in Algebra 43 (2015), no. 10, 4258-4282, • Kevin De Laet and Lieven Le Bruyn, The geometry of representations of 3-dimensional Sklyanin algebras, Algebras and Representa- tion Theory 18 (2015), no. 3, 761-776, • Kevin De Laet, Character series and quaternionic Sklyanin algebras, arXiv preprint arXiv:1412.7001 (2014), • Pieter Belmans, Kevin De Laet, and Lieven Le Bruyn, The point variety of quantum polynomial rings, Journal of Algebra 463 (2016), 10-22, Elsevier • Kevin De Laet, Quotients of degenerate Sklyanin algebras, Journal of Algebra and Its Applications (2016), World Scientiﬁc • Kevin De Laet, Constructing G-algebras, arXiv preprint arXiv:1605.09265 (2016).

xx Major revisions

The following major revisions in this thesis have been made following the comments of the jury during the preliminary defence:

• the deﬁnitions of G-deformations and G-isomorphisms up to degree k in Chapter1 have been altered, • in Chapter2, Section 2.4 there are some new theorems showing the primeness of the constructed algebras, and

• in general, more references to books and papers about Cliﬀord algebras and Koszul algebras are given.

xxi

Conventions and notations

In order to read this thesis, one needs a basic knowledge about group theory, representation theory, algebraic geometry and noncommutative ring theory. The following books are suggested for the readers’ beneﬁt:

• Claudio Procesi, Lie Groups: An Approach through Invariants and Representations, Springer Science & Business Media (2007),

• Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathe- matics 52, Springer Science & Business Media (1977),

• John C. McConnell, James Christopher Robson and Lance W. Small, Noncommutative noetherian rings, American Mathematical Soc. 30 (2001),

• Constantin Nastasescu and Freddy Van Oystaeyen, Methods of Graded Rings, Springer Science & Business Media (2004).

In addition, the following papers gives a summary about noncommutative projective geometry:

• J Toby Staﬀord, Noncommutative projective geometry, Algebra- Representation Theory (2001), 415-417, Springer,

• Daniel Rogalski, An introduction to Noncommutative Projective Geometry, arXiv preprint arXiv:1403.3065 (2014).

Throughout this thesis, the following conventions are used.

• C will be the ﬁeld of complex numbers.

• Every algebra A will be a C-algebra and A will be an associative, unital, ﬁnitely generated algebra.

xxiii • The group GLa(F) for F a ﬁeld will be the general linear group of degree a over F. If V is an a-dimensional vector space over F with a < ∞, then GL(V ) will be the group of linear automorphisms of V (which is of course isomorphic to GLa(F)). Additionally, the groups SLa(F), PGLa(F), respectively PSLa(F) are the special linear group, the projective general linear group, respectively the projective special linear group of degree a over F. If F = Fpn for some n n n prime number p, then the abbreviations GLa(p ), SLa(p ), PGLa(p ) n and PSLa(p ) are used instead of GLa(Fpn ), SLa(Fpn ), PGLa(Fpn ) or PSLa(Fpn ).

• If n = 0, then GLn = {1}, the trivial group.

• The groups Sn and An will be the symmetric and alternating group on n letters.

• Let V be a ﬁnite dimensional F-vector space. Then F∗ will be embedded in GL(V ) as homotheties.

n n−1 • The set V(I ) ⊂ A or P for I ⊂ C[a1, ... , an] an ideal is the Zariski-closed subset determined by I , it will be clear from the context if the projective or aﬃne variety is used if I is homogeneous.

• The Zariski-open subset D(I ) for I an ideal I ⊂ C[a1, ... , an] will be equal to An \ V(I ) or Pn−1 \ V(I ), it will be clear from the context if it is an open subset of aﬃne space or of projective space. If I = (a), then D(a) is used instead of D(I ).

• The group Zn = Z/nZ will be the cyclic group of order n for n ∈ N. k To emphasize the ﬁeld structure if n = p with p prime, Fpk will be used.

• The vector space Ma×b(F) will be the a × b-matrices over F. If a = b, then Ma(F) = Ma×a(F), in which F itself is embedded as scalar matrices.

• The following subsets of Ma×b(F) and Ma(F) are deﬁned:

s Ma×b(F) = {M ∈ Ma×b(F) : rank(M) = min{a, b}}

and s Ma(F) = GLa(F).

• The projective variety Grass(m, n) with 0 ≤ m ≤ n will be the projective variety parametrizing m-dimensional vector spaces in Cn. s Alternatively, Grass(m, n) = Mm×n(C)/GLm(C).

xxiv • For an algebra A and elements x, y ∈ A, set {x, y} = xy + yx. Similarly, for an algebra A and elements x, y ∈ A, set [x, y] = xy − yx.

• If G is a group, then for g, h ∈ G, let [g, h] = ghg −1h−1. The derived subgroup [G, G] is the subgroup of G generated by all [g, h], g, h ∈ G. This is always a normal subgroup and the quotient Gab = G/[G, G] is the abelianization of G.

• The algebra A is Z-graded if A has a decomposition as C-vector space A = ⊕k∈ZAk such that

∀k, l ∈ Z, ∀a ∈ Ak , ∀b ∈ Al : ab ∈ Ak+l .

If Ak = 0 whenever k < 0, then A is positively graded. If A0 = C, k then A is connected. If ∀k ∈ N, Ak = (A1) , then A is generated in degree 1.

• A Z-gradation on an algebra A is the same as saying that C∗ acts on A as automorphisms, by the equivalence

∗ n a ∈ An ⇔ ∀t ∈ C : t · a = t a.

These two equivalent deﬁnitions will be used simultaneously.

• A Z-graded algebra A is strongly graded if ∀k, l ∈ Z : Ak Al = Ak+l . Equivalently, A is strongly graded if ∀k ∈ Z : 1 ∈ Ak A−k .

• Let A = ⊕m∈ZAm and B = ⊕m∈ZBm be graded algebras. Assume that f : A / B is a graded algebra morphism, then for every k ∈ Z, the map fk is the unique map fk : Ak / Bk making the following diagram commute

f A / B O

ik Πk ? fk Ak / Bk

The linear map ik is the natural inclusion Ak / A and Πk is

the natural projection map B / / Bk .

• For V a ﬁnite dimensional vector space over C, the following associative algebras are deﬁned:

∞ ⊗k ⊗0 – the tensor algebra over V : T (V ) = ⊕k=0V with V = C,

xxv – the polynomial ring over V : C[V ] = T (V )/(V ∧ V ), with V ∧V the vector space generated by the antisymmetric tensors v ⊗ w − w ⊗ v for v, w ∈ V , and – the wedge algebra over V : ∧V = T (V )/(S 2(V )), with S 2(V ) the vector space generated by the symmetric tensors v ⊗ w + w ⊗ v for v, w ∈ V .

• If ρV : G / GL(V ) is a representation of G, then ρV is assumed to be faithful. This way, G can be identiﬁed with ρV (G).

• The vector space Derσ(A) will be the σ-derivations of A for σ ∈ Aut(A). If A is graded and σ preserves this gradation, then the σ-derivations of A become a graded vector space

σ σ Der (A) = ⊕m∈Z Der (A)m

with

σ σ Der (A)m = {f ∈ Der (A): f (An) ⊂ An+m∀n ∈ Z}.

• If A is an algebra on which a group G acts as algebra automorphisms, then AutG (A) are the algebra automorphisms of A that commute with this action of G.

• For A a positively graded algebra such that ∀k ∈ N : dimC Ak < ∞, HA(t) will be the Hilbert series of A, that is,

∞ X k HA(t) = dimC Ak t . k=0

• The Euler’s totient function φ will be used, that is, φ is the function φ φ : N \{0} / N \{0} deﬁned by

φ(n) = |{k ∈ N : 1 ≤ k ≤ n, gcd(k, n) = 1}|.

• The n×n-matrix D = diag(t1, t2, ... , tn) will be the diagonal matrix deﬁned by

Dij = δij ti , 1 ≤ i, j ≤ n.

• The n-dimensional maximal subtorus of GLn(C) will be denoted by ∗ n Tn = (C ) . Most of the time it will be assumed that this group is indeed embedded in GLn(C).

xxvi • For Tn, let χej be the character deﬁned by χej (t1, ... , tn) = tj . The characters of Tn form an abelian group under the tensor product, n isomorphic to Z . For a1, ... , an ∈ Z, the abbreviation

n O ⊗a χPn = χ i i=1 ai ei ei i=1 is used.

• If A is an algebra, Z(A) will be the center of A. Similar for a group G, Z(G) will be the center of G. • If V is a G-representation of a reductive group G with decomposi- k ej tion V = ⊕j=1Sj , then it is assumed that

1. each Sj is a simple G-module,

2. ∀1 ≤ i < j ≤ k : HomG (Si , Sj ) = 0 and

3. ∀1 ≤ j ≤ k : ej 6= 0. • Let V be a G-representation, then V G will be the subspace ﬁxed by all elements of G.

• Let X1, ... , Xn be noncommuting variables of degree 1. Then the noncommutative algebra C−1[X1, ... , Xn] is the algebra deﬁned by

C−1[X1, ... , Xn] = ChX1, ... , Xni/({Xi , Xj }, 1 ≤ i < j ≤ n).

• If φ ∈ Aut(A) for some algebra A, then Aφ is the twist of A with respect to φ. • If G ⊂ Aut(A) is a subgroup, then AG is the subalgebra of A consisting of the G-invariant elements. • Let A be a positively graded, connected algebra, ﬁnitely generated in degree 1. Then A is a quantum algebra of global dimension n if and only if there exists a basis of A1 {x1, ... , xn} and elements ∗ qij ∈ C for 1 ≤ i < j ≤ n such that the deﬁning relations of A can be written as

∀1 ≤ i < j ≤ n : xi xj − qij xj xi = 0.

xxvii

Chapter 1

Construction of G-algebras

This chapter contains the necessary theory on how to construct G-algebras and G-deformations. In particular, it shows that the constructed algebras are determined by (inverse limits of) projective varieties. Examples like twisting and Ore extensions are discussed in order to construct new G- algebras. In the last part of this chapter this theory is illustrated with an example: the Sn+1-deformations of the polynomial ring C[V ] are determined, with V the natural permutation representation of Sn+1. In later chapters, the following special cases are considered:

• in chapter2, the reductive group is G = H3, the Heisenberg group of order 27 and the H3-representation V will be V1, the Schr¨odinger representation of H3 as deﬁned in AppendixA,

• in chapter3, G = Hp o Z2 and V = V1 will be a p-dimensional simple representation. The ‘basis’ algebra will not be C[V ] but the quantum polynomial ring C−1[x0, ... , xp−1],

3 • in chapter4, G = Hn will be the Heisenberg group of order n and V = V1, the n-dimensional Schr¨odingerrepresentation as deﬁned for general n ≥ 2 in AppendixA and

n+1 • in chapter5, G = Tn+1 and V = ⊕i=1 χei .

1 1.1 G-deformations

1.1.1 Definition. Let G be a reductive group (that is, each finite dimensional G-representation is semi-simple). A graded connected algebra A, finitely generated in degree 1 is called a G-algebra if G acts on it by gradation preserving automorphisms.

This implies that for each degree k ∈ N, Ak is a ﬁnite dimensional representation of G. Let V = A1, then there exists a homogeneous subspace R of T (V ) such that T (V )/(R) = A. From now on, A is a G-algebra.

In this setting, as G is reductive, each graded component Ak has a decomposition in simple G-representations.

1.1.2 Deﬁnition. Let A = T (V )/(R) and B = T (W )/(R0) be two G-algebras with A1 = V , B1 = W and let k ≥ 2 be an integer.

∼k • A is G-isomorphic up to degree k to B (notation: A =G B) if and ∼ only if ∀0 ≤ i ≤ k : Ai = Bi as G-representations. ∼ ∼k • A is G-isomorphic to B (notation: A =G B) if and only if A =G B as G-representations for each k ∈ N.

∼k In particular, if B =G A, then B is also a quotient of T (V ) with V = A1. Usually, G-algebras G-isomorphic to a specific G-algebra A will be studied, which justifies the following definition.

1.1.3 Deﬁnition. Let A be a G-algebra and k ≥ 2 an integer.

• B = T (V )/(R0) is a G-deformation of A up to degree k for k ≥ 2 ∼k 0 if B =G A and R = (g1, ... , gm) with 2 ≤ deg(gi ) ≤ k for each i ∈ {1, ... , m}. ∼ • B is a G-deformation of A if B =G A.

As follows from the deﬁnition, the relations that will be considered are always of degree larger than or equal to two. Let π : T (V ) / / A be the natural epimorphism. If A is a G-algebra, then A determines for each k ≥ 2 a short exact sequence of G-morphisms

πk 0 / ker(πk ) / T (V )k / Ak / 0 .

e a Let A =∼ ⊕nk S ik and T (V ) =∼ (⊕nk S ik ) ⊕ W as G-modules with k ik =1 ik k ik =1 ik k naturally 0 < eik ≤ aik and Wk a submodule of T (V )k such that the

2 f following holds: Hom (A , W ) = 0. Then ker(π ) =∼ (⊕nk S ik ) ⊕ W G k k k ik =1 ik k with fik = aik − eik . 0 ∼k If B = T (V )/(R ) such that B =G A, then B determines for each 2 ≤ j ≤ k a G-subrepresentation of T (V )j , isomorphic to ker(πj ). Such subrepresentations are determined by

⊗j EmbG (ker(πj ), V ) f ⊗j = ker(πj ) / V : f injective, G-linear / AutG (ker(πj )).

Due to Schur’s lemma, using the decompositions from above, this variety is isomorphic to

 nj  Y s M ( )/GLfi ( ) × AutG (Wk , Wk )/ AutG (Wk , Wk )  aij ×fij C j C  ij =1

nj ∼ Y = Grass(fij , aij ).

ij =1

It follows that a G-deformation B up to degree k of an algebra A determines a unique point in

k nj Y Y Grass(fij , aij ).

j=2 ij =1 1.1.4 Theorem. Let k ≥ 2 be a natural number. Then the set of G- deformations up to degree k of A has the structure of a projective variety.

Proof. The proof is by induction. For any degree k, put a e T (V ) =∼ ⊕nk S ik , A =∼ ⊕nk S ik . k ik =1 ik k ik =1 ik

Let fik = aik − eik be the multiplicities of the simple representations in ker(πk ) with as before πk being the natural projection map πk : T (V )k / Ak . Let Vk be the variety parametrizing G-deformations up to degree k of A. First consider k = 2. Then from the above discussion it follows that the set of G-deformations up to degree 2 of A are parametrized by Qn2 Grass(f , a ), which is clearly a projective variety. i2=1 i2 i2 Assume now that Vk−1 is a projective subvariety of

k−1 nj Y Y Grass(fij , aij ).

j=2 ij =1

3 Then a point (P , ... , P , P ) ∈ Qk Qnj Grass(f , a ) with P ⊂ 2 k−1 k j=2 ij =1 ij ij i T (V )i is an element of Vk if and only if (P2, ... , Pk−1) ∈ Vk−1 and for ⊗l ⊗k−i−l each 2 ≤ i ≤ k − 1 and each 0 ≤ l ≤ k − i, V ⊗ Pi ⊗ V ⊂ Pk . This is clearly a closed condition, so Vk is indeed closed.

As in the theorem, let Vk be the variety parametrizing G-deformations of A up to degree k.

1.1.5 Remark. It may be possible that Vk is disconnected, which makes it unfortunate to call these algebras G-deformations of A. However, I don’t know of an example, so it may be that Vk is always connected, although this seems doubtful. There are natural morphisms coming from projection maps between prod- ucts of Grassmannians

ψk+1 ψk ψk−1 ... / Vk / Vk−1 / ... such that V = lim V is the variety defining G-deformations of A. ←− k 1.1.6 Definition. A connected variety Z defines G-deformations up to degree k of a G-algebra A if Z can be embedded in Vk for some k ≥ 2 by some morphism α : Z / Vk such that the point corresponding to the algebra A is in Im(α). If for each point z ∈ Z, the algebra ∼ A(z) = T (V )/(α(z)) has the property A(z) =G A, then Z defines G- deformations of A. 1.1.7 Proposition. Let A be a G-algebra and let C be a smooth projective irreducible curve defining G-deformations up to degree k such that there exists an open subset U ⊂ C parametrizing G-deformations of A, let αk : C / Vk be the corresponding embedding. Then C naturally defines G-deformations of A, more specifically, there exists a natural embedding α : C / V extending αk .

Proof. The following diagram is commutative

ψk+2 ψk+1 ψk ... / Vk+1 / Vk / ... , b O αk αk+1 ? C with αk an embedding and αk+1 a rational morphism. Due to [53, Propo- sition 2.1], αk+1 can be extended to a morphism of C into Vk+1 which is also denoted by αk+1. As ψk+1 ◦ αk+1 coincides with αk on an open (and hence dense) subset of C, they coincide on C. As αk is an injection, αk+1 is also an injection.

4 By induction, there are commuting triangles ∀k ≥ 2 with αk an embedding for all k large enough, so there indeed exists an embedding of C into V.

1.1.8 Example. Let A = C−1[x, y, z], put V = Cx+Cy +Cz and consider the group G generated by the permutation matrices classically embedded in GL3(C) and the order 3 element 1 0 0  2πI 0 ω 0  , with ω = e 3 . 0 0 ω2

∼ −1 Then G = H3 o Z2, with the action of Z2 = hti deﬁned by t · e1 = e1 , −1 t · e2 = e2 . As G-representations, there are the decompositions V ⊗ V =∼ W ⊕2 ⊕ P and V ∧ V =∼ P with P and W simple, pairwise non-isomorphic 3- dimensional representations, so the G-deformations of A correspond to 1 ∼ 1 points of P . These algebras parametrized by V2 = P have relations  A(yz + zy) + Bx 2 = 0,  A(zx + xz) + By 2 = 0, A(xy + yx) + Bz2 = 0.

As will be explained in Chapter3, generically these are Artin-Schelter (AS) regular graded Cliﬀord algebras over a polynomial ring in three variables (as deﬁned in [11]). There are four points where the corresponding algebra doesn’t have (1 − t)−3 as Hilbert series:

S = {[0 : 1], [1 : 1], [1 : ω], [1 : ω2]} with ω3 = 1, ω 6= 1.

For example, in [0 : 1], the relation [{x, y}, z] and cyclic permutations of this relation are not implied by the relations x 2 = y 2 = z2 = 0, although they are implied for all other points of P1. So a natural extension of the 1 rational map α3 : P / V3 to the point [0 : 1] is by adding the cyclic permutations of [{x, y}, z] as extra relations.

A consequence of a connected variety Z determining G-deformations is that for each i ∈ N the function βi : Z / N , z / dimC A(z)i is constant. A natural question to ask follows: if Z deﬁnes G-deformations up to degree k for some natural number k ≥ 2 and if the condition

∀i ∈ N : βi (z) = dimC Ai holds, does Z then deﬁne G-deformations of A?

5 1.1.9 Lemma. Let A be a G-algebra with π : T (V ) / / A the natural epimorphism. Then for each m ∈ N there exists a subrepresentation ⊗m Wm ⊂ V such that πm|Wm : Wm / Am is an isomorphism.

Proof. This is just a consequence of the fact that G is a reductive group, which implies that for each m ∈ N, πm has a section, that is, a G- ⊗m morphism sm : Am / V such that πm ◦ sm = I dAm . Then the subrepresentation Wm = Im(sm) will do.

∼ jm ej Let Am = ⊕j=1Sj be the decomposition of A as G-module and let Wm be as in the previous lemma. For each 1 ≤ j ≤ jm and each 1 ≤ k ≤ ej , there are G-generators vj,k of Wm such that

∼ ∼ •∀ 1 ≤ j ≤ jm, ∀1 ≤ k < l ≤ ej : Sj = CGvj,k = CGvj,l as G- modules and the map fk,l : CGvj,k / CGvj,l deﬁned by

∀a ∈ CG : fk,l (avj,k ) = avj,l is a G-isomorphism, and

• the following decomposition of Wm as G-modules holds:

jm ej Wm = ⊕j=1 ⊕k=1 CGvj,k .

Consequently, by the fact that sm is a section, it holds that

jm ej Am = ⊕j=1 ⊕k=1 CGπm(vj,k ).

Let Xm = {vj,k : 1 ≤ j ≤ jm, 1 ≤ k ≤ ej } be the set of these generators. 1.1.10 Theorem. Let Z be an irreducible variety deﬁning G-deformations up to degree k of a G-algebra A with k ≥ 2. For z ∈ Z, let A(z) be the corresponding algebra. Assume that

∀z ∈ Z : HA(z)(t) = HA(t).

Then Z deﬁnes G-deformations of A or equivalently, ∼ ∀z ∈ Z : A(z) =G A.

Proof. For z ∈ Z, let πz be the canonical epimorphism z Lk z π : T (V ) / / A(z) and let R(z) = ker( j=2(π )j ) be the vector z space generating the kernel of π . Let z0 be the point of Z corresponding to A itself, that is, A(z0) = A.

6 Let K(z) = (R(z)) be the kernel of πz , which is a graded ideal of T (V ).

Assume that the theorem is not true and let z1 ∈ Z be a point ful- ∼ filling A(z0) =6 A(z1) as graded G-modules. Then there exists a m ∈ ∼ N minimal such that A(z0)m =6 A(z1)m with m ≥ 3. By the previ- ⊗m ous lemma, there are G-submodules (W0)m,(W1)m ⊂ V such that z0 z1 (π )m :(W0)m / A(z0)m and (π )m :(W1)m / A(z1)m are G-isomorphisms. Let X0,m, respectively X1,m be sets of G-generators of (W0)m and (W1)m fulfilling the previous requirements. Define the following subsets of Z

z (π )m U0 = {z ∈ Z : the map (W0)m / A(z)m is a G-isomorphism},

z (π )m U1 = {z ∈ Z : the map (W1)m / A(z)m is a G-isomorphism}.

Then these two subsets are equal to

z U0 = {z ∈ Z : the set {π (w): w ∈ X0,m} is linearly independent}, z U1 = {z ∈ Z : the set {π (w): w ∈ X1,m} is linearly independent}.

U0 and U1 are both Zariski open, for the fact that the complements of U0, respectively U1 are the sets

M ⊗m C0 = {z ∈ Z : K(z)m ⊕ CGw 6= V }, w∈X0,m M ⊗m C1 = {z ∈ Z : K(z)m ⊕ CGw 6= V }, w∈X1,m which can be described as a certain matrix not having rank nm, which puts polynomial restrictions on the points of Z. In addition, U0 6= ∅ and U1 6= ∅, for z0 ∈ U0 and z1 ∈ U1. As Z is irreducible, U0 ∩ U1 6= ∅. Let ∼ y ∈ U0 ∩ U1, then from the deﬁnition of U0 and U1, one has (W0)m = (W1)m. But this is impossible, for this would imply that ∼ ∼ ∼ (A(z0))m = (W0)m = (W1)m = (A(z1))m, which is a contradiction.

A trivial consequence of this theorem follows. 1.1.11 Corollary. Let Z be a connected variety deﬁning G-deformations up to degree k for some k ≥ 2 of a G-algebra A such that for each point of Z, the Hilbert series of the corresponding algebra ∼ stays constant. Then for every 2 points x, y ∈ Z, one has A(x) =G A(y).

7 1.1.1 Symmetries on Vk

Assume that A = T (V )/(R) is a G-algebra as before and let

−1 H = NGL(V )(G) = {h ∈ GL(V )|hρV (g)h ∈ G}.

−1 By convention, ρV is injective. Let ρV be the inverse function with 0 domain the image of ρV . Let H be the maximal element with respect to inclusion of the set

T = {H00 ⊂ H subgroup : A is a H00-algebra}.

Note that T 6= ∅ for G ∈ T . There exists a morphism 0 −1 −1 H / Aut(G) deﬁned by h 7→ ϕh, ϕh(g) = ρV (hρV (g)h ). Let W be a G-representation, then one can twist the representation

ρW : G / GL(W ) of G with ϕh

ϕh ρW G / G / GL(W ), which deﬁnes a new representation ρW ◦ϕh of G, possibly non-isomorphic to W (except if V = W of course). Denote this new G-representation by W h. It is obvious that if W is simple, then W h is also simple. 0 By deﬁnition, A is an H -algebra. Assume that S ⊂ ker(πk ) is a simple G-module. Then for each h ∈ H0, the subrepresentation S h = hS is also contained in ker(πk ) by assumption. 1.1.12 Proposition. Let A = T (V )/(R) be a G-algebra and H, H0 as 0 before. Then H acts on Vk such that points in the same orbit correspond to isomorphic algebras.

Proof. Let h ∈ H0 and B a G-deformation of A. Deﬁne a new action of G on B by g · v = hgh−1v, then B is a G-algebra under this action, with the degree 1 part isomorphic to V . Taking as generators of B the elements yi = hxi , for some basis x1, ... , xm of V , one can decompose the relations of B with respect to the generators yi in the same way as one did for the generators x1, ... , xm. But these 2 decompositions will be the same as H0 acts on the set of simple representations of G and A was also a H0-algebra.

One would expect that the centralizer of G in GL(V ) acts trivially on Vk , but this will not be the case in general. However, if V is a simple representation, then this is obviously true. 0 If A = C[V ] or A = ∧V , then H = H and there is an action of H on Vk for each k ≥ 2.

8 1.1.13 Example. Let V = χe1 ⊕ χe2 = Cx1 + Cx2 and G = T2. Then 1 1 the T2-deformations of C[V ] are parametrized by P with [a : b] ∈ P corresponding to the algebra

T (V )/(ax1x2 − bx2x1).

The normalizer of T2 ⊂ GL(V ) is the semidirect product T2 o Z2. The group T2 acts trivially on this moduli space, so the action boils down to 1 the action of Z2 on P deﬁned by s · [a : b] = [b : a], which indeed gives isomorphic algebras.

1.2 Character series

A useful way to decode decompositions of G-algebras in hC∗, Gi-modules can be found in [24] and [51, Chapter 9, Section 4.3], although the focus there lies on Sn-representations. 1.2.1 Deﬁnition. Let g ∈ G and A a G-algebra. Deﬁne the character series chA(g, t) to be the formal power series

X n chA(g, t) = χAn (g)t . n∈N

1.2.2 Example. Naturally, chA(1, t) = HA(t). In general, if g = λIn, then chA(g, t) = HA(λt).

In case G is a ﬁnite group with m conjugacy classes, then the decomposition of A in hC∗, Gi-representations can be decoded by an element f (t) of C[[t]]m with

f (t) = (chA(g1, t), chA(g2, t), ... , chA(gm, t)), with each gi a representative of the ith conjugacy class. In the case that A is a Koszul algebra (deﬁned in AppendixB), the following holds (see for reference [30]):

1.2.3 Proposition. Let g ∈ G and A be a Koszul G-algebra, then

chA(g, t)chA! (g, −t) = 1.

This proposition follows directly from the exactness of the Koszul complex.

9 1.3 Twisting

The main G-deformations studied here are those of the polynomial ring C[V ]. Although the variety EmbG (V ∧ V , V ⊗ V ) can be a completely arbitrary product of Grassmannians, the dimension of Vk can be bound from below using twisting as deﬁned in for example [68]. 1.3.1 Deﬁnition. Let B be a graded algebra and β ∈ Aut(B) an au-

∗ tomorphism that preserves the gradation (equivalently β ∈ AutC (B)). Then the twist Bβ is deﬁned as the graded associative algebra with un- derlying vector space the elements of B and multiplication rule

k ∀a ∈ Bk , b ∈ Bl : a ∗β b = aβ (b).

1.3.2 Proposition. Let Vk be the variety corresponding to ∼ n ei G-deformations up to degree k of C[V ] and let V = ⊕i=1Si be the decomposition of V in simple G-representations. Then

n X 2 dim Vk ≥ −1 + ei . i=1

∼ Qn Proof. Let α ∈ AutG (V ) = i=1 GLei (C). Then the map

−1 fα = I d ⊗ α : V ⊗ V / V ⊗ V is also a G-isomorphism, and fα(V ∧ V ) is the vector space of deﬁning α ∼ relations of C[V ] . Therefore fα(V ∧ V ) = V ∧ V as G-representation. As C[V ] has the property

∗ ∀α, β ∈ Aut(C[V ]) : fα(V ∧ V ) = fβ(V ∧ V ) ⇒ ∃λ ∈ C : α = λβ,

Qn ∗ we have that each α ∈ ( i=1 GLei (C))/C deﬁnes a G-deformation of C[V ].

It can be the case that G-deformations up to degree 2 of C[V ] are (generically) twists of C[V ], as the next example will show. 1.3.3 Example. Let G = T2 again be the 2-dimensional torus and let

V = χe1 ⊕ χe2 . Then there is the decomposition V ⊗ V ∼ χ ⊕ χ ⊕ χ2 , = 2e1 2e2 e1+e2 ∼ V ∧ V = χe1+e2 .

So the G-deformations up to degree 2 of C[V ] are parametrized by P1. The twists of C[V ] commuting with the action of T2 are given by T2 itself, so the twists give a 1-dimensional family of G-deformations of C[V ].

10 In fact, every point of P1 deﬁnes a G-deformation of C[V ], as for each point p = [a : b] ∈ P1 and A(p) = Chx, yi/(axy − byx), the Hilbert −2 series is HA(p)(t) = (1 − t) and using Corollary 1.1.11, one concludes that 1 ∼ ∀p ∈ P : A(p) =G C[V ].

If one is interested in whether there are twists of A which are also G- algebras but not necessarily G-deformations, then there is an improvement upon the previous theorem.

∗ 1.3.4 Theorem. Let A = T (V )/(R) be a G-algebra and φ ∈ AutC (A) such that ∀g ∈ G :[φ, g] ∈ Z(GL(V )). φ φ ∼ Then the twist A is also a G-algebra with (A )1 = V .

φ φ Proof. Let R be the relations of A and let φk be the vector space automorphism on V ⊗k deﬁned by k−1 φk (xi1 ... xik ) = xi1 φ(xi2 ) . . . φ (xik ) φ with xj ∈ V and extending this linearly. From the construction of A , it follows that −1 φ f ∈ Rk ⇔ φk (f ) ∈ Rk . −1 φ One needs to show that, for any g ∈ G, g · φk (f ) ∈ Rk . Let [φ, g] = λ ∈ C∗. Then the following holds: k g · φ−1(f ) = λmφ−1(g · f ), m = k k 2

−1 φ and as g · f ∈ Rk , g · φk (f ) ∈ Rk .

φ ∼ However, in general it is not true that A =G A.

1.3.5 Example. Let D4 = H2 be the Heisenberg group of order 8 (which is the same as the dihedral group of order 8) and take the Schr¨odinger representation of H2, that is, V = Cx + Cy, deﬁned by the matrices 0 1 1 0 e 7→ , e 7→ . 1 1 0 2 0 −1

Then the induced map H2 / PGL2(C) has as kernel the group gener- ∼ ated by [e1, e2] and there is the isomorphism H2/([e1, e2]) = Z2 × Z2. So for example e2 satisﬁes the condition of the previous theorem. Therefore, let A = C[V ], then Ae2 =∼ Chx, yi/(xy + yx). ∼ ∼ But C(xy − yx) =6 C(xy + yx) as H2-representations, as C(xy − yx) = ∼ ∼ e2 χ1,1 while C(xy + yx) = χ0,1. So A =6 A as graded H2-module.

11 1.4 Ore extensions

Another way to make G-algebras is by using Ore extensions. Recall that an Ore extension of A is of the form A[t; σ, δ] with σ ∈ Aut(A) and δ ∈ Derσ(A). As the algebras discussed here are generated in degree 1, σ σ has to preserve the gradation and δ ∈ Der (A)2. 1.4.1 Proposition. Let A = T (V )/(R) be a quadratic G-algebra, χ a representation of degree 1 of G, σ ∈ AutG (A) gradation preserving and σ δ ∈ Der (A)2 such that

−1 ∀g ∈ G, ∀x ∈ A1 : δ(g(x)) = χ (g)g(δ(x)). (1.1)

Then A[t; σ, δ] is a G-algebra such that deg(t) = 1 and Ct =∼ χ. Con- ∼ versely, if A[t; σ, δ] is a G-algebra such that Ct = χ, then σ ∈ AutG (A) σ and δ ∈ Der (A)2 fulﬁls the above property.

Proof. The relations of an Ore extension of a quadratic algebra A are of the form ∀x ∈ A1 : tx − σ(x)t − δ(x) = 0. In order for G to act on such an extension such that Ct =∼ χ, g(tx − σ(x)t − δ(x)) has to be calculated for each x ∈ A1. the following holds: g(tx − σ(x)t − δ(x)) = g(t)g(x) − g(σ(x))g(t) − g(δ(x)) = χ(g)tg(x) − σ(g(x))χ(g)t − χ(g)δ(g(x)) = χ(g)(tg(x) − σ(g(x))t − δ(g(x))) = 0. For the other implication, let A[t; σ, δ] be a G-algebra such that deg(t) = 1 and Ct =∼ χ. Observe that the extra relations to get from A to A[t; σ, δ] lie in the ﬁnite dimensional vector space Ct ⊗ V ⊕ V ⊗ Ct ⊕ A2, which is also a direct sum of G-representations. These relations then form a G-subrepresentation if and only if the 2 maps

f1 : Ct ⊗ V / V ⊗ Ct

f2 : Ct ⊗ V / A2 with f1 mapping t ⊗ v to σ(v) ⊗ t and f2 mapping t ⊗ v to δ(v) ∈ A2 are G-morphisms. So σ ∈ AutG (A) and δ fulﬁls the requirements of the proposition.

Unfortunately, sometimes the only Ore extensions of C[V ] one can make are those with δ = 0.

1.4.2 Proposition. If σ ∈ C \{0, 1} and if there is no G-submodule of σ V isomorphic to χ, then there exist no 0 6= δ ∈ Der (C[V ])2 fulﬁlling requirement 1.1.

12 n−1 Proof. Take V = ⊕i=0 Cxi . From the deﬁnition of a σ-derivation, one needs that ∀0 ≤ i, j ≤ n − 1

σ(xi )δ(xj ) + xj δ(xi ) = δ(xi xj ) = δ(xj xi ) = σ(xj )δ(xi ) + xi δ(xj ).

From which it follows that

(σ(xi ) − xi )δ(xj ) = (σ(xj ) − xj )δ(xi ).

Let µ = σ − 1, then it follows that

µxi δ(xj ) = µxj δ(xi ).

By the fact that C[V ] is an UFD, it follows that δ(xi ) = vxi for some v ∈ V . By requirement 1.1, one sees that for each g ∈ G and each x ∈ V

vg(x) = δ(g(x)) = χ−1(g)g(δ(x)) = χ−1(g)g(v)g(x), from which it follows that

g(v) = χ(g)v.

So if χ is not a subrepresentation of V , then necessarily δ = 0.

1.5 Sn+1-deformations of C[V ]

Let V = S ⊕ T be the n + 1-dimensional permutation representation of Sn+1 with S n-dimensional and T the trivial representation.

1.5.1 Proposition. Let n ≥ 2. The Sn+1-deformations up to degree 2 of C[V ] are parametrized by P2.

Proof. There is the following decomposition of Sn+1-modules:

V ⊗ V = S ⊗ S ⊕ S ⊕2 ⊕ T , V ∧ V = S ∧ S ⊕ S.

The tensor product S ⊗ S decomposes as S ∧ S ⊕ T ⊕ S ⊕ W with W simple, W 6= S, S ∧ S according to [26, Exercise 4.19] if n 6= 2, if n = 2 then W = 0. In both cases, the Sn+1-deformations of C[V ] are parametrized by

⊕3 ∼ 2 EmbSn+1 (S ∧ S ⊕ S, S ∧ S ⊕ S ) = P .

13 n Let V = ⊕i=0Cxi such that

∀σ ∈ Sn+1 : σ(xi ) = xσ(i) Pn and put yi = x0 − xi , 1 ≤ i ≤ n. Let v = i=0 xi be an element generating V Sn+1 . Then for p = [α : β : γ] ∈ P2, the relations of a Sn+1-deformation A(p) of C[V ] are given by

( Pn αyi v + βvyi + γyi (n − 1)yi − 2 j=1,j6=i yj = 0, 1 ≤ i ≤ n,

[yi , yj ] = 0, 1 ≤ i < j ≤ n.

The theorem follows.

From the relations, it follows generically that these algebras are Ore extensions of the commutative ring generated by y1, ... , yn. Unfortunately for 2 β a point [α : β : γ] ∈ P , if α 6= −1 and γ 6= 0, the algebra C[y1, ... , yn] is not a commutative polynomial ring, as the corresponding automorphism of C[y1, ... , yn] does not have eigenvalue 1 as in Proposition 1.4.2.

Therefore, the Sn+1-deformations of C[V ] are parametrized by 2 lines, one corresponding to the diﬀerential polynomial rings Pn C[y1, ... , yn][v; δa] with δa(yi ) = ayi (n − 1)yi − 2 j=1,j6=i yj and one corresponding to a skew polynomial ring C[y1, ... , yn][v; σa] with σa(yi ) = ayi . 1.5.2 Theorem. The Koszul algebras of global dimension n + 1 with Hilbert which are Sn+1-deformations of C[V ] and are domains correspond to points of V(xy) ⊂ A2, with one line corresponding to quantum algebras and the other line corresponding to diﬀerential polynomial rings.

Proof. Both lines deﬁne Ore extensions of the polynomial ring C[S], so the results of [47] can be applied. The points at inﬁnity will not be domains, as there will be zero divisors in degree 1.

The two strata of importance correspond to the algebras with relations ( yi v − avyi = 0, 1 ≤ i ≤ n,

[yi , yj ] = 0, 1 ≤ i < j ≤ n, and

( Pn vyi − yi v = cyi (n − 1)yi − 2 j=1,j6=i yj , 1 ≤ i ≤ n,

[yi , yj ] = 0, 1 ≤ i < j ≤ n.

14 1.5.1 Classiﬁcation of the simple objects in Proj(A)

Let A be an AS regular Sn+1-deformation of C[V ]. In order to calculate the simple elements of Proj(A) as deﬁned in AppendixB, it is useful that in both cases the sequence y1, y2, ... , yn is a normalizing, regular sequence.

The Skew polynomial case

Here the results of Chapter5 can be applied, where the quantum polynomial rings will be studied in greater detail. 1.5.3 Theorem. The noncommutative algebra A is a twist of the polynomial ring C[V ]. Consequently, Proj(A) =∼ Pn.

∗ β Proof. Take β(yi ) = yi , βv = av with a ∈ C . Then the twist C[V ] has relations

yi ∗β yj = yi yj = yj ∗β yi , yi ∗β v = ayi v = av ∗β yi , 1 ≤ i < j ≤ n. which corresponds to one of the lines in P2 of interest.

The diﬀerential polynomial ring case

By rescaling v, assume that the derivation is deﬁned by

 n  X δ(yi ) = yi (n − 1)yi − 2 yj  = yi fi , i = 1 ... n. j=1,j6=i

1.5.4 Theorem. The simple objects of Proj(A) are parametrized by n n ∗ 2 −1 lines through one point in P = P((A1) ). Each point on such a line corresponds to a point module of A. If n is even, the only point module with (non-trivial) ﬁnite dimensional simple quotients is the intersection n point of these lines. If n is odd, then there are n+1 -lines parametrizing 2 C∗-orbits of 1-dimensional representations, which are the only lines with non-trivial simple quotients. The equations for these lines are determined by V(yi yj (yi − yj ) : 1 ≤ i < j ≤ n) There are no other simple objects in Proj(A).

Proof. Let P ∈ Proj(A) be a simple object, that is, P is a graded module p with Hilbert series (1−t) for some p ∈ N, p ≥ 1 and each graded quotient of P is ﬁnite dimensional. As each yi is normalizing, either yi ∈ Ann(P)

15 −1 or P corresponds to a simple A[yi ]0-representation, see for example ∼ [44]. Assume that each yi ∈ Ann(P), then P is a A/(y1, ... , yn) = C[v]- module and therefore P =∼ C[v] as A-module.

Assume now that y1 ∈/ Ann(P), then P corresponds to a simple repre- −1 −1 −1 sentation of A[y1 ]0. Let vj = yj y1 , 2 ≤ j ≤ n, w = vy1 . Then the −1 relations of A[y1 ]0 become vj vk − vk vj = 0 and for 2 ≤ k ≤ n one ﬁnds −1 −1 wvk =vy1 yk y1  n  −1 X −1 −1 =(y1 v − (n − 1) − 2 yj y1 )yk y1 j=2 n −1 −1 X =y1 vyk y1 − (n − 1)vk + 2 vj vk j=2

 n  −1 X −1 =y1 (yk v + yk (n − 1)yk − 2 yj )y1 − (n − 1)vk j=1,j6=k n X + 2 vj vk j=2 n n 2 X X =vk w + (n − 1)(vk − vk ) − 2 vj vk + 2 vj vk j=1,j6=k j=2

=vk w + (n + 1)vk (vk − 1).

−1 Consequently, the 1-dimensional representations of A[y1 ]0 correspond n to the Zariski closed subset V(ak (ak − 1) : 2 ≤ k ≤ n) ⊂ A , which is the union of 2n−1 lines. −1 Now the same can be done for the algebra (A[y2 ])0, which will give an additional 2n−1 lines. However, there are already 2n−1 − 2n−2 lines found −1 n−2 for A[y1 ]0 (those not annihilated by y1), so this adds 2 new lines. By induction, there is a total of

n−1 X 2j = 2n − 1 j=0 lines. In Pn, each of these lines goes through the point [0 : 0 : ... : 0 : 1], which will be the intersection point. In order to prove that there are no fat point modules (simple objects p −1 with Hilbert series 1−t , p > 1), it will be shown that A[y1 ]0 has no p- −1 ∼ dimensional representations for p > 1. This will be enough, for A[y1 ]0 = −1 −1 A[yi ]0 for any 1 ≤ i ≤ n. After rescaling v, A[y1 ]0 is isomorphic to the diﬀerential polynomial ring C[v2, ... , vn][w; δ] with δ(vi ) = vi (vi −1).

16 First, a lemma is needed. 1.5.5 Lemma. Let ρ be a simple representation of the algebra

Chx, ui/(ux − xu − x(x − 1)). Then ρ is 1-dimensional and ρ(x) = 0 or ρ(x) = 1.

Proof. If this is not so, then ρ(x) − 1 and ρ(x) are invertible. For Y = 1 − ρ(x)−1 and U = ρ(u), one ﬁnds

YU − UY = ρ(u)ρ(x)−1 − ρ(x)−1ρ(u) = −(1 − ρ(x)−1) = −Y , so the couple (U, Y ) forms a representation of the 2-dimensional Heisen- berg Lie algebra. If the lemma is not true, Y is also invertible. Let a be an eigenvector of U with eigenvalue α. The following then holds:

UYa = (YU + Y )a = (α + 1)Ya.

But ρ is ﬁnite dimensional, so Y has a non-trivial kernel, which is a contradiction.

Let ρ0 = {a ∈ ρ : ρ(x)a = 0} and ρ1 = {a ∈ ρ : ρ(x)a = a} and take elements a ∈ ρ0, b ∈ ρ1. A calculation shows

ρ(x)ρ(u)a = ρ(u)ρ(x)a − ρ(x)(ρ(x) − 1)a = 0, ρ(x)ρ(u)b = ρ(u)ρ(x)b − ρ(x)(ρ(x) − 1)b = ρ(u)b.

So ρ = ρ0 and ρ1 = 0 or vice versa as ρ is simple, in both cases ρ(x(x − 1)) = 0 and ρ is actually a simple representation of C[x, u]/(x(x − 1)). The lemma follows.

As the vj , 2 ≤ j ≤ n commute with each other, let a be an eigenvector of each vj with eigenvalue either 1 or 0 for each j, so vj a = αj a with αj = 0 −1 or αj = 1 for some simple representation ρ of A[y1 ]0. One ﬁnds

vj wa = wvj a + vj (vj − 1)a = wvj a.

−1 As A[y1 ]0a = ρ, it follows that the images of w and vj commute, which means that ρ is only simple if ρ is 1-dimensional. The last thing to check is which of these point modules have simple quotients. This amounts to check which of these point modules have [1] a ﬁnite orbit under the shift functor Proj(A) / Proj(A) . Equiv- alently, as A is AS regular, it is enough to check which orbits of the associated automorphism φ on the point variety are ﬁnite, see for example [54]. It is clear that under φ the intersection point is sent to

17 itself, as it is the only singular point of the point variety. Let P be a point module, not the intersection point, then P corresponds to a set T ⊂ {1, ... , n}, T 6= {1, ... , n} such that yi ∈ Ann(P) ⇔ i ∈ T . In addition, yi , yj ∈/ Ann(P) ⇒ yi − yj ∈ Ann(P). Let [a1 : ... : an : r] be n the corresponding point of P , so ai = 0 ⇔ i ∈ T and i, j 6∈ T ⇒ ai = aj and r arbitrary. One needs to ﬁnd [b1 : ... : bn : s] such that the following equations hold

( Pn rbi − ai s = ai (n − 1)bi − 2 j=1,j6=i bj , 1 ≤ i ≤ n,

ai bj − aj bi = 0, 1 ≤ i < j ≤ n.

From the second set of equations, it follows that bi = ai ∀1 ≤ i ≤ n. It may be assumed that ai = bi = 1 if i 6∈ T . Consequently, the ﬁrst set of equations are fulﬁlled if i ∈ T . If i 6∈ T , then it follows that

 n  X s = r − (n − 1) − 2 aj  = r − n − 1 + 2|T | j=1,j6=i

If |T |= 6 n+1 , this shows that φ is a translation on 1 , which ﬁxes one 2 P[ai :r] n+1 point (the intersection point of all the lines). However, if |T | = 2 (so n has to be odd), then φ ﬁxes each point on the corresponding line, so this will give a line parametrizing C∗-orbits of 1-dimensional representations of A.

For n = 2, 3, the previous theorem can be made more explicit. ∼ 3 2 2 1.5.6 Example. As S3 = D3 = he1, e2 : e1 = e2 = 1, e2e1e2 = e1 i, the diﬀerential polynomial ring A is isomorphic to the quotient of Chx, y, ti by the relations  xy − yx = 0,  xt − tx = y 2, yt − ty = x 2. In this case, the point modules can be computed by the method of multilinearization as the zeroset of the polynomial   −y0 x0 0 3 3 det −t0 −y0 x0 = y0 − x0 . −x0 −t0 y0

This is indeed the union of three lines through the point [0 : 0 : 1]. Take for example the line x0 = y0, then the automorphism defined by this algebra on V(x0 − y0) is given by [x0 : x0 : t0] 7→ [x0 : x0 : t0 + x0], which is clearly an automorphism of infinite order and fixes only one point.

18 Figure 1.1: Point variety of C[x, y][t; δ]

The fact that all 1-dimensional simple representations come from the intersection point follows immediately as one has 2 2 A/([x, t], [y, t]) =∼ C[a, b, c]/(a , b ). ∼ 1.5.7 Example. For n = 3, there is the isomorphism S4 = (Z2 × Z2) o S3. Let e1, e2 be the natural generators of Z2 × Z2. If V = S ⊕ T is the permutation representation of S4, then there exists a basis {vi,j : 0 ≤ i, j ≤ 1} of V such that i j e1 · vi,j = (−1) vi,j , e2 · vi,j = (−1) vi,j .

As S4-module, the decomposition is given by Cv0,0 ⊕ CS4 · v1,0. Using this basis, the differential polynomial ring has relations (under a suitable isomorphism)  [v1,0, v0,1] = [v1,1, v1,0] = [v0,1, v1,1] = 0,  v1,0v0,0 − v0,0v1,0 = v0,1v1,1, v0,1v0,0 − v0,0v0,1 = v1,0v1,1,  v1,1v0,0 − v0,0v1,1 = v1,0v0,1. Again using the method of multilinearization, one sees that the point modules are parametrized by 7 lines, given by the Zariski closed subset in P3 defined by 2 2 2 2 2 2 V V1,1(V1,0 − V0,1), V1,0(V0,1 − V1,1), V0,1(V1,0 − V1,1) . The point modules with 1-dimensional simple quotients can again be easily found, by taking the quotient ∼ A/([A1, A1]) = C[a, b, c, d]/(bc, bd, cd) . This ring is clearly the coordinate ring of 3 affine planes, intersecting 2 by 2.

Chapter 2

3-dimensional Sklyanin algebras

The Sklyanin algebras are perhaps the most famous noncommutative algebras studied in noncommutative algebraic geometry following [7] and [6]. In this chapter the 3-dimensional Sklyanin algebras are studied using the representation theory of H3. In particular, it is shown that (for the terminology, I refer to this chapter)

• the Sklyanin algebras are H3-deformations of the polynomial ring C[V ], with V a 3-dimensional simple representation of H3 (see AppendixA for the representation theory of Hn in general),

• the existence of a central element c3 of degree 3 in a Sklyanin algebra Aτ (E) can be derived from the fact that c3 is normal and (c3) = ∩P∈E Ann(P), that is, c3 annihilates each point module of a Sklyanin algebra and the ideal generated by c3 is the intersection of all annihilators of all point modules and

• there exists a connection between c3 and the superpotential of the algebra A−2τ (E).

In addition, the space V4 of H3-deformations up to degree 4 of C[V ] is studied. This space is isomorphic to the blow-up of P2 in O, with

O = {[1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1]} ∪ {[a : b : c]: a3 = b3 = c3}.

There is an open subset U ⊂ V4 that deﬁnes H3-deformations of C[V ] that is strictly larger than P2 \O.

21 In the last section, the representation theory of Aτ (E) in the case that τ is a torsion point is discussed. A partial resolution of the singularity of the center of Aτ (E) is constructed using the method of a noncommutative blow-up.

2.1 Deﬁnitions and basic properties

2.1.1 Deﬁnition. The 3-dimensional Sklyanin algebras are the quotients S([a : b : c]) of the tensor algebra Chx0, x1, x2i by the relations  ax x + bx x + cx 2 = 0,  1 2 2 1 0 2 ax2x0 + bx0x2 + cx1 = 0,  2 ax0x1 + bx1x0 + cx2 = 0, for [a : b : c] ∈ P2 \ V(abc((a3 + b3 + c3)3 − 27a3b3c3)).

The notation S(p) will be used for the quotient of Chx0, x1, x2i by these three quadratic relations for any point p ∈ P2, not just the Sklyanin algebras.

2.1.2 Proposition. The Sklyanin algebras are H3-deformations of C[V ] for V = V1 the Schr¨odingerrepresentation of H3.

∼ ∗ 3 3 Proof. It is easy to see that V ⊗ V = (V ) = V2 . As V ∧ V is always 2 a subrepresentation of V ⊗ V and it is 3-dimensional, V2 = P[a:b:c]. To a point p = [a : b : c] corresponds the vector space of relations generated by  ax x + bx x + cx 2 = 0,  1 2 2 1 0 2 ax2x0 + bx0x2 + cx1 = 0,  2 ax0x1 + bx1x0 + cx2 = 0.

These are (for generic values of [a : b : c]) the relations of the 3- dimensional Sklyanin algebras. As the Hilbert series of Sklyanin algebras is constant and equal to (1 − t)−3, the proposition has been proved by Corollary 1.1.11.

There is the inclusion of groups SL2(F3) ⊂ NGL(V )(H3). By Proposition 1.1.12, the group SL2(F3) acts on V2 such that orbits correspond to isomorphic algebras. Under the SL2(F3)-action, there are the following 2πI special orbits (again, let ω = e 3 ):

22 •O ([0 : 0 : 1]) = {[0 : 0 : 1], [1 : 1 : 1], [1 : 1 : ω], [1 : 1 : ω2]}. The corresponding algebras are isomorphic to Chx, y, zi/(x 2 = y 2 = z2 = 0). •O ([1 : 0 : 0]) = {[1 : 0 : 0], [0 : 1 : 0], [ω : 1 : 1], [ω2 : 1 : 1], [1 : ω : 1], [1 : ω2 : 1], [ω : ω2 : 1], [ω2 : ω : 1]}. The corresponding algebras are isomorphic to Chx, y, zi/(xy = yz = zx = 0). This algebra in turn is a twist of Chx, y, zi/(x 2 = y 2 = z2 = 0). Let O = O([0 : 0 : 1]) ∪ O([1 : 0 : 0]). From these isomorphisms, it follows that each of these algebras has the following property. 2.1.3 Proposition. Let p ∈ O, then there exists a unique l ∈ 1 such PF3 φ that for each point φ ∈ l and each lift φ ∈ H3, the twist S(p) is isomorphic to S(p).

Proof. Twisting an H3-algebra with an automorphism of H3 corresponds 2 to the projective action of H3 on V2 = P . It is therefore easily seen that

• [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] are fixed by e2, 2 2 • [1 : 1 : 1], [ω : ω : 1] and [ω : ω : 1] are fixed by e1, −1 • [1 : 1 : ω], [1 : ω : 1] and [ω : 1 : 1] are fixed by e1 e2 and 2 2 2 • [1 : 1 : ω ], [1 : ω : 1] and [ω : 1 : 1] are fixed by e1e2. ∼ Each of these elements of H3/([e1, e2]) = Z3 × Z3 corresponds to a point of 1 , as required. PF3 The lines dual to these points correspond to algebras isomorphic to quantum polynomial rings  x x − λx x = 0,  1 2 2 1 x2x0 − λx0x2 = 0,  x0x1 − λx1x0 = 0, ∗ for some λ ∈ C or to twists with elements of H3 of these algebras.

Let x = x0, y = x1, z = x2. Associated to a Sklyanin algebra is an elliptic curve E and a point τ ∈ E. The elliptic curve and the point τ are given by E = V(abc(X 3 + Y 3 + Z 3) − (a3 + b3 + c3)XYZ), τ = [a : b : c]. The elliptic curve parametrizes the point modules of E and addition with τ corresponds to the shift functor P 7→ P[1]. Let Aτ (E) be the 3- dimensional Sklyanin algebra associated to E and τ ∈ E. Some well known facts regarding the 3-dimensional Sklyanin algebras (most if not all is proven in [6], [4], [3] or [7]):

23 1. they are noetherian domains,

2. they are Koszul algebras, with Hilbert series (1−t)−3 (in particular, they have global dimension 3),

3. they are ﬁnite modules over their center if and only if the point τ is a torsion point. In this case, they have PI-degree n = ord τ and their center is isomorphic to

n C[u1, u2, u3, g]/(Φ(u1, u2, u3) − c3 ),

with deg u1 = deg u2 = deg u3 = n, deg c3 = 3 and Φ a homogeneous polynomial of degree 3n corresponding to the elliptic curve E 0 = E/hτi.

4. If τ is not torsion, then the center of Aτ (E) is generated by a degree 3 element c3. ∼ 5. Aτ (E)/(c3) = Oτ (E), the twisted homogeneous coordinate ring associated to E and the automorphism on E deﬁned by τ.

Properties (1) and (2) and the existence of a central element c3 of degree 2 2 3 ﬁxed by H3 also hold for all p ∈ P \O. In particular, all points in P \O correspond to AS regular algebras and the point [1 − 1 : 0] corresponds to the commutative polynomial ring.

2.1.4 Theorem. The H3-deformations up to degree 2 of C[V ] that also 2 deﬁne H3-deformations are parametrized by P \O.

Proof. This follows directly from Corollary 1.1.11.

2.2 The superpotential and the center

2.2.1 The superpotential

∼ From Theorem 2.1.4, it follows that S(τ)n = C[V ]n as H3-modules for each n ∈ N and each τ ∈ U. This is in particular true for n = 3. It is ⊗3 not hard to calculate the following decompositions of C[V ]3 and V :

∼ 2 M C[V ]3 = χ0,0 ⊕ χa,b, 2 (a,b)∈Z3,(a,b)6=(0,0) ⊗3 ∼ M 3 V = χa,b. 2 (a,b)∈Z3

24 This implies that for the AS-regular algebras, the quotient map

⊗3 H3 H3 πτ :(V ) / / Aτ (E)3 with τ = [a : b : c] has a 1-dimensional kernel. It is easily checked that the degree 3 element of (V ⊗3)H3 generating the kernel is given by

a(zxy + xyz + yzx) + b(yxz + zyx + xzy) + c(x 3 + y 3 + z3), which plays the role of superpotential associated to this algebra.

2.2.1 Deﬁnition. A superpotential associated to a Koszul algebra with relations I ⊂ V ⊗ V of global dimension n for some n ∈ N is an ele- ! ∗ n−2 ⊗j ment generating the 1-dimensional vector space (V )n = ∩j=0 V ⊗ I ⊗ V n−2−j .

This shows that there is a natural rational morphism

ψ 2 ⊗3 H3 2 P = V2 / P((V ) ) = P .

The natural question is whether ψ extends to the entire V2. This boils H3 down to proving that for each p ∈ O, S(p)3 is 2-dimensional.

2.2.2 Proposition. Let p ∈ O and let l = [a : b] be the point of P1 ∼ F3 corresponding to the elements φ ∈ H3/[H3, H3] = Z3 × Z3 such that φ S(p) is isomorphic to S(p). Then S(p)3 decomposes as

∼ M 2 M S(p)3 = χx ⊕ χx , ⊥ 2 ⊥ x∈l x∈Z3\l

⊥ with l the line in Z3 × Z3 perpendicular to the line corresponding to l with respect to the usual Euclidean inner product.

Proof. It is enough to prove this for the two algebras Chx, y, zi/(x 2 = y 2 = z2 = 0) and Chx, y, zi/(xy = yz = zx = 0) with twist-stabilizer the group generated by e2, which corresponds to the element [0 : 1] ∈ 1 ∼ . Afterwards the action of SO2( 3) = 2 × 2-action, naturally em- PF3 F Z Z bedded in SL2(F3) can be applied. The proposition will be proved for A = Chx, y, zi/(x 2 = y 2 = z2 = 0), the other case is completely similar. This algebra is a monomial algebra, with basis in degree 3

{zxy, xyz, yzx, yxz, zyx, xzy, yxy, zyz, xzx, zxz, xyx, yzy}.

a b It is then easy to see that χA3 (e1 e2 ) = 0 if a 6= 0, for e1 does not ﬁx any elements of this basis. The element [e1, e2] ∈ H3 ﬁxes every element (for

25 this basis consists of elements of degree 3). For e2, a calculation shows that

2 2 2 χA3 (e2)=1+1+1+1+1+1+ ω + ω + ω + ω + ω + ω = 3.

2 ∼ L 2 L and similarly for e . It follows that A3 = ⊥ χx ⊕ 2 ⊥ χx . 2 x∈l x∈Z3\l 2.2.3 Corollary. The map ψ can be naturally extended. 2.2.4 Remark. A diﬀerent proof of Proposition 2.2.2 can be obtained using character series. The non-regular algebras are still Koszul algebras, with Koszul dual either isomorphic to A = C[x, y, z]/(xy, yz, zx) or the twist of A by the automorphism e1, the algebra B = Chx, y, zi/(yx, zy, xz, x 2, y 2, z2), with Cx + Cy + Cz =∼ V ∗. For example, the ﬁrst algebra has the property

n n n ∀n ∈ N \{0} : An = Cx + Cy + Cz , from which it follows that  ∼ A0 = χ0,0  ∼ ∗ An = V , n ≡ 1 mod 3 A =∼ V , n ≡ 2 mod 3  n  ∼ An = χ0,0 ⊕ χ1,0 ⊕ χ2,0, n ≡ 0 mod 3, n 6= 0.

From this it follows that

 2 ch ([e , e ], t) = 1 + 3ω t ,  A 1 2 1−ω2t chA(e1, t) = 1,  3t3 chA(e2, t) = 1 + 1−t3 .

One can then calculate the decomposition of A3 using Koszul duality for G-algebras using these results by Proposition 1.2.3.

2.2.2 The center

Fix a point τ = [a : b : c] ∈ P2 and the associated Sklyanin algebra A = S(τ). From the original paper [6] of Artin, Tate and Van den Bergh, it follows that A has a normal element of degree 3, cA,3 ∈ A3, which was later proved to be central by a computer calculation. In this section, it is discussed how the Heisenberg action can be used to prove that cA,3 is central, not just normal. In addition, this subsection also shows that ∼ CcA,3 = χ0,0 as H3-module.

2.2.5 Lemma. The vector space CcA,3 is a subrepresentation of A3.

26 Proof. The ideal (cA,3) is the intersection of the annihilators of all point modules of A. As the point modules form an H3-set, it follows that this ideal is a H3-submodule of A.

From this lemma it follows that there exists a character χ of H3 such that for every element g ∈ H3, g · cA,3 = χ(g)cA,3.

2.2.6 Theorem. If [α : β : γ] = 2τ ∈ E, then cA,3 is a scalar multiple of β(zxy + xyz + yzx) + α(yxz + zyx + xzy) + γ(x 3 + y 3 + z3).

Proof. Let B = A/(cA,3). From Theorem 2.1.4 it follows that

∼ 2 M A3 = χ0,0 ⊕ χa,b 2 (a,b)∈Z3,(a,b)6=(0,0) and it was already observed that

⊗3 ∼ M 3 V = χa,b. 2 (a,b)∈Z3

If Π : T (V ) / / B is the canonical epimorphism, then it is enough to

⊗3 H3 H3 show that the map Π3 :(V ) / / B3 has a 2-dimensional kernel. Consider the following basis of (V ⊗3)H3

I1 = zxy + xyz + yzx,

I2 = yxz + zyx + xzy, 3 3 3 I3 = x + y + z .

Then a linear combination a1I1 + a2I2 + a3I3 belongs to ker(Π3) if and only if its multi-linearization vanishes on the variety X =∼ E deﬁned as {(P, Q, R) ∈ E × E × E : Q − P = R − Q = τ} ⊂ 2 × 2 × 2 . P[x0:y0:z0] P[x1:y1:z1] P[x2:y2:z2] If [α : β : γ] = 2τ, then three of the elements in the deﬁning ideal I (X ) ⊂ C[x0, y0, z0, x1, y1, z1, x2, y2, z2] of X are given by

f1 = αy0z2 + βz0y2 + γx0x2,

f2 = αz0x2 + βx0z2 + γy0y2,

f3 = αx0y2 + βy0x2 + γz0z2.

But then the element x1f1 + y1f2 + z1f3 is also in I (X ), which is the multi-linearization of

3 3 3 wA,3 = β(zxy + xyz + yzx) + α(yxz + zyx + xzy) + γ(x + y + z ).

27 As −2τ 6= τ if and only if 3τ 6= O, the element cA,3 = π(wA,3) for π : T (V ) / / A the canonical epimorphism is the extra relation one needs to get from A to B.

∼ Consequently, this proof shows that CcA,3 = χ0,0. The proof of this theorem also works if the point variety of A is a triangle instead of an elliptic curve, with some minor modiﬁcations.

2.2.7 Theorem. The degree 3 element cA,3 is a central element of A.

Proof. If cA,3 is normal, then there exists a gradation preserving automorphism σ of A such that for every u ∈ A, the following holds:

ucA,3 = cA,3σ(u).

Let g ∈ H3 and u ∈ A1, then it holds that

g · (ucA,3) = (g · u)(g · cA,3)

= (g · u)(cA,3)

= cA,3σ(g · u),

g · (ucA,3) = g · (cA,3σ(u))

= (g · cA,3)(g · σ(u)))

= cA,3(g · (σ(u))).

∼ ∗ This implies that σ ∈ AutH3 (V ) = C . As σ(cA,3) = cA,3, it follows that σ is multiplication with ωk for some 0 ≤ k ≤ 2. 3 3 3 Let U = V2 \ O ∪ V(abc, a + b + c ) and for p = [a : b : c] ∈ U, ∗ let Rp be the corresponding embedding of V in V ⊗ V . Let Wp = Rp ⊗V ⊗V +V ⊗Rp ⊗V +V ⊗V ⊗Rp. For k ∈ Z3, deﬁne the following functions

gk k U / N , p 7→ dimC(Wp + C[cS(p),3, ω x]) − dimC Wp.

Then gk (p) is either 0 or 1 and {gk = 0} is a closed subset. It is also open, because {gk = 0} = U \({gk+1 = 0} ∪ {gk+2 = 0}), indices taken in Z3. So U = {gk = 0} for some k ∈ Z3 and {gk+1 = 0} ∪ {gk+2 = 0} = ∅. The point [1 : 1 : 0] is an element of {g0 = 0}, so U = {g0 = 0} and σ is indeed the identity map.

∼ 2.2.8 Remark. To prove this theorem, it is unnecessary that CcA,3 = χ0,0, although it is useful if one wants to ﬁnd a real description of this central element.

28 2.3 Birational maps of the projective plane coming from H3-deformations of C[V ]

2 Take τ = [a : b : c] ∈ P = V2. From Corollary 2.2.3 it follows that such τ corresponds to a unique 1-dimensional subspace of (V ⊗3)H3 by associating to this point the kernel of the map

⊗3 H3 H3 πτ :(V ) / / S(τ)3 .

The constructed map

⊗3 H3 V2 / P((V ) ) is an isomorphism.

One can now look at the map πτ restricted to other isotypical compo- ⊗3 ⊗3 nents of V . For 0 ≤ i, j ≤ 2, let Wij ⊂ V be such an isotypical ∼ 3 component, with Wij = χij as H3-representations. As before, to τ one can associate the kernel of the map πτ : Wij / S(τ)3 , which is generically a 2-dimensional subspace of Wij and therefore deﬁnes a point 2 ∼ ∼ 2 of EmbH3 (χij , Wij ) = Grass(2, 3) = (P )ij . However, from the previous sections it follows that for each point (i, j) ∈ Z3 × Z3 \{(0, 0)}, there are 3 points of V2 where this kernel is 1-dimensional instead of 2-dimensional. Let pij be the unique point of these 3 points lying on the line a = b. Let 2 fij be the rational map V2 / (P )ij constructed this way.

2.3.1 Theorem. The map fij is the standard Cremona transformation that is not deﬁned at the H3-orbit of pij and contracts the 3 lines between these points to a point. Alternatively, fij corresponds to the linear system deﬁned by the degree 2 part of the Koszul dual of S(pij ) as subset of C[x ∗, y ∗, z∗].

Proof. It is enough to prove this for the speciﬁc case (i, j) = (1, 0), in the general case the SL2(F3)-action can be applied. A basis for W10 is given by

3 3 2 3 v1 = x + ωy + ω z , 2 v2 = zxy + ωxyz + ω yzx, 2 v3 = yxz + ωzyx + ω xzy.

Let Rp be the quadratic relations associated to p, then the isotypical component of Rp ⊗ V + V ⊗ Rp corresponding to χ10 is the vector space

29 generated by

3 f1 =axyz + bxzy + cx + aωyzx + bωyxz + cωy 3 + aω2zxy + bω2zyx + cω2z3, 3 f2 =ayzx + bzyx + cx + aωzxy + bωxzy + cωy 3 + aω2xyz + bω2yxz + cω2z3.

Writing these 2 elements in the basis v1, v2, v3, one sees that

2 cv1 + aω v2 + bωv3, 2 cv1 + aωv2 + bω v3.

The rational map f10 then becomes

2 V2 / P ,[a : b : c] 7→ [bc : ac : ab].

This is indeed the standard Cremona transformation for P2.

The 9 points in V2 corresponding to twists of the commutative polynomial ring C[V ] have the following property: each birational morphism fij is deﬁned at these points, but they always lie on one of the contracted lines.

2.3.2 Theorem. For a point p of V2, there are the following equivalences:

• p belongs to the domain of each fij and is never contracted if and only if S(p) is a Sklyanin algebra,

• p belongs to the domain of each fij and is contracted by 2 functions if and only if S(p) is either a quantum algebra or a twist of a quantum algebra with an element of H3, with point variety the union of 3 lines,

• p belongs to the domain of each fij but is contracted by all functions if and only if S(p) is a twist of the polynomial ring C[V ] by an element of H3 and ﬁnally

• p does not belong to the domain of exactly 2 functions if and only if S(p) is not regular if and only if S(p) is not a domain.

One might expect that, if S(p) is a quantum algebra, then the twist g S(p) is also a quantum algebra for g ∈ H3. This is however not the case.

30 2.3.3 Proposition. The twist of the quantum polynomial algebra Chx, y, zi/(yz − qzy, zx − qxz, xy − qyx) for q ∈ C∗ with the automorphism e1 ∈ H3 is not isomorphic to a quantum polynomial algebra except in the case that q3 = 1.

Proof. The relations of

e1 Aq = (Chx, y, zi/(yz − qzy, zx − qxz, xy − qyx)) are  zy − qx 2 = 0,  xz − qy 2 = 0, yx − qz2 = 0.

If Aq was a quantum algebra, then it should have non-trivial 1-dimensional representations. Let ρ : Aq / C be such a representation. By the Heisenberg action and the C∗-action, assume that ρ(x) = 1. Let Y = ρ(y), Z = ρ(z). Then the triple (q, Y , Z) fulﬁls the relations

ZY = q, Z = qY 2, Y = qZ 2, from which it follows that Z 3 = Y 3 = 1. But then q is necessarily also a third root of unity. If q is a third root of unity, then Chx, y, zi/(yz −qzy, zx −qxz, xy −qyx) 2 is a twist of the polynomial ring C[x, y, z] with either 1,e2 or e2 . In this case the algebra Aq is a twist of a twist of a polynomial ring, such that the composition of the associated automorphisms has order 3 in PGL(V ). But this again implies that up to isomorphism, Aq is a twist of the polynomial ring by a diagonalizable automorphism, which is again a quantum algebra.

2.4 Quotients of non-regular quadratic algebras

2 In the projective plane P[a:b:c], there are 12 points where the corresponding algebra is not AS-regular: the SL2(F3)-orbit of [1 : 0 : 0] (containing 8 elements) and the orbit of [0 : 0 : 1] (containing 4 elements). All these 1+t algebras have as Hilbert series 1−2t and are clearly not domains, for a detailed description of these algebras, see [57] and [66]. However, it seems that these algebras have a 1-dimensional family of quotients that ‘behave’ like the 3-dimensional AS-regular algebras in the following sense:

• the Hilbert series is the same,

31 • the character series is the same for each element of H3 and

• there exists a central element of degree 3, ﬁxed by the H3-action.

Let us consider the following example: take the Cliﬀord algebra C over the polynomial ring C[u0, u1, u2] with associated quadratic form   0 u2 u1 u2 0 u0 . u1 u0 0 In term of generators and relations, this algebra has three generators x0, x1, x2 of degree 1 and three generators of u0, u1, u2 of degree 2, with relations  x 2 = 0, 0 ≤ i ≤ 2,  i [ui , xj ] = 0, 0 ≤ i, j ≤ 2,  {xi , xi+1} = ui+2, 0 ≤ i ≤ 2, indices taken in Z3. However, from these relations, one observes that the generators u0, u1, u2 are redundant, so C is generated in degree 1 and the relations become

( 2 xi = 0, 0 ≤ i ≤ 2 [{xi , xi+1}, xi+2] = 0, 0 ≤ i ≤ 2.

This algebra is a quotient of the algebra S([0 : 0 : 1]) by 2 elements of degree 3 (adding two commutation relations of degree 3 automatically implies the third relation). ∼ 2.4.1 Theorem. The following holds: C =H3 C[V ].

Proof. Deﬁne on C an action of H3 by

e1 · xi = xi−1 e1 · ui = ui−1, i 2i e2 · xi = ω xi e2 · ui = ω ui .

C is a free module of rank 8 over C[u0, u1, u2] with basis

{1, x0, x1, x2, x1x2, x2x0, x0x1, x0x1x2}. It is then easy to compute that under these conditions, C is a graded algebra with character series equal to the polynomial ring in 3 variables with the standard action of H3.

In particular, this means that the natural epimorphism

π : S([0 : 0 : 1]) = Chx, y, zi/(x 2, y 2, z2) / / C

32 has as kernel an ideal generated by two elements of degree 3. The C-vector space generated by these two elements decomposes as H3- representation into χ1,0 ⊕ χ−1,0, which is the ‘H3-surplus’ of S([0 : 0 : 1] in degree 3. Recall that for the commutative polynomial ring C[x, y, z], the degree 3 part decomposes as H3-representation into ∼ M C[x, y, z]3 = χ0,0 ⊕ χi,j . (i,j)∈Z3×Z3 Considering the representations of C, the following holds. 2.4.2 Theorem. The algebra C is Azumaya over every point of the sheme Spec(Z(C)) except over the trivial ideal (u0, u1, u2).

Proof. C is not Azumaya over a point if and only if the associated quadratic form after specialization is of rank ≤ 1 (see [33]). Taking 2 the 2 × 2-minors of the quadratic form, this only happens if ui is mapped to 0 for all 0 ≤ i ≤ 2.

This means that, considering simple representations, C has more in common with the Sklyanin algebras associated to points of order 2 than the quantum algebra C−1[x, y, z], although the last one is AS-regular. In addition, although C is not a domain, the following is true. (For general results regarding prime rings, see [34, Chapter 4]). 2.4.3 Theorem. The Cliﬀord algebra C is a prime ring.

Proof. Let R = C[u0, u1, u2], K = Frac(R) and take D = K ⊗R C. Then D is a ﬁnite dimensional algebra of dimension 8 over K. If we can prove ∼ that D = M2(L) with L a quadratic extension over K, then we are done, for then D is a central simple algebra, which implies that C is prime. Let 2 L = K[g]/(g − u0u1u2). Take the elements u1u2 0 1 u0u2 0 0 g I 1 X0 = , X1 = , X2 = ∈ M2(L). g 0 0 g 1 0 u2 1 −I Then these three elements fulﬁl the relations

( 2 Xi = 0, 0 ≤ i ≤ 2, {Xi , Xi+1} = ui+2, 0 ≤ i ≤ 2. ∼ Consequently, D = M2(L) as a K-algebra (as both algebras are of dimension 8 over K and there exists a K-algebra morphism

D / M2(L) which is surjective) and D is a central simple algebra. From this it follows that C is a prime ring.

33 In the next section, a 1-dimensional family of quotients of S([0 : 0 : 1]) by degree 3 elements will be studied, such that each quotient has the same character series up to degree 4 as the polynomial ring C[x, y, z]. It will be shown that there is an open subset in this 1-dimensional family that corresponds to algebras with Hilbert series (1 − t)−3.

2.4.1 The Hilbert series

Fix the algebra A = S([0 : 0 : 1]). The decomposition of the degree 3 part of A3 in H3-modules becomes ∼ X A3 = χi,j ⊕ χ0,0 ⊕ χ1,0 ⊕ χ2,0. i,j∈Z3

In particular, the multiplicity of χ1,0 and χ2,0 is 2 in A3. This means that the variety parametrizing quotients of A with the right character series up to degree 3 is given by P1 × P1.

Let Ip be the ideal of A generated by 2 2 v1 = A1(zxy + ωxyz + ω yzx) + B1(yxz + ωzyx + ω xzy), 2 2 v2 = A2(zxy + ω xyz + ωyzx) + B2(yxz + ω zyx + ωxzy), 1 1 with p = ([A1 : B1], [A2 : B2]) ∈ P × P . Let Wp = Cv1 + Cv2. One of the similarities one would like is whether there are quotients such that the Hilbert series is (1 − t)−3, in particular, the Hilbert series has to be correct up to degree 4. As dim A4 = 24 and dim C[x, y, z]4 = 15, one needs that dim(Ip)4 = 9. The following holds:

dim(Ip)4 = dim Wp ⊗ V + dim V ⊗ Wp − dim V ⊗ Wp ∩ Wp ⊗ V .

One needs to find Wp such that dim V ⊗ Wp ∩ Wp ⊗ V is 3-dimensional. However, as this vector space is an H3-representation and isomorphic to e V for some e ∈ N, it is enough to find Wp such that (V ⊗ Wp ∩ Wp ⊗ V )he2i is 1-dimensional, as V has a unique 1-dimensional subspace fixed by e2.

2.4.4 Lemma. A/Ip has the correct Hilbert series up to degree 4 if and 1 1 only if p lies on the line V(A1B2 − A2B1) = ∆ ⊂ P × P .

Proof. The elements ﬁxed by e2 in Wp ⊗ V + V ⊗ Wp lie in the vector space generated by 2 xv1 = A1xzxy + A1ω xyzx + B1xyxz + B1ωxzyx, 2 xv2 = A2xzxy + A2ωxyzx + B2xyxz + B2ω xzyx, 2 v1x = A1zxyx + A1ωxyzx + B1yxzx + B1ω xzyx, 2 v2x = A2zxyx + A2ω xyzx + B2yxzx + B2ωxzyx.

34 he2i Then (V ⊗ Wp ∩ Wp ⊗ V ) 6= 0 if and only if the following matrix has rank ≤ 3:  2  A1 0 A1ω B1 0 B1ω 2 A2 0 A2ω B2 0 B2ω   2 .  0 A1 A1ω 0 B1 B1ω  2 0 A2 A2ω 0 B2 B2ω The ﬁrst 4 × 4-minor is equal to 2 2 2 A1B2A1A2(ω −ω)−A2B1A1A2(ω −ω) = (ω −ω)A1A2(A1B2 −A2B1).

If A1 = 0, then B1 = 1 and the 4 × 4-minor given by the columns (2, 3, 4, 5) becomes  0 0 1 0   0 A2ω B2 0    .  0 0 0 1  2 A2 A2ω 0 B2 2 Taking the determinant of this matrix, one gets A2ω. So this also implies A1B2 − A2B1 = 0. A similar result is true if A2 = 0.

If A1B2 − A2B1 = 0, then one immediately checks that  2  A1 0 A1ω B1 0 B1ω 2 −B2 B1 −B2 B1 A2 0 A2ω B2 0 B2ω   2 = 0. −A2 A1 −A2 A1  0 A1 A1ω 0 B1 B1ω  2 0 A2 A2ω 0 B2 B2ω

As either one of the rows of the ﬁrst matrix is not 0, dim(V ⊗Wp ∩Wp ⊗ V )he2i ≥ 1 in this case. The only points where this inequality is possibly strict is when either both A1 and B2 are 0 or both A2 and B1 are 0, this follows from taking the determinants of    2 A2 0 A2ω B2 0 B2ω 2  0 A1 A1ω  and  0 B1 B1ω  . 2 0 A2 A2ω 0 B2 B2ω

But if for example A1 = 0, then necessarily A2 = 0, but B2 and A2 can not be 0 at the same time. Similar results hold for the other cases, so the lemma has been proved.

This means that the only points to be considered lie on the diagonal ∆ ⊂ 1 × 1 . From now on, let t = β for [α : β] ∈ 1 and let P[A1:B1] P[A2:B2] α P It be the ideal in A generated by the elements 2 2 (v1)t = α(zxy + ωxyz + ω yzx) − β(yxz + ωzyx + ω xzy), 2 2 (v2)t = α(zxy + ω xyz + ωyzx) − β(yxz + ω zyx + ωxzy),

The next obvious question is whether these algebras parametrized by P1 indeed have Hilbert series (1 − t)−3. For C∗ = P1 \{0, ∞} this is true.

35 2.4.5 Lemma. The Cliﬀord algebra C corresponds to taking the quotient for the value t = 1.

Proof. It is enough to prove that the relation [{x, y}, z] belongs to the vector space C(v1)1 + C(v2)1. This means that the following matrix should have rank 2  1 ω ω2 −1 −ω −ω2  1 ω2 ω −1 −ω2 −ω  , −1 1 0 1 −1 0 which is indeed true.

Later it will be shown that all these algebras can be embedded in a smash product C#Z if t 6= 0, ∞, from which the next theorem will follow. −3 2.4.6 Theorem. Each algebra A/It has Hilbert series (1 − t) if t 6= 0, ∞.

2.4.2 Point modules of A/It

In [57] the point modules of A were classiﬁed. The point modules of Tt = A/It can be calculated using these results. Recall that point modules of A were determined in the following way: let V = V(XYZ) ⊂ P2 be the union of 3 lines and let q0, q1, q2 be the intersection points. Then a point module P of A depends on a point sequence p0p1p2 ... fulﬁlling the following requirements:

• for every i ∈ N, pi ∈ V,

• if pi 6= qj for any j, then pi+1 is the intersection point not lying on the same line as pi ,

• if pi = qj for some j, then pi+1 is any point on the line opposite of qj .

If one sets P = ⊕n∈NCen, then the point pi = [ai : bi : ci ] corresponds to deﬁning an action of A on P by

x · ei = ai ei+1, y · ei = bi ei+1, z · ei = ci ei+1.

2.4.7 Theorem. The point modules of Tt for t 6= 0, ∞ are parametrized by 6 lines, call this set W. The isomorphism φ induced by sending a point module P to P[1] on W is such that φ2 ﬁxes the intersection points and sends each line of W to itself.

36 p

φ2(p) q

Figure 2.1: The automorphism φ

Proof. Let P be a point module of Tt with associated point sequence p0p1p2 .... Let pi pi+1pi+2 be a subtriple of this sequence.

• Assume that pi is one of the intersection points. Using the Heisen- berg action, one may assume that pi = [1 : 0 : 0]. Then pi+1 = [0 : α : β]. – Assume that [0 : α : β] 6= [0 : 1 : 0] and [0 : α : β] 6= [0 : 0 : 1]. Then pi+2 = pi and the relations of Tt are trivially fulﬁlled.

– Assume that [0 : α : β] = [0 : 1 : 0]. Then pi+2 = [a : 0 : b]. But from the degree 3 relations it follows that b = 0 and so pi+2 = pi . – The case [0 : α : β] = [0 : 0 : 1] is similar to the previous case.

• Assume that pi is not one of the intersection points. Again using the Heisenberg-action, one may assume that pi = [0 : α : β], pi+1 = [1 : 0 : 0] and pi+2 = [0 : γ : δ]. But then it follows from the degree 3 relations that δα = tγβ or diﬀerently put, φ2 is an isomorphism of V(XYZ) ﬁxing the intersection points.

From the proof of this theorem one can notice that if t is a primitive nth root of unity, then φ is of order 2n. ∗ The point module corresponding to q0q1q0q1 ... corresponds to the C - family of 2-dimensional simple representations coming from the quotient

2 2 Tt /(z) = Chx, yi/(x , y ). Similar results hold for the Heisenberg-orbit of this point module.

37 2.4.3 The central element

One of the many similarities one wants for these new algebras is that there exists a central element of degree 3, ﬁxed by the action of the Heisenberg group.

2.4.8 Theorem. For every element t ∈ ∆ ⊂ P1 ×P1 there exists a degree 3 central element in (Tt )3 ﬁxed by H3.

Proof. Using for example MAGMA, one checks that

gt = (zxy + xyz + yzx) − t(yxz + zyx + xzy) is central in Tt .

One also checks that gt acts trivially on each point module. These observations show that the constructed quotients have indeed much in common with the AS-regular algebras, as each point module of the generic AS- regular algebra is also annihilated by the unique central element in degree 3.

2.4.4 An analogue of the twisted coordinate ring

One can now calculate the Hilbert series of Mt = Tt /(gt ).

1−t3 2.4.9 Theorem. The Hilbert series of Mt is (1−t)3 except if t = 0, ∞.

Proof. Adding the relation (zxy + xyz + yzx) − t(yxz + zyx + xzy), an easier basis for the degree 3 relations of Mt can be found. Taking linear combinations, the relations can be written as  zxy = tyxz,  xyz = tzyx, yzx = txzy, for t 6= 0. For t = 1, the corresponding algebra is a quotient of the Cliﬀord algebra by the square root of the determinant of the associated quadratic form. As T1 is a free module of rank 8 over R = C[{y, z}, {z, x}, {x, y}], 2 a polynomial ring in 3 variables and g1 ∈ R, it follows that g1 is not a zero divisor of T1. This implies that the Hilbert series of M1 = T1/(g1) 1−t3 is (1−t)3 . But then the Hilbert series is correct for all t 6= 0, ∞, as all these algebras are monomial algebras with the same basis as the quotient of the Cliﬀord algebra.

2.4.10 Corollary. gt is not a zero divisor of Tt .

38 Proof. This follows directly from Hilbert series considerations and from Theorem 2.4.6.

There exists an easy basis for Mt if t 6= 0, ∞.

2.4.11 Lemma. A monomial w ∈ Mt is 0 if and only if any letter is both on an even and an odd place in w.

Proof. If w is a monomial with not one variable on both an even and an odd place, then w can never be 0, as the 3 degree 3 relations preserve the even and odd places of the variables. So the combinations x 2, y 2, z2 can never occur in w. Suppose now that x occurs on both an even and an odd place. There is then a submonomial of w of the form xzyzy ... yx or xyzyz ... zx. But then the last x can be brought back to place 2 in the submonomial, so one gets x 2 and w becomes 0. Similar results are true for y and z.

If one now takes a monomial which is not 0 in Mt , then it follows that either all odd or all even places have the same variable, say x. But then one can ‘jump’ over this variable to get 1 variable to the left and the other to the right. Say one wants to make a basis of the degree n part of Mt , with n even. Take 2 variables, say for example x and y, take the classical basis of [x, y] n and put a z between each variable. Put z ﬁrst C 2 on the odd places and then the even places. Apply this procedure 3 times (one for each variable), but remember to discard 6 elements (for xzxz ..., yzyzyzy ... and others have been counted twice). For example, a basis for (Mt )4 is given by

xyxy xyzy zyzy, xzxz xzyz yzyz, yxyx yxzx zxzx, yxyz zxzy xyxz.

If n is odd, one ﬁrst takes the classical basis of [x, y] n−1 and add z in C 2 the odd places and afterwards one takes the classical basis of [x, y] n+1 C 2 and add z in the even places, do this procedure 3 times, but again discard 6 elements. A basis of (Mt )5 for example is given by

xyxyx xyxzx xzxzx yxyxz yxzxz, yxyxy yxyzy yzyzy xyxyz xyzyz, zxzxz zxzyz zyzyz xzxzy xzyzy.

In the even cases, one then ﬁnds a linearly independent set consisting of n n 3( 2 + 1) + 3( 2 + 1) − 6 = 3n elements, so this is a basis in even degree.

39 n+1 n−1 In odd degree, there are 3( 2 + 1) + 3( 2 + 1) − 6 = 3n elements, so also in this case the constructed linearly independent set is a basis.

The next question is to determine the center of Mt .

2.4.12 Proposition. Mt has no central elements in odd degree.

2 2 Proof. Mt maps to the three graded Clifford algebras Chy, zi/(y , z ), Chx, zi/(x 2, z2) and Chx, yi/(x 2, y 2). This means that any central element of odd degree, say w, has to have all variables in its monomials, as each of these quotients doesn’t have central elements in odd degree, so w has to belong to the kernel of each of these algebra morphisms. So say that w has as one of its monomials xyxy ... xzx, with all x on the odd places. For w to belong to the center, one needs to get the first y in yxyxy ... xzx to the last place. This is however impossible, as the first y is in a odd place and the last place is even. Similar results hold for monomials of the form yxyx ... zy and zxzx ... yz. For monomials of the form yx ... xz with x on every even place, the first x in xyxyx ... xz has to be brought to the last place, but again this is impossible as the first place is odd and the last place is even. Similar results hold for xy ... yz (y at even place) and yz ... zx (z at even place). This means that w does not contain any monomials, so w = 0. 2.4.13 Theorem. If t is a primitive nth root of unity, then the elements (x + y)2n,(x + z)2n and (y + z)2n generate the center.

Proof. It is clear that these three elements belong to the center of Mt , as one sees that

(x + y)2x = x(x + y)2,(x + y)2y = y(x + y)2,(x + y)2nz = z(x + y)2n.

The other claimed elements are central by the fact that the center is stable under the action of the Heisenberg group. (2) Consider the second Veronese subalgebra Pt = Mt with generators xy, yx, xz, zx, yz, zy. Then the relations are  (xy)(yx) = 0,  (xy)(xz) = t−1(xz)(xy),  (xy)(zx) = 0,  (xy)(yz) = 0,  (xy)(zy) = t(zy)(xy), together with Heisenberg orbits (in particular, any word containing 3 diﬀerent couples is necessarily 0). It then follows that the center of Pt is generated by the generators of degree 2 of Pt to the nth power. Take a

40 degree 2n central element w of Mt and suppose it contains the monomial (xy)na(xz)nb. It follows that y(xy)na(xz)nb = (yx)nay(xz)nb = (yx)na(zx)nby. This monomial can only be in w if and only if the monomial element (yx)na(zx)nb also occurs in w, with the same coeﬃcient. But this sum of 2 monomials is equal to (xy + yx)na(xz + zx)nb. Similar results hold for the other monomials, so the theorem has been proved.

From now on, assume that t is a primitive nth root of unity. The relation (xy +yx)n(yz +zy)n(zx +xz)n = 0 holds, so the dimension of the center is ≤ 2.

2.4.14 Lemma. Pt is a ﬁnite module over its center.

Proof. Let B be the positively graded ring generated in degree 1 by the 6 elements {uxy , uyx , uxz , uzx , uyz , uzy } and relations  u u = tu u with (ijk) a cyclic permutation of (xyz)  ij kj kj ij −1 uij uik = t uik uij with (ijk) a cyclic permutation of (xyz)  uij ukl = ukl uij i 6= k, l 6= j. Then B is a ﬁnite module over its center. The ring map

B / / Pt deﬁned by uij 7→ ij is an epimorphism, so Pt is a ﬁnite module over its center.

2.4.15 Theorem. Mt is a ﬁnite module over its center.

Proof. Mt is a finite module over Pt , which in turn is a finite module over n 0 n 0 its center. The center of Pt is generated by u = (xy) , u = (yx) , v = n n n 0 n (xz) , v = (zx) , w = (yz) , w = (zy) . The center of Mt is then 0 0 0 generated by u + u , v + v , w + w . Now, Z(Pt ) is generated as a 0 0 0 module over Z(Mt ) by the elements 1, u, u , v, v , w, w , as for example uav 0b = (u + u0)(v + v 0)(ua−1v 0b−1) so by induction one concludes that Mt is a finite module over its center.

41 2.4.16 Corollary. The Krull dimension of the center is two.

Proof. It was already shown that the Krull dimension of the center is ≤ 2. If it was < 2, then Mt would not be a ﬁnite module over its center, as the GK dimension of Mt is 2.

Although Mt is not a prime ring (for the center of Mt is not a domain), it will be shown in Theorem 2.4.21 that the ring Tt is indeed prime (for any ∗ t ∈ C ) and in Theorem 2.4.22 it will be shown that Tt is noetherian. Therefore, using these results, we can give the PI-degree of Mt .

2.4.17 Theorem. The PI-degree of Mt is 2n.

Proof. In Theorem 2.4.7 we found the point modules of Tt and we observed that the orbit of a point module under the shift functor consists of 2n elements if t is of order n in C∗, except in the case that the point module was equal to N = q0q1q0q1 ..., H3-orbits of N or shifts of N. In these special cases, the number of elements in the orbit was two. In addition, the central element gt annihilated each point module.

By the fact that Tt is a prime noetherian ring if t 6= 0, ∞ (as we will see in Theorem 2.4.21 and Theorem 2.4.22), the statements of [54, Section 1, Section 2] can be applied. From this, it follows that shift-equivalence ∗ classes of point modules of Tt correspond to 1-dimensional C -families of simple representations of Tt of dimension the number of elements in the orbit. Consequently, we have found a 2-dimensional family of 2n-dimensional simple representations of Tt , which are annihilated by gt , which therefore are actually representations of Mt . If X is the set of point modules and φ is the automorphism associated to the shift functor, then it is easy to ∼ see that X /hφi = Proj(Z(Mt )), so the PI-degree is indeed 2n.

2.4.5 Connection with the Cliﬀord algebra

∗ Let C be the Clifford algebra C = T1 and let λ ∈ C , for the moment not a root of unity. Define an action of the infinite cyclic group Z =∼ hαi on the Clifford algebra C by

α(x) = λ−1x, α(y) = y, α(z) = λz and take (the localization of) an Ore extension C = C#Z = C[t, t−1; α]. The algebra C itself is Z3-graded, which can be naturally extended to a 4-grading on C by setting deg t = (0, 0, 0, 1). Z Z4

42 Consider the elements x 0 = x, y 0 = yt, z0 = zt−1. The following relations then hold:

z0x 0y 0 = zt−1xyt = λzxy, x 0y 0z0 = xytzt−1 = λxyz, y 0z0x 0 = ytzt−1x = λyzx, y 0x 0z0 = ytxzt−1 = yxz, z0y 0x 0 = zt−1ytx = zyx, x 0z0y 0 = xzt−1yt = xzy.

0 0 0 This means that x , y , z are solutions for the relations of Tλ. In addition, one calculates that

z0x 0y 0 + x 0y 0z0 + y 0z0x 0 − λ(y 0x 0z0 + z0y 0x 0 + x 0z0y 0) =λ((zxy + xyz + yzx) − (yxz + zyx + xzy)), which is also central in C. From Theorem 2.4.3 one can deduce that

2.4.18 Theorem. The ring C is prime.

Proof. Assume that C, so let vCw = 0 with v 6= 0 6= w elements of C. By putting the lexicographic ordering on Z4, which is a total order and taking

f = vmin{(a,b,c,d):v(a,b,c,d)6=0}, g = wmin{(a,b,c,d):w(a,b,c,d)6=0}, it follows that f Cg = 0 and f 6= 0 6= g. There are elements f 0, g 0 ∈ C and k, l ∈ Z such that

f = f 0tk , g = g 0tl .

From Theorem 2.4.3 it follows that there exists a Z3-homogeneous element h ∈ C such that f 0hg 0 6= 0. But then

f hg = f 0tk hg 0 = λr f 0hg 0 6= 0, which is a contradiction. Therefore, C is prime.

2.4.19 Proposition. The Hilbert series of the algebra Qλ generated by x 0, y 0, z0 is equal to (1 − t)−3.

43 Proof. The algebra C = C[t, t−1; α] is deﬁned by generators and relations by

−1 C = Chx, y, z, t, t i/(I ), I = (x 2, y 2, z2,[{x, y}, z], [{y, z}, x], tx − λ−1xt, ty − yt, tz − λzt).

3 S Fix a -homogeneous basis of C of monomials, say 3 Wa,b,c . Z (a,b,c)∈N Then a Z4-homogeneous basis of C is determined by [ Wa,b,c,d , 4 (a,b,c,d)∈N with d Wa,b,c,d = {f t : f ∈ Wa,b,c }. 3 Both Qλ and C are Z -graded and it is easy to see that

−1 k ∀f ∈ C[t, t ; α]a,b,c : f t ∈ Qλ ⇔ k = b − c.

But this implies that, if Wa,b,c is a basis of monomials for Ca,b,c , then

0 b−c Wa,b,c = {f t : f ∈ Wa,b,c } is a basis of (Qλ)a,b,c . Consequently, the multigraded Hilbert series of Qλ and C are the same, but then so are their Hilbert series.

2.4.20 Corollary. The algebra Tλ can be embedded in C, with the image a subalgebra with Hilbert series (1 − t)−3.

0 0 0 Proof. As x , y , z fulﬁl the relations of Tλ, there exists a natural morphism −1 σ : Tλ / C[t, t ; α]

0 0 0 deﬁned by x 7→ x , y 7→ y , z 7→ z , with image the algebra Qλ. Let J = ker σ. Both Tλ and Qλ have a central element of degree 3, gλ and its image σ(gλ). σ(gλ) is not a zero divisor of Qλ (as it is not a zero −1 divisor in C[t, t ; α]), therefore the quotient of Qλ by this element has 1−t3 Hilbert series (1−t)3 . In addition, it was already shown that the Hilbert 1−t3 series of Tλ/(gλ) is (1−t)3 in Theorem 2.4.9. So the diagram

σ Tλ / / Qλ

! ! Mλ,

44 is commutative, which implies that J ⊂ (gλ). Let a be an homogeneous element of J of smallest degree. Then a = bgλ for some b ∈ Tλ homogeneous. But then

0 = σ(a) = σ(b)σ(gλ), which would imply b ∈ J, which would contradict the minimality of the degree of a. Therefore, J = 0.

From this, it follows that Theorem 2.4.6 has been proved. From Theorem 2.4.3 and Corollary 2.4.20 we can now deduce

∗ 2.4.21 Theorem. For any λ ∈ C , the ring Tλ is prime. ∼ Proof. Assume that Tλ = Qλ is not a prime ring and take two non- zero elements v, w ∈ Qλ such that vQλw = 0. Take the lexicograph- 3 P ical order on Z , which is a total order. Writing v = (a,b,c)∈ 3 va,b,c P Z and w = 3 wa,b,c as a sum of homogeneous elements, let f = (a,b,c)∈Z 3 vmin{(a,b,c):va,b,c 6=0} and g = wmin{(a,b,c):wa,b,c 6=0}. Then f and g are Z - homogeneous, f 6= 0 6= g and f Q g = 0. Let deg 3 (f ) = (f , f , f ) and λ Z x y z deg 3 (g) = (g , g , g ). Z x y z From the proof of Proposition 2.4.19, it follows that there exist f 0, g 0 ∈ C such that f = f 0tfy −fz , g = g 0tgy −gz , with f 0 6= 0 6= g 0. From Theorem 2.4.3 it follows that there exists an h ∈ C, which we may assume is Z3-homogeneous such that f 0hg 0 6= 0. 0 Let deg 3 (h ) = (h , h , h ). Then again from the proof of Proposition Z x y z hy −hz 2.4.19, it follows that ht is an element of Qλ. But then we ﬁnd

0 = f hthy −hz g = f 0tfy −fz hthy −hz g 0tgy −gz = λr f 0hg 0tk for some k, r ∈ Z. As t is invertible in C, it follows that this can only 0 0 be 0 if f hg is 0, which is a contradiction. Consequently, Tλ is a prime ring.

∗ 2.4.22 Theorem. For any λ ∈ C , the ring Tλ is a (left and right) noetherian ring.

Proof. This theorem is proved in 3 steps:

• the second Veronese of Tλ/(gλ) is noetherian.

• Tλ/(gλ) is noetherian.

• Tλ is noetherian.

45 First step: In Lemma 2.4.14, it was proved that the second Veronese subalgebra of the quotient Tλ/(gλ) is a quotient of a quantum polynomial ring, which is noetherian. The fact that λ was a root of unity was not relevant in that proof, so even in the general case, the second Veronese subalgebra is indeed noetherian. Second step: Tλ/(gλ) is a ﬁnite module over its second Veronese subalgebra, so it is noetherian. Third step: The conditions of [6, Lemma 8.2] are satisﬁed for the algebras B = Tλ/(gλ), A = Tλ and g = gλ. So Tλ is indeed a (left and right) noetherian ring.

As in the 3-dimensional Sklyanin case, it follows that Tλ is a prime noetherian ring for any λ ∈ C∗.

2.4.6 Roots of unity

When λ is a primitive root of unity of order n, it follows that C[t, t−1; α] is a prime, noetherian PI-ring. In the case that λ is a primitive nth root of unity, there exists a lift of the three linearly independent central elements of degree n of Tλ/(gλ). For simplicity sake, we will assume that (n, 3) = 1. 2.4.23 Proposition. The element (x 0z0 + z0x 0)n belongs to the center of C[t, t−1; α].

Proof. The following holds:

x 0z0 + z0x 0 = xzt−1 + zt−1x = (xz + λzx)t−1.

The elements xz and zx are ﬁxed by α, so one ﬁnds that

((xz + λzx)t−1)n = (xz + zx)nt−n which is indeed central.

n This implies that (xz + zx) belongs to the center of Tλ. Then one can n n use the Heisenberg action in Tλ to see that (yz + zy) and (xy + yx) are also central in Tλ. From [59, Lemma 3.6], the following theorem can be deduced:

2.4.24 Theorem. The center of Tλ with λ a primitive nth root of unity is generated by one element of degree three gλ and three algebraically independent elements of degree 2n, say u, v, w, with one relation of the 2n ∗ form uvw = αgλ for some α ∈ C . Tλ is also a ﬁnite module over its center.

46 Proof. All the conditions of the mentioned lemma are satisﬁed: gλ is regular and the image of C[u, v, w] in Mλ is equal to Z(Mλ). Moreover, as in Mλ the unique relation in the center is uvw = 0, it follows that the 2n only relation is of the form uvw = αgλ . The element α is not 0 as it follows from Theorem 2.4.21 that the center is a domain.

Representation theory

Fix λ a primitive root of unity of order n, n 6= 1, (n, 3) = 1. Let Q = Qλ = Tλ be the subalgebra of C coming from a quotient of S([0 : 0 : 1]). Let P = C/(tn − 1).

2.4.25 Theorem. The center of P is isomorphic to the commutative algebra C[a, b, c, d, e]/(ae − d 2, bc − en).

Proof. The center of C is generated by (y + z)2,(x + z)2,(x + y)2 and

g = zxy + xyz + yzx − (yxz + zyx + xzy) with one relation of the form (x +y)2(y +z)2(x +z)2 = αg 2 for some α ∈ C∗. This follows from (see for example [33]) the fact that C is a Clifford algebra over a polynomial ring in three variables with an associated 3 × 3 symmetric matrix with determinant not a square. So the center of C is isomorphic to C[u, v, w, g]/(uvw − g 2). Then the center of P is isomorphic to the subring of C[u, v, w, g]/(uvw − g 2) fixed by the action n of Zn = hαi/hα i, whose action is defined by the rule

α(u) = λu, α(v) = v, α(w) = λ−1w, α(g) = g.

From this it follows that (C[u, v, w, g]/(uvw − g 2))Zn is generated by a = v, d = g, b = un, c = w n, e = uw, who satisfy the relations ae − d 2, bc − en as claimed.

2.4.26 Corollary. The PI-degree of P is 2n.

Proof. The algebra C is a free module of rank n over C[tn, t−n]. The Clif- ford algebra C[tn, t−n] is a module of rank 4 over its center Z(C)[tn, t−n], which in turn is a module of rank n over its ring of invariants Z(C)Zn [tn, t−n]. So the PI-degree of C is 2n. As every simple representation of P is a simple representation of C, it follows that the PI-degree of P is smaller or equal to 2n. However, as tn is central, each simple representation of C factors through a tn − µ for some µ ∈ C∗. Due to the Z4-gradation, it follows that C/(tn − µ) =∼ C/(tn − 1) =∼ P. Consequently, the PI-degree of P is 2n.

47 2.4.27 Proposition. The PI-degree of Q is 2n.

Proof. The PI-degree of Q is at least 2n, as there are simple representations of dimension 2n coming from the quotient of Q by the degree 3 central element gλ. As Z(P) is a ﬁnite Z(Q)-module of rank n, every simple 2n-dimensional representation of P induces a representation of Q and the induced morphism

Spec(Z(P)) / Spec(Z(Q)) is surjective. But then there is an open subset of Spec(Z(Q)) corresponding to simple 2n-dimensional representations. This proves the claim.

In fact, by using the (C∗)3-action on Q as gradation preserving algebra automorphisms, the following proposition is easy to prove:

2.4.28 Proposition. The Azumaya locus of Spec(Z(Q)) is equal to the variety Spec(Z(Q)) minus 3 lines, the two-by-two intersections of the 3 planes V(u),V(v) and V(w).

∗ 3 Proof. The hH3,(C ) i-action on Q divides max(Z(Q)) into 4 orbits:

• one orbit of dimension 3: the orbit of ((yz +zy)n −1, (zx +xz)n − n 1, (xy + yx) − 1, gλ − 1),

• one orbit of dimension 2: the orbit of ((yz +zy)n −1, (zx +xz)n − n 1, (xy + yx) , gλ),

• one orbit of dimension 1: the orbit of ((yz + zy)n − 1, (zx + n n xz) ,(xy + yx) , gλ), and

• one orbit of dimension 0: the orbit of ((yz +zy)n,(zx +xz)n,(xy + n yx) , gλ).

The ﬁrst orbit is open and therefore intersects Azu2n(Q), but then this orbit is contained in Azu2n(Q). The orbit of dimension two comes from simple representations of Mλ of dimension 2n, so this orbit also belongs to Azu2n(Q).

The orbits of dimension 1 and 0 can not belong to Azu2n(Q), as there are (at least) 1-dimensional families of simple representations of dimension 2 coming from the quotient Q/(z) = Chx, yi/(x 2, y 2).

48 2.4.7 The algebra S([1 : 0 : 0])

In this case, S([1 : 0 : 0]) is a Zhang-twist (see [68]) of S([0 : 0 : 1]) by the automorphism x 7→ z 7→ y 7→ x. Recall from Chapter1 that if A is a graded algebra with a graded automorphism φ, then the Zhang twist is deﬁned as the algebra with the same generators but with multiplication i rule a∗b = aφ (b) if a ∈ Ai . A Zhang twist of a graded algebra preserves the Hilbert series.

2.4.29 Proposition. The algebra S([1 : 0 : 0]) has a 1-dimensional family of quotients parametrized by C∗ such that these quotients have the right Hilbert series. The relations of degree 3 are deﬁned by

2 3 3 2 3 (v1)t = (zyx + ωxzy + ω yxz) − t(y + ωz + ω x ), 2 3 2 3 3 (v2)t = (zyx + ω xzy + ωyxz) − t(y + ω z + ωx ).

3 3 3 The central element becomes gt = (zyx + xzy + yxz) − t(y + z + x ).

Proof. The following holds:

x ∗ x ∗ x = xzy, y ∗ y ∗ y = yxz, z ∗ z ∗ z = zyx, z ∗ y ∗ x = zxy, y ∗ x ∗ z = yzx, x ∗ z ∗ y = xyz and the other 3 monomials become 0. It is then clear that the relations from the proposition are really the Zhang twists of the relations in S([0 : 0 : 1]). If one calculates (z ∗y ∗x +x ∗z ∗y +y ∗x ∗z)−t(y ∗y ∗y +z ∗z ∗z +x ∗x ∗x) in S([1 : 0 : 0]), then one ﬁnds the element

zxy + xyz + yzx − t(yxz + zyx + xzy), which is central in the original quotient of S([0 : 0 : 1]). As this element is also ﬁxed under φ, the proposition has been proved.

2.4.8 The controlling variety

In order to get algebras with the Hilbert series up to degree 4 the same as the Hilbert series up to degree 4 of C[x, y, z], the point [0 : 0 : 1] was replaced with a P1. One of course hopes that the corresponding variety 2 V4 parametrizing these algebras is just the blow-up of P in this point.

2.4.30 Theorem. The variety V4 parametrizing H3-deformations up to degree 4 of the polynomial ring C[V ] is the blow-up of P2 in the 12 points of O.

49 Proof. The variety V3 is a subvariety of

∗ ∗ 3 3 EmbH3 (V ,(V ) ) × EmbH3 (χ0,0, χ0,0)× Y 2 3 ∼ 2 10 EmbH3 (χa,b, χa,b) = (P ) . (a,b)∈Z3×Z3\{(0,0)}

(a,b) Let W be the isotypical component of any H3-representation W corre- 2 10 sponding to the character χa,b, then the conditions for (R2, R3) ∈ (P ) to be a point of V3 (with R2 corresponding to relations of degree 2 and R3 corresponding to relations of degree 3) can be reformulated as R2 ⊗ V + V ⊗ R2 ⊂ R3, which can be decomposed in smaller conditions

(a,b) (a,b) (R2 ⊗ V + V ⊗ R2) ⊂ R3 .

In the case of (a, b) = (0, 0), it was already shown that any R2 completely 3 determines a point of EmbH3 (χ0,0, χ0,0) and vice versa, so this component can be discarded.

As the diﬀerent components of R3 don’t interact with each other, each projection to

∗ ∗ 3 2 3 EmbH3 (V ,(V ) ) × EmbH3 (χa,b, χa,b) can be studied separately. Let (V3)a,b be such a projection. It will be shown that the variety ∗ ∗ 3 2 3 ∼ 2 2 (V3)1,0 ⊂ EmbH3 (V ,(V ) ) × EmbH3 (χ1,0, χ1,0) = P × P parametrizing good quotients with the right multiplicity of χ1,0 in degree 2 3 is the blow-up of P in the H3-orbit of [0 : 0 : 1]. Let R correspond to the relations  ayz + bzy + cx 2 = 0,  azx + bxz + cy 2 = 0, axy + byx + cz2 = 0.

Take the following generators of (V ⊗ V ⊗ V )1,0:

3 3 2 3 f1 = x + ωy + ω z , 2 f2 = zxy + ωxyz + ω yzx, 2 f3 = yxz + ωzyx + ω xzy. Let φ be the subspace of V ⊗ V ⊗ V generated by ( Af1 + Bf2 + Cf3,

Df1 + Ef2 + F f3,

50 such that the matrix ABC DEF has rank 2. Decomposing V ⊗ R + R ⊗ V in H3-representations, one ﬁnds that the couple (R, φ) belongs to (V3)1,0 if and only if the following matrix has rank 2  c aω2 bω   c aω bω2 M =   , ABC  DEF where the ﬁrst row corresponds to (V ⊗ R)(1,0) and the second row to (R ⊗ V )(1,0). Assume that c = 1, putting a = 1 or b = 1 gives similar results. Put a01 = AE − BD, a20 = CD − AF , a12 = BF − CE. Then M has rank 2 if and only if

2 a12 + aωa20 + bω a01 = 0, 2 a12 + aω a20 + bωa01 = 0.

From this it follows that aa20 = ba01, which is indeed the equation for the blow-up of P2 in [a : b : c] = [0 : 0 : 1]. The same result holds for 2 the points [1 : 0 : 0] and [0 : 1 : 0], call (V3)1,0 the blow-up of P in these 3 points.

One could now do the same for the representation χ2,0 to get (V3)2,0, which is also isomorphic to P2 blown-up in the same 3 points. So up to 2 degree 3, the variety becomes (V3)1,0 ×P (V3)2,0. However, in order to get the dimension in degree 4 equal to 15, it was shown in Lemma 2.4.4

2 that one needs to take the ‘diagonal’ ∆ ⊂ (V3)1,0 ×P (V3)2,0, which is of course just the blow-up of P2 in 3 points.

For the other 8 projections, the action of SL2(F3) can be used, which also acts on V4 to conclude that

2 • ∆ ⊂ (V3)0,1 ×P (V3)0,2 corresponds to the blow-up of the H3-orbit of [1 : 1 : 1],

2 • ∆ ⊂ (V3)1,1 ×P (V3)2,2 corresponds to the blow-up of the H3-orbit of [1 : 1 : ω],

2 • ∆ ⊂ (V3)1,2 ×P (V3)2,1 corresponds to the blow-up of the H3-orbit of [1 : 1 : ω2].

2 So V4 itself is the blow-up of P in 12 points.

51 2.4.9 The case t = 0, ∞

There are still 24 points in V4 where the Hilbert series of the corresponding algebras explodes. Still, it would be useful to know what the Hilbert series of these algebras are. As all these algebras are isomorphic to each other (or isomorphic to a Zhang twist) by courtesy of the SL2(F3)-action, it is enough to calculate the Hilbert series of the algebra

2 2 2 2 2 C = Chx, y, zi/(x , y , z , zxy + ωxyz + ω yzx, zxy + ω xyz + ωyzx).

2.4.31 Proposition. The element g0 = zxy + xyz + yzx fulﬁls the conditions g0x = xg0 = g0y = yg0 = g0z = zg0 = 0.

Proof. Using the Heisenberg action, it is enough to prove this for x. Then it is an easy computer calculation with for example MAGMA.

It is therefore enough to compute the Hilbert series of

2 2 2 A = C/(g0) = Chx, y, zi/(x , y , z , xyz, yzx, zxy). This algebra has the advantage that 2 words w and w 0 can only be equal to each other if and only if w = w 0 = 0. From this it follows that each An has a unique monomial basis Bn on which e1 acts by a permutation of order 3, without ﬁxed points. This implies that

#{w ∈ Bn : w begins with x} = #{w ∈ Bn : w begins with y}

= #{w ∈ Bn : w begins with z}.

From Lemma 2.4.4 it follows that the Hilbert series starts with the terms

2 3 4 HA(t) = 1 + 3t + 6t + 9t + 15t + ...

P∞ n 2.4.32 Theorem. Let f (t) = HA(t) = n=0 ant be the Hilbert series of HA(t) with an = dim An. Then the coeﬃcients of f (t) fulﬁl the recurrence relation an = 2an−1 − an−3 for n ≥ 4.

Proof. For n = 4, there is nothing to prove by Lemma 2.4.4. Let n ≥ 5. Let fx : An−1 / An be the linear map deﬁned by w 7→ xw. It then follows that an = 3 dim Im(fx ). So one has to calculate ker(fx ). Due to the relations, ker(fx ) is the subspace of An−1 spanned by the words beginning with x or yz. The space spanned by the words beginning with x has dimension

52 an−1 3 . The space spanned by the words beginning with yz is the image of the map fyz : An−3 / An−1 .

The kernel of fyz is the subspace spanned by the monomials in An−3 2an−3 an−3 starting with x or z. So dim Im(fyz ) = an−3 − 3 = 3 . This gives

an = 3 dim Im(fx )

= 3(an−1 − dim ker(fx )) a a = 3(a − n−1 + n−3 n−1 3 3 = 2an−1 − an−3.

2.4.33 Corollary. For n ≥ 3, there is the recurrence relation an = an−1 + an−2.

Proof. n = 3 is trivial. By induction, assume that an−1 = an−2 + an−3. But then one gets

an = 2an−1 − an−3 = an−1 + (an−2 + an−3) − an−3 = an−1 + an−2.

2.4.34 Corollary. 3 H (t) = − 2 A 1 − (t + t2) Unfortunately, this is not exactly the Fibonacci sequence. However, if he1i one takes B = A , the subalgebra of A ﬁxed by the automorphism e1, then one gets bn = dim Bn = Fn with Fn the nth Fibonacci number starting from F0 = 1, F1 = 1.

2.5 Representations of Sklyanin algebras of global dimension 3 at torsion points

2.5.1 Graded Cayley-Hamilton algebras

Let R be a graded C-algebra, generated by ﬁnitely many homogeneous elements x1, ... , xm where deg(xi ) = di ≥ 0, which is a ﬁnite module over its center Z(R). Following [50, Section 2.3] or [38], R is called a Cayley-Hamilton algebra of degree n if there is a Z(R)-linear gradation tr preserving trace map R / Z(R) such that for all a, b ∈ R one has

53 • tr(ab) = tr(ba) • tr(1) = n

• χn,a(a) = 0 where χn,a(t) is the n-th Cayley-Hamilton identity expressed in the traces of powers of a. Maximal orders in a central simple algebra of dimension n2 are examples of Cayley-Hamilton algebras of degree n. In particular, a 3-dimensional Sklyanin algebra A associated to a couple (E, τ) with p a torsion point of order n, and the corresponding blow- up algebra B = A ⊕ mt ⊕ m2t2 ⊕ ... with m = (x, y, z) are finitely generated graded Cayley-Hamilton algebras of degree n equipped with the (gradation preserving) reduced trace map (see for example [60]). If R is a finitely generated graded Cayley-Hamilton algebra of degree n then one defines trepn R to be the affine scheme of all n-dimensional trace preserving representations, that is of all algebra morphisms

φ R / Mn(C) such that ∀a ∈ R : φ(tr(a)) = Tr(φ(a)), where Tr is the usual trace map on Mn(C). Isomorphism of representations deﬁnes a PGLn(C)-action of trepn R and a result of Artin [2, §12.6] asserts that the closed orbits under this action, that is, the points of the

GIT-quotient scheme trepn R/PGLn(C), are precisely the isomorphism classes of n-dimensional trace preserving semi-simple representations of R. The reconstruction result of Procesi [50, Theorem 2.6] asserts that in this setting

spec Z(R) ' trepn R/PGLn(C). ∗ The gradation on R deﬁnes an additional C -action on trepn R com- ss muting with the PGLn(C)-action. With trepn R the Zariski open subset of all semi-stable trace preserving representations φ : R / Mn(C) is denoted, that is, such that there is an homogeneous central element c of positive degree such that φ(c) 6= 0. There is the following graded version of Procesi’s reconstruction result, see amongst others [14, Section 8],

ss ∗ proj Z(R) ' trepn R/(PGLn(C) × C ). ∗ ss As a PGLn(C)×C -orbit is closed in trepn R if and only if the PGLn(C)- orbit is closed, one sees that points of proj Z(R) classify one-parameter families of isoclasses of trace-preserving n-dimensional semi-simple representations of R. In case of a simple representation such a one-parameter family determines a graded algebra morphism

−1 R / Mn(C[t, t ])(0, ... , 0, 1, ... , 1, ... , e − 1, ... , e − 1) | {z } | {z } | {z } m0 m1 me−1

54 Pe−1 where e is the degree of t, the mi are natural numbers with i=0 mi = n and where one follows [43] in deﬁning the shifted graded matrix algebra Mn(S)(a1, ... , an) for S a graded ring by taking its homogeneous part of degree i to be   Si Si−a1+a2 ... Si−a1+an

Si−a2+a1 Si ... Si−a2+an    .  . . .. .   . . . . 

Si−an+a1 Si−an+a2 ... Si

∗ The PGLn(C)×C -stabilizer subgroup of any of the simple representations φ in this family is then isomorphic to µe where the cyclic group µe has ∗ generator (gζ , ζ) ∈ PGLn(C)×C where ζ is a primitive e-th root of unity and e−1 e−1 gζ = diag(1, ... , 1, ζ, ... , ζ, ... , ζ , ... , ζ ), | {z } | {z } | {z } m0 m1 me−1 ss see [14, lemma 4]. If, in addition, φ is a smooth point of trepn R then the normal space

ss N(φ) = Tφtrepn R/TφPGLn(C).φ to the PGLn(C)-orbit decomposes as a µe -representation into a direct sum of 1-dimensional representations

N(φ) = Ci1 ⊕ ... ⊕ Cid

k where the action of the generator on Ck is by multiplication with ζ . Alternatively, φ determines a (necessarily smooth) point [φ] ∈ spec Z(R) 1 and because N(φ) is equal to ExtR (Sφ, Sφ) and because R is Azumaya 1 in [φ] it coincides with ExtZ(R)(S[φ], S[φ]) (where S[φ] is the simple 1- dimensional representation of Z(R) determined by [φ]) which is identical to the tangent space T[φ]specZ(R). The action of the stabilizer subgroup 1 µe on ExtR (Sφ, Sφ) carries over to that on T[φ]specZ(R). The one-parameter family of simple representations also determines a point φ ∈ projZ(R) and an application of the Luna slice theorem [39] asserts that for all t ∈ C there is a neighborhood of (φ, t) ∈ projZ(R)×C which is ´etaleisomorphic to a neighborhood of 0 in N(φ)/µµe , see [14, Thm. 5].

2.5.2 From Proj(A) to trepnA

As explaind in AppendixB, one studies in noncommutative algebraic geometry the category Proj(A) which is the quotient category of all

55 graded left A-modules modulo the subcategory of torsion modules. In the case of 3-dimensional Sklyanin algebras the linear modules, that is those with Hilbert series (1 − t)−1 (point modules) or (1 − t)−2 (line 2 2 ∗ modules) were classiﬁed in [7, Section 6]. Identify P with Pnc = P(A1), then

• point modules correspond to points on the elliptic curve

2 E / Pnc ,

2 • line modules correspond to lines in Pnc .

In addition, if τ ∈ E is a torsion point of order not divisible by 3, there are fat point modules to consider. These were classiﬁed by Artin [3, Section 2 3] and are relevant in the study of projZ(A) = Pc . Observe that the reduced norm map N relates the diﬀerent manifestations of P2 and the elliptic curve E with its isogenous curve E 0 = E/hτi.

2 ∗ N 2 Pnc = P(A1) / Pc O O

./hτi E / E 0 = E/hτi

2 0 Points Π ∈ Pc − E determine fat points FΠ with graded endomor- −1 phism ring isomorphic to Mn(C[t, t ]) with deg(t) = 1, and hence determine a one-parameter family of simple n-dimensional representations ss ∗ in trepn A with trivial PGLn(C) × C -stabilizer subgroup. There is an eﬀective method to construct FΠ , see [35, Section 3]. Write Π as the intersection of two lines V(z)∩V(z0) and let V(z0)∩E 0 = 0 {q1, q2, q3} be the intersection points with the elliptic curve E . Then by lifting the qi through the isogeny to n points pij ∈ E one sees that one 0 2 2 ∗ 2 can lift the line V(z ) to n lines in Pnc = P(A1), that is, there are n 0 one-dimensional subspaces Cl ⊂ A1 with the property that CN(l) = Cz . The fat point corresponding to Π is then the shifted quotient of a line module determined by l A F ' [n]. ρ Al + Az On the other hand, if q is a point on E 0, then lifting q through the isogeny results in an orbit of n points of E, {r, r + τ, r + [2]τ, ... , r + [n − 1]τ}. If P is the point module corresponding to r ∈ E, then the graded module corresponding to q is

Fq = P ⊕ P[1] ⊕ P[2] ⊕ ... ⊕ P[n − 1]

56 and the corresponding graded endomorphism ring is isomorphic to the −1 graded matrix algebra Mn(C[t, t ])(0, 1, 2, ... , n − 1) where deg(t) = n and hence corresponds to a one-parameter family of simple n-dimensional ss ∗ representations in trepn A with PGLn × C -stabilizer subgroup a cyclic group of order n

1   ζ  µ = h(  , ζ)i n  ..   .  ζn−1

with ζ a primitive n-th root of unity. In fact, a concrete matrix representation can be given of these simple modules. Assume that r − [i]τ = 2 [ai : bi : ci ] ∈ Pnc , then the graded module Fq corresponds to the quiver- representation

3? 1K an−1 a0 bn−1 b0 cn−1 c0 1 + 1 C O [

c1 b1 a1

1 1 Ó

1 s

57 −1 and the map A / Mn(C[t, t ])(0, 1, 2, ... , n − 1) sends the generators x, y and z to the degree one matrices   0 0 ...... an−1t a0 0 ...... 0     .. .   0 a1 . .  ,    ......   . . . .  0 0 ... an−2 0   0 0 ...... bn−1t b0 0 ...... 0     .. .   0 b1 . .  ,    ......   . . . .  0 0 ... bn−2 0   0 0 ...... cn−1t c0 0 ...... 0     .. .   0 c1 . .  .    ......   . . . .  0 0 ... cn−2 0 2.5.1 Lemma. The three matrices deﬁne a simple n-dimensional repre- ∗ sentation of A for each choice of t = t0 ∈ C .

Proof. The point modules of A are given by the elliptic curve E and the automorphism is determined by summation with the point τ = [a : b : c]. 2 Choose r ∈ E and let ri = r − [i]τ = [ai : bi : ci ] ∈ Pnc . Then by deﬁnition of point modules and the associated automorphism,  aa b + bb a + cc c = 0,  i+1 i i+1 i i+1 i abi+1ci + bci+1bi + cai+1ai = 0,  aci+1ai + bai+1ci + cbi+1bi = 0.

∗ Therefore, a quick calculation shows that for each t0 ∈ C , these 3 matrices deﬁne a n-dimensional representation of A. This representation is not an extension of the trivial representation for the center should then be mapped to 0, which is not the case. Using [67, Theorem 3.7], one concludes that this representation is indeed simple. 2.5.2 Theorem. Let A be a 3-dimensional Sklyanin algebra corresponding to a couple (E, τ) where τ is a torsion point of order n and assume that (n, 3) = 1. Consider the GIT-quotient

π : trepnA / / specZ(A) = trepnA/PGLn(C).

58 Then the following holds:

ss 2 1. trepn A is a smooth variety of dimension n + 2, 2. A is an Azumaya algebra away from the isolated singularity (0, 0, 0, 0) ∈ specZ(A), and

3. the nullcone π−1((0, 0, 0, 0)) contains singularities.

Proof. It was already discussed that

2 ss ∗ Pc = projZ(A) = trepn A/(PGLn(C) × C ) classiﬁes one-parameter families of semi-stable n-dimensional semi-simple 2 representations of A. To every point ρ ∈ Pc there was a one-parameter family of simple representations associated, so all semi-stable A-modules are in fact simple as the semi-simpliﬁcation Mss of a semi-stable repre- ss sentation still belongs to trepn A. But then, all non-trivial semi-simple A-representations are simple and therefore the GIT-quotient

ss trepnA / / specZ(A) \{(0, 0, 0, 0)} = trepn A/PGLn(C) is a principal PGLn(C)-fibration in the étaletopology. This proves (1). The second assertion follows as principal PGLn(C)-fibrations in the étale topology correspond to Azumaya algebras. For (3), if trepnA would be smooth, the algebra A would be Cayley-smooth as in [38]. There it is shown that the only type of central singularity that can arise for Cayley- smooth algebras with a 3-dimensional center is the conifold singularity. This is not the case as for the conifold singularity there need to be at least 2 simple representations lying above τ, but there is only one.

⊕e1 In general, if R is Cayley-smooth in m ∈ specZ(R) and if M = S1 ⊕ ⊕ek ...⊕Sk is an isotypical decomposition of the corresponding semi-simple representation M, then one knows that the tangent space TM (trepnR) is the vector space of all (trace preserving) algebra maps ψ

ψ R / Mn(C[]) / / Mn(C) such that the composition with the canonical epimorphism to Mn(C) is the representation φM determined by M. Likewise, the normal space NM to the PGLn(C)-orbit coincides with the vector space of all trace tr preserving extensions ExtR (M, M). From [38, §4.2] recall that this vector space, together with the natural action of Stab(M) = (GLe1 (C) × ∗ ... × GLek (C))/C (1e1 , ... , 1ek ), is given by the representation space

59 • • • rep(Q , αM ) of a (marked) quiver setting (Q , αM ) where Q is a di- rected graph on k vertices, corresponding to the distinct simple components Si of M where some of the loops may be marked, the dimension vector αM = (e1, ... , ek ) encodes the multiplicities of the simple components in M and the representation space is the usual quiver-representation space modulo the requirement that matrices corresponding to marked loops are required to have trace zero. This allows us to compute a defect against R being Cayley-smooth in m. With notations as before, this defect is

k tr 2 X 2 defectm(R) = dimC ExtR (M, M) + (n − ei ) − dim trepnR. i=1

For example, if Rm is an Azumaya algebra over Z(R)m, then R is Cayley- smooth in m if and only if m is a smooth point of specZ(R). In the previous section, two diﬀerent types of simple n-dimensional were deﬁned: representations of A corresponding to whether or not the max- 0 2 imal ideal m lies over a point of E ⊂ Pc or not. Still, their marked quiver-settings are the same (as A is Azumaya in this point and m is a smooth point of the center). In order to distinguish between the two types one has to bring in the extra C∗-action coming from the gradation and turn these (marked) quiver-settings into weighted quiver-settings as in [14].

∗ If the PGLn(C) × C -orbit of the simple representation M corresponding to m determines a point not lying on the elliptic curve E 0, its stabiliser subgroup is trivial, whereas if it determines a point in E 0 the stabiliser subgroup is a cyclic group of order n with generator

1   ζ  µ = h(  , ζ)i n  ..   .  ζn−1 where ζ is a primitive n-th root of unity. This can be easily veriﬁed using the quiver-representation description of the matrices given before. As a consequence, this ﬁnite group acts on the normal-space to the orbit and hence all three loops correspond to a one-dimensional eigenspace i for the µn-action with eigenvalue ζ for some i. To encode this extra information one can weight the corresponding loop by i. 2.5.3 Example (Quaternionic Sklyanin algebras). One can see that 3- dimensional Sklyanin algebras Aλ determined by a point τ of order 2

60 have deﬁning equations (see Chapter3 for more information)  xy + yx = λz2,  yz + zy = λx 2, zx + xz = λy 2,

3 3 3 with −27λ 6= (2 − λ ) , λ 6= 0. An alternative description of Aλ is as a Cliﬀord algebra (see [11] or [33] for some general theory regarding Cliﬀord algebras) over C[u, v, w] (with u = x 2, v = y 2 and w = z2) associated to the rank 3 bilinear form determined by the symmetric matrix

 2u λw λv  λw 2v λu λv λu 2w

The center Z(Aλ) is generated by u, v, w and the square root of the determinant of the matrix which gives the equation of the elliptic curve 0 2 E in Pc λ2(u3 + v 3 + w 3) − (4 + λ3)uvw = 0

A 2-dimensional simple representation M1 of Aλ corresponding to the point [0 : 0 : 1] ∈ P2 − E 0 is given by the matrices 0 λ 0 0 1 0 x 7→ y 7→ z 7→ 0 0 1 0 0 −1

0 A simple representation M2 corresponding to the point [1 : −1 : 0] ∈ E is given by the matrices

0 1 0 −1 0 0 x 7→ y 7→ z 7→ 1 0 1 0 −λ 0

Note that M2 corresponds to the orbit {[1 : −1 : 0], [1 : 1 : −λ]} of 2 points on the elliptic curve E ⊂ Pnc given by the equation λ(x 3 + y 3 + z3) − (λ3 − 2)xyz = 0

The tangent space in M2 to trep2(Aλ) is determined by trace-preserving maps

0 1 a a x 7→ + 1 2 1 0 a3 −a1 0 −1 b b y 7→ + 1 2 1 0 b3 −b1 0 0 c c z 7→ + 1 2 −λ 0 c3 −c1

61 satisfying the three quadratic deﬁning relations of Aλ. The -terms of these equalities give the independent linear relations  −λ2c = a − a + b + b ,  2 2 3 2 3 λ(a2 + a3 + b2) = c2 − c3,  c2 + c3 − λa2 = λ(b2 − b3), which implies that this tangent space is indeed (as required) of dimension ∗ 6. The additional µ2-stabilizer for the PGL2(C) × C -action is generated by 1 0 , −1 0 −1 which acts on a trace zero matrix by sending it to

a b 1 0 −a −b 1 0 −a b 7→ = . c −a 0 −1 −c a 0 −1 c a

Observe that the three linear equations above are ﬁxed under this action so correspond to eigenspaces of weight 0. Hence the tangent space together with the action of the stabiliser subgroup can be encoded by the weighted quiver setting

1 1 1 ! Ó 0C 1 pa where unadorned loops correspond to weight zero. To compute the tangent space to the PGLn(C)-orbit, the subspace of the tangent space given by the -terms of

a b a b (1 + )(φ (x), φ (y), φ (z))(1 − ) 2 c d M2 M2 M2 2 c d has to be determined, which gives the three dimensional subspace consisting of matrix triples

b − c a − d b + c d − a −λb 0 ( , , ) d − a c − b d − a −b − c λ(a − d) λb and under the action of µ2 this space is spanned by one eigenvector of weight zero a − d and two of weight one b and c, whence the tangent space to the orbit can be represented by the weighted quiver-setting

62 1 1 ! Ó 1 p and hence the weighted quiver-setting corresponding to the normal space is represented by 1 ! Ó N ↔ 1 p

Having a precise description of the center Z(A), one doesn’t need such tangent space computations, even for general order n Sklyanin algebras:

∗ 2.5.4 Lemma. If S is a simple A-representation with PGLn(C)×C -orbit 0 determining a fat point Fq with q ∈ E , then the normal space N(S) to ∗ the PGLn(C)-orbit decomposes as representation over the PGLn(C) × C - stabilizer subgroup µn as C0 ⊕ C0 ⊕ C3, or in the terminology of [14], the associated local weighted quiver is

3 ! Ó 1 p

Proof. From [59, Theorem 4.7], the center Z(A) can be represented as

0 0 0 C[x , y , z , c3] Z(A) = n 0 0 0 (c3 − cubic(x , y , z ))

0 0 0 where x , y , z are of degree n (the reduced norms of x, y, z) and c3 is the canonical central element of degree 3. The simple A-representation S determines a point s ∈ specZ(A) such that c3(s) = 0. Again, as A tr is Azumaya over s, N(S) = ExtA (S, S) coincides with the tangent space 0 0 0 Ts specZ(A). Gradation deﬁnes a µn-action on Z(A) leaving x , y , z 3 invariant and sending c3 to ζ c3. The stabilizer subgroup of this action in s is clearly µn and computing the tangent space gives the required decomposition.

2.5.3 The non-commutative blow-up

Consider the augmentation ideal m = (x, y, z) of the 3-dimensional Sklyanin algebra A corresponding to a couple (E, τ) with τ a torsion point of order n. Deﬁne the non-commutative blow-up algebra to be the graded algebra B = A ⊕ mt ⊕ m2t2 ⊕ ... ⊂ A[t] with degree zero part A and where the commuting variable t is given degree 1. Note that B is a graded subalgebra of A[t] and therefore is

63 again a Cayley-Hamilton algebra of degree n. Moreover, B is a ﬁnite module over its center Z(B) which is a graded subalgebra of Z(A)[t]. Observe that B is generated by the degree zero elements x, y, z and by the degree one elements X = xt, Y = yt and Z = zt. Apart from the Sklyanin relations among x, y, z and among X , Y , Z these generators also satisfy commutation relations such as X x = xX , X y = xY , X z = xZ and so on. ss The variety trepn B again denotes the Zariski open subset of trepnB consisting of all trace-preserving n-dimensional semi-stable representations, that is, those on which some central homogeneous element of Z(B) of strictly positive degree does not vanish.

ss 2 2.5.5 Theorem. The variety trepn B is smooth of dimension n + 3.

∗ ss Proof. As before, there is a PGLn(C) × C -action on trepn B with corresponding GIT-quotient

ss ∗ projZ(B) ' trepn B/(PGLn(C) × C )

Composing the GIT-quotient map with the canonical morphism (taking the degree zero part) projZ(B) / / specZ(A) there is a projection map ss γ trepn B / / specZ(A).

Let p be a maximal ideal of Z(A) corresponding to a smooth point, then the graded localization of B at the degree zero multiplicative subset Z(A) − p gives −1 Bp ' Ap[t, t ]

−1 whence Bp is an Azumaya algebra over Z(A)[t, t ] and therefore over the open subset specZ(A) \{(0, 0, 0, 0)} the projection function γ is ∗ a principal PGLn(C) × C -ﬁbration and in particular the dimension of ss 2 trepn B is equal to n + 3. ss This further shows that possible singularities of trepn B must lie in the nullcone γ−1((0, 0, 0, 0)) and as the singular locus is Zariski closed, it is enough to prove smoothness in points of closed PGLn(C)-orbits in γ−1((0, 0, 0, 0)). Such a point φ must be of the form

x 7→ 0, y 7→ 0, z 7→ 0, X 7→ K, Y 7→ L, Z 7→ M.

By semi-stability, (K, L, M) deﬁnes a simple n-dimensional representation ∗ of A and its PGLn(C) × C -orbit deﬁnes the point [det(K) : det(L): 2 det(M)] ∈ Pc , hence one may assume for instance that K is invertible.

64 ss The tangent space Tφtrepn B is the linear space of all trace-preserving algebra maps B / Mn(C[]) of the form

x 7→ 0 + U,y 7→ 0 + V , z 7→ 0 + W , X 7→ K + R,Y 7→ L + S, Z 7→ M + T and one has to use the relations in B to show that the dimension of this space is at most n2 + 3. As (K, L, M) is a simple n-dimensional representation of the Sklyanin algebra, the triple (R, S, T ) depends on at most n2 + 2 parameters. Further, from the commutation relations in B the following equalities are deduced (using the assumption that K is invertible)

• xX = X x ⇒ UK = KU, • xY = X y ⇒ UL = KV ⇒ K −1UL = V , • xZ = X z ⇒ UM = KW ⇒ K −1UM = W , • Y x = yX ⇒ LU = VK ⇒ LK −1U = V , • Zx = zX ⇒ MU = WK ⇒ MK −1U = W .

These equalities imply that K −1U commutes with K, L and M. As (K, L, M) is a simple representation, which implies that K, L and M gen- −1 erate Mn(C) as C-algebra it follows that K U = λ1n for some λ ∈ C. But then it follows that U = λK, V = λL, W = λM and so the triple (U, V , W ) depends on at most one extra parameter, ss 2 showing that Tφtrepn B has dimension at most n + 3, ﬁnishing the proof. 2.5.6 Remark. The statement of the previous theorem holds in a more ss + general setting, that is, trepn B is smooth whenever B = A ⊕ A t ⊕ (A+)2t2 ⊕ ... with A a positively graded algebra that is Azumaya away from the maximal ideal A+ and Z(A) smooth away from the origin.

Unfortunately this does not imply that

ss ∗ projZ(B) = trepn B/(PGLn(C) × C ) ∗ is smooth as there are closed PGLn(C) × C -orbits with non-trivial stabilizer subgroups. This happens precisely in semi-stable representations φ determined by x 7→ 0, y 7→ 0, z 7→ 0, X 7→ K, Y 7→ L, Z 7→ M

65 with [det(K) : det(L) : det(M)] ∈ E 0. In which case the matrices (K, L, M) can be brought into the form   0 0 ...... an−1t a0 0 ...... 0     .. .   0 a1 . .  ,    ......   . . . .  0 0 ... an−2 0   0 0 ...... bn−1t b0 0 ...... 0     .. .   0 b1 . .  ,    ......   . . . .  0 0 ... bn−2 0   0 0 ...... cn−1t c0 0 ...... 0     .. .   0 c1 . .     ......   . . . .  0 0 ... cn−2 0

∗ and the stabilizer subgroup is the cyclic subgroup of PGLn(C) × C

1   ζ  µ = h(  , ζ)i. n  ..   .  ζn−1

2.5.7 Lemma. If φ is a representation as above, then the normal space N(φ) to the PGLn(C)-orbit decomposes as a representation over the ∗ PGLn(C) × C -stabilizer subgroup µn as C0 ⊕ C0 ⊕ C3 ⊕ C−1, that is, the associated local weighted quiver is

# Ó C 1 c 3 −1

Proof. The extra tangential coordinate λ determines the tangent vectors of the three degree zero generators

x 7→ 0 + λK, y 7→ 0 + λL, z 7→ 0 + λM

66 and so the generator of µn acts as follows

1  1   ζn−1   ζ    .(λ(K, L, M)).    ..   ..   .   .  ζ ζn−1 = ζn−1λ(K, L, M) and hence accounts for the extra component C−1. 2.5.8 Theorem. The canonical map

projZ(B) / / specZ(A) is a partial resolution of singularities, with singular locus E 0 = E/hτi 2 in the exceptional ﬁber and all the singularities are of type C × C /Zn. In other words, the isolated singularity of specZ(A) ‘sees’ the elliptic curve E 0 and the isogeny E / / E 0 deﬁning the 3-dimensional Sklyanin algebra A.

Proof. The GIT-quotient map

ss trepn B / / projZ(B)

∗ 0 is a principal PGLn(C) × C -bundle away from the elliptic curve E in the exceptional ﬁber whence projZ(B) \ E 0 is smooth. The application to the Luna slice theorem of [14, Theorem 5] asserts that for any point 0 φ ∈ E / projZ(B) and all t0 ∈ C there is a neighborhood of (φ, t0) ∈ projZ(B) × C which is ´etaleisomorphic to a neighborhood of 0 in N(φ)/µµn. From the previous lemma it follows that

2 N(φ)/µµn ' C × C × C /Zn

2 n 3 where C[C /Zn] ' C[u, v, w]/(w − uv ), ﬁnishing the proof.

Chapter 3

Graded Cliﬀord algebras with an action of Hp

In this chapter, graded Cliﬀord algebras (see [36, Section 4]) on which the ﬁnite Heisenberg group of order p3 acts are studied, with p an odd prime. These algebras are G-deformations of C−1[x0, ... , xp−1], with G = Hp o Z2, where Z2 = hϕi acts on Hp by −1 −1 ϕ(e1) = e1 , ϕ(e2) = e2 .

The corresponding variety V2 parametrizing G-deformations up to degree p−1 2 of the quantum polynomial ring C−1[x0, ... , xp−1] will be P 2 . It will be shown that there are always exactly p + 1 points in V2 such that the corresponding algebra is isomorphic to C−1[x0, ... , xp−1]. In terms of the geometry of elliptic curves with level p-structure, ϕ cor- −1 responds to the automorphism E / E . This suggest a connection between these algebras and p-dimensional Sklyanin algebras at torsion points of order 2. In the next chapter it will be explained that this is indeed the case. For p = 3, these G-deformations of C−1[x0, x1, x2] will in fact be Sklyanin algebras (generically), while in higher dimensions, there will be a 1-dimensional variety in V2 parametrizing Sklyanin algebras at points of order 2. For p = 5, these results will be used in the next chapter to understand the simple representations of Aτ (E) in this particular case.

3.1 Deﬁnition

In general, ﬁnding all G-deformations of an algebra A is an impossible task, even if A is just the commutative polynomial ring. Therefore, it is

69 perhaps useful to study algebras with extra structure, for example graded Clifford algebras on which some group G acts. This is the subject of this chapter: the study of p-dimensional graded Clifford algebras with p an odd prime on which Hp acts with the degree 1 part isomorphic to the Schrödingerrepresentation. Such algebras correspond to quotients of p−1 hx0, ... , xp−1i parametrized by elements (a1 : ... : a p−1 : a0) ∈ 2 , C 2 P with corresponding relations (indices as always taken in Zp) with 1 ≤ i < p−1 j ≤ 2 , 0 ≤ k ≤ p − 1, ( a {x , x } = a x 2, 0 i+k −i+k i k (3.1) aj {xi+k , x−i+k } = ai {xj+k , x−j+k }.

Notice that the second set of equations follows from the ﬁrst set if a0 6= 0, but they have to be included if a0 = 0. As always, Hp acts on these algebras by the rule

i e1 · xi = xi−1, e2 · xi = ω xi , 0 ≤ i ≤ p − 1,

2πI with ω = e p .

3.1.1 Deﬁnition. A graded algebra generated by x0, ... , xp−1 and relations ( 2 a0{xi+k , x−i+k } = ai xk , aj {xi+k , x−i+k } = ai {xj+k , x−j+k }. will be called a Hp-Cliﬀord algebra. Such an algebra will be denoted by

C(a1 : ... : a p−1 : a0) or C(a1, ... , a p−1 ) 2 2

p−1 p−1 for [a1 : ... : a p−1 : a0] ∈ 2 or (a1, ... , a p−1 ) ∈ 2 . 2 P 2 A

3.1.2 Remark. A Hp-Clifford algebra is not necessarily a Clifford algebra, but it always has a quotient which is a Clifford algebra.

3.2 Some additional results on graded Clif- ford algebras

Recall from AppendixB, Section B.4 (or for a more general reference, see [11]) that a graded Cliﬀord algebra depends on the following data: a commutative polynomial ring R = C[y1, ... , yn] with deg(yi ) = 2 for all 1 ≤ i ≤ n and a symmetric matrix M = (mij )ij ∈ Mn(R) with deg(mij ) = 2 for all 1 ≤ i, j ≤ n. The graded Cliﬀord algebra associated to this data is then the algebra with generators x1, ... , xn, y1, ... , yn and relations

{xi , xj } = mij ,[xi , yj ] = 0, [yi , yj ] = 0 ∀1 ≤ i, j ≤ n,

70 with deg(xi ) = 1 for all 1 ≤ i ≤ n. Let CM be this graded Clifford 2 algebra. Consider the elements in v ∈ (CM )1 such that v = 0. These n−1 ∗ elements define a projective variety in P = P((CM )1), defined by a quadric system.

3.2.1 Deﬁnition. A quadric system {q1, ... , qk } is called base-point free if k ∩j=1V(qj ) = ∅.

The following theorem can be found in [15].

3.2.2 Theorem. A graded Cliﬀord algebra CM is quadratic, Auslander- regular of global dimension n, satisﬁes the Cohen-Macaulay property and has as Hilbert series (1−t)−n if and only if the associated quadric system is base-point free. If this is the case, CM is also a noetherian domain and AS regular.

Consider now an algebra C generated in degree one with generators Pn 2 x1, ... , xn and relations {xi , xj } = k=1 aijk xk , 1 ≤ i < j ≤ n for some complex numbers aijk . One would like to know if this algebra is indeed an AS regular graded Cliﬀord algebra. If this is indeed true, then 2 2 x1 , ... , xn are linearly independent, for the global dimension of C should be n. To check whether there are degree 1 elements that become 0, set Pn v = l=1 uk xk . One ﬁnds that

n 2 X 2 X 2 v = (uk + aijk ui uj )xk . k=1 1≤i

So the degree 1 elements of C that have the property v 2 = 0 are parametrized (up to C∗-action) by

n 2 X ∩k=1V(uk + aijk ui uj ). 1≤i

These equations together with the commutation relations [ui , uj ] are the ∗ equations of the Koszul dual of C with ui = xi . From this and Theorem 3.2.2, one deduces:

3.2.3 Proposition. Let C be an algebra generated in degree 1 with Pn 2 generators x1, ... , xn and relations {xi , xj } = k=1 aijk xk , 1 ≤ i < j ≤ n for some complex numbers aijk . Then C is an AS regular algebra with Hilbert series (1 − t)−n if and only if its Koszul dual is ﬁnite dimensional.

71 3.3 p = 3

When p = 3, the algebras studied here have the following relations for [1 : t] = [S : T ] ∈ P1 xy + yx = tz2, yz + zy = tx 2, zx + xz = ty 2.

The representation variety and the normal space in points of these algebras were already studied in Section 2.5.

3.3.1 Theorem. For generic values of t, the H3-Cliﬀord algebra C(t) is a Sklyanin algebra associated to the elliptic curve

E ↔ t(x 3 + y 3 + z3) + (2 − t3)xyz, O = [1 : −1 : 0] and translation by the point [1 : 1 : −t], which is a point of order 2.

There are however 7 values for t ∈ C where the corresponding H3-Cliﬀord algebra is not a Sklyanin algebra:

t = 2, 2ω, 2ω2, −1, −ω, −ω2, 0, with ω being a primitive third root of unity. The noncommutative algebra corresponding to t = ∞ is the algebra with relations

x 2 = y 2 = z2 = 0 and is clearly not regular, so there is a total of 8 points in P1 where C(t) is not a Sklyanin algebra. When t = 0, 2, 2ω, 2ω2, the corresponding algebra is still regular, but in the other 4 cases the Koszul dual C(t)! does not deﬁne the empty set. For t = −1, −ω, −ω2, ∞, C(t)! deﬁnes a set of three points

t = −1 ←→{(1 : 1 : 1), (1 : ω : ω2), (1 : ω2 : ω)}, t = −ω ←→{(1 : 1 : ω2), (1 : ω : ω), (1 : ω2 : 1)}, t = −ω2 ←→{(1 : 1 : ω), (1 : ω : 1), (1 : ω2 : ω2)}, t = ∞ ←→{(1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1)}.

As in Chapter2, there exists an action of SL2(F3) on the moduli space P1 such that algebras in the same orbit are isomorphic (coming from Proposition 1.1.12). However, the element −I2 ∈ SL2(F3) will act trivially 1 on P , so it will be a PSL2(F3)-action (due to the fact that not only H3, but the semidirect product H3 o Z2 acts on these algebras, see example 1.1.8). Using the following generators

0 −1 0 1 U = , V = , 1 0 −1 1

72 2 3 ∼ with U = V = I2, the action of PSL2(F3) = A4 is given by the following M¨obiustransformations:

−t + 2 −ω2t + 2 U ↔ U0(t) = , V ↔ V 0(t) = . t + 1 ω2t + 1

Using this action, the sets {0, 2, 2ω, 2ω2} and {∞, −1, −ω, −ω2} corre- 1 spond to the action of PSL2( 3) on . F PF3 In this case, the action of PSL2(F3) is not a projectiﬁcation of a 2- dimensional representation, because all PSL2(F3)-representations of dimension 2 would be direct sums of 1-dimensional representations. This is impossible because then every element of order 2 would act trivially and this is clearly not the case.

3.4 p = 5

2πi From now on, ω = e 5 . Similar to the case of p = 3, there exists a right ∼ 2 action of PSL2(F5) = A5 on P that gives isomorphic algebras in the moduli space. The algebra C(A : B : C) will correspond to the quotient of Chx0, x1, x2, x3, x4i by the relations

2 C{x1+k , x4+k } = Axk , 0 ≤ k ≤ 4, (3.2) 2 C{x2+k , x3+k } = Bxk , 0 ≤ k ≤ 4, (3.3)

B{x1+k , x4+k } = A{x2+k , x3+k }, 0 ≤ k ≤ 4. (3.4)

The algebra C(a, b) will correspond to C(A : B : 1). Every point with 2 C = 0 lies in the A5-orbit of a point in D(C) ⊂ P . For C(a, b) regular, 2 2 2 2 2 the associated quadratic form Q over C[x0 , x1 , x2 , x3 , x4 ] will be

 2 2 2 2 2 2x0 bx3 ax1 ax4 bx2 2 2 2 2 2 bx3 2x1 bx4 ax2 ax0   2 2 2 2 2 Q = ax1 bx4 2x2 bx0 ax3  .  2 2 2 2 2 ax4 ax2 bx0 2x3 bx1  2 2 2 2 2 bx2 ax0 ax3 bx1 2x4

3.4.1 The nonregular algebras

Using Theorem 3.2.2 and Proposition 3.2.3, one can determine the non- regular algebras. It follows that the following equations need to be anal-

73 ysed:

 2 Cu0 + Au1u4 + Bu2u3 = 0,  Cu2 + Au u + Bu u = 0,  1 2 0 3 4 2 Cu2 + Au3u1 + Bu4u0 = 0, (3.5)  2 Cu3 + Au4u2 + Bu0u1 = 0,  2 Cu4 + Au0u3 + Bu1u2 = 0.

For these 5 equations, one has to determine [A : B : C] such that the only solution is given by the trivial solution.

3.4.1 Theorem. Generically, the H5-Cliﬀord algebra C(A : B : C) is a regular graded Cliﬀord algebra.

Proof. Let V ⊂ P2 × A5 deﬁned by the equations of 3.5 and consider the projection map π1 restricted to V

π1 V / P2 .

Then above the point (A : B : C) = (0 : 0 : 1), the ﬁbre is 1 point. But then there is a Zariski open subset U ⊂ P2 such that

−1 ∀x ∈ U : dim π1 (π1(x)) = 0.

As C∗ acts on each ﬁbre by its action on A5, this is equivalent to

−1 ∀x ∈ U : π1 (π1(x)) = {(x, 0)}.

So each point of U corresponds to a regular algebra.

3.4.2 Remark. The same proof holds for p any odd prime with some minor modiﬁcations.

Generically, C(a1 : ... : a p−1 : a0) is a regular graded Cliﬀord algebra for 2 any p an odd prime. In the case p = 5 however, one can really determine which points give non-regular graded algebras.

3.4.3 Theorem. The non-regular algebras correspond to the points on the following 6 lines and a conic section:

V(C), V(C + A + B), V(C + ωA + ω4B), V(C + ω4A + ωB), V(C + ω2A + ω3B), V(C + ω3A + ω2B), V(AB + C 2).

74 Proof. The image of the projection map

π2 V / A5 is deﬁned by

 u2 u2 u2 u2 u2  5 0 1 2 3 4 {(u0, u1, u2, u3, u4) ∈ A : rank u1u4 u2u0 u3u1 u4u2 u0u3 ≤ 2}. u2u3 u3u4 u4u0 u0u1 u1u2 Let M be the deﬁning matrix of this variety. The projective quotient of this variety is S15, which is the closure of the union of all elliptic 4 curves H5-embedded in P = P(V ) and which is intensely studied in [29, Chapter IV, Section 5]. From these results, it follows that M has rank 2 in all these points except in 30 points, call this set X , which is the hH5, PSL2(F5)i-orbit of (1 : 0 : 0 : 0 : 0). In N = S15 \ X , the map

N / P2 deﬁned by taking the kernel of Mx for M evaluated in x ∈ N has as image the conic section V(AB + C 2). For the point (1 : 0 : 0 : 0 : 0), one observes that the ﬁbre of π2 over this point is the line V(C). The other lines are found similarly.

The 6 lines of this theorem form an orbit under PSL2(F5) (as expected). This means that, if one wants to analyse how ‘far’ these algebras are from being AS regular by describing the base-point set, it is enough to study the points on the line C = 0 and on the conic section AB + C 2 = 0. 3.4.4 Theorem. For a point on the line C = 0, the algebra C(A : B : C)! determines the H5-orbit of (1 : 0 : 0 : 0 : 0) as Zariski-closed set, except in the following 7 cases:

• For k = 0, ... , 4, the point (1 : −ωk : 0) is the intersection of the line C = 0 and the line C + ω3k A + ω−3k B = 0. At the point (1 : −ωk : 0), the projective variety determined by C(1 : −ωk : 0)! is given by 10 points. These 10 points form the union of 2 H5-orbits, each orbit consisting of 5 elements. Representatives of these 2 orbits are given by (1 : 0 : 0 : 0 : 0) and (1 : 1 : ω−2k : ω−k : ω−2k ). The Hilbert series of the commutative algebra C(C : A : B)! is given by 1 + 4t + 5t2 . 1 − t • For (A : B : C) = (1 : 0 : 0), there are ﬁve lines, given by 4 ! ∪i=0V(ui , ui+1, ui+2). The Hilbert series of C(1 : 0 : 0) is given by 1 + 3t + t2 . (1 − t)2

75 The corresponding conﬁguration with its vertices the 5 points in the H5-orbit of (1 : 0 : 0 : 0 : 0) is given by Figure 3.1.

(1 : 0 : 0 : 0 : 0)

(0 : 0 : 0 : 0 : 1) (0 : 1 : 0 : 0 : 0)

(0 : 0 : 0 : 1 : 0) (0 : 0 : 1 : 0 : 0)

Figure 3.1: First conﬁguration

• For (A : B : C) = (0 : 1 : 0), there are again ﬁve lines, now given by 4 the Zariski closed subset ∪i=0V(ui , ui+1, ui+3). The Hilbert series of C(0 : 1 : 0)! is again given by

1 + 3t + t2 . (1 − t)2

The corresponding conﬁguration is given by Figure 3.2.

76 (1 : 0 : 0 : 0 : 0)

(0 : 0 : 0 : 0 : 1) (0 : 1 : 0 : 0 : 0)

(0 : 0 : 1 : 0 : 0) (0 : 0 : 1 : 0 : 0)

Figure 3.2: Second conﬁguration

For a point on the conic section V(AB +C 2), the relations from equation 3.5 determine a smooth genus 1 curve Ea, except when A = 0, B = 0 or

(A : B : C) = (ωk (ω2 + ω3): ω−k (ω + ω4) : 1), k = 0, ... 4, (A : B : C) = (ωk (ω + ω4): ω−k (ω2 + ω3) : 1), k = 0, ... 4.

These 12 points are of course the intersection points with the 6 lines −1 from equation 3.4.3. For every aﬃne point (a, a ) excluding the 12 special points, the point Oa = (0 : 1 : a : −a : −1) and its H5-orbit lies on the curve Ea. Taking this point to be the neutral point Oa of Ea, Ea is an elliptic curve, and the H5-orbit of Oa gives an embedding of Z5 × Z5 in E. It follows from [29] that every point on V(AB + C 2) determines an elliptic curve E, except for 12 points, which are the intersection points of the 6 lines from equation 3.4.3.A PSL2(F5)-orbit on this conic section determines an isomorphism class of elliptic curves, but changes the chosen generators of E[5] = {P ∈ E : 5P = P + P + P + P + P = O}. These new generators however have the same Weil pairing as the original generators (as they determine the representation of H5 in degree 1 of the homogeneous coordinate ring, which is unchanged) and therefore the 2 conic section V(AB + C ) − {12 points} with the PSL2(F5)-action is a model for the modular curve X (5), that is, the curve parametrizing elliptic curves with level 5 structure. The extra 12 points give the natural compactiﬁcation of X (5). Summarizing, the following theorem holds:

77 3.4.5 Theorem. For the points on the curve

V(AB + C 2) − {12 points}, the corresponding algebra C(A : B : C)! is the homogeneous coordinate 4 ring of an elliptic curve E embedded in P with O = Oa = (0 : 1 : a : −a : −1). Furthermore, the point (A : B : C) determines an embedding of Z5 × Z5 in E and a ﬁxed Z5 basis {e1 · O, e2 · O}. Conversely, every point in X (5) determines a unique point on V(AB +C 2) and thus V(AB +C 2) is a model for X (5), with the 12 additional intersection points the points needed to make the compactiﬁcation X (5).

Now, choose a line from Theorem 3.4.3. This line will intersect V(AB + 2 C ) in exactly 2 points, P1 and P2. The commutative algebra corresponding to P1 will describe the union of 5 lines that intersect 2 by 2. The intersection points will be a H5-orbit of 5 elements and the configuration will be like figure 3.1. The point P2 will also determine the union of 5 lines that intersect 2 by 2, with the intersection points the same as for P1, but now the configuration will be like figure 3.2.

3.4.2 The Koszul property

This section is the proof of the following Theorem:

3.4.6 Theorem. Every H5-Cliﬀord algebra C(A : B : C) is Koszul, except for the points on the 6 lines given by the PSL2(F5)-orbit of C = 0 that do not lie on the conic section V(AB + C 2).

Proof. When C(A : B : C) is regular (the generic case), these algebras are Koszul by deﬁnition of being regular, quadratic and having global dimension n. When the Koszul dual is an elliptic curve, it follows from the Koszulity of the homogeneous coordinate ring of an elliptic curve (see [48, Corollary 2]) that the algebras of the form (choosing representatives such that the sum of the indices is 0)

−1 x x + x x = ax 2, x x + x x = x 2 1 4 4 1 0 2 3 3 2 a 0

−1 are indeed Koszul whenever the point (a, a ) determines an elliptic curve. For the 12 points on V(AB +C 2) that determine 5 lines, Koszulity follows from the fact that in this case C(A : B : C) is isomorphic to C(1 : 0 : 0). The relations of C(1 : 0 : 0)! are given by 5 monomials of degree 2 (and the commutator relations of course). Then [17, Theorem 3.15] gives us that C(1 : 0 : 0)! is indeed Koszul, but then C(1 : 0 : 0) is also Koszul.

78 These are all the Koszul algebras. Assume that (A : B : C) lies on 1 of the 6 lines from 3.5, excluding the 12 points lying on the conic section and the 15 points at the intersections and that C(A : B : C) is Koszul. Then C(A : B : C) has as Hilbert series

1 + t = 1 + 5t + 15t2 + 35t3 + 70t4 + 130t5 + ... 1 − 4t + 5t2 − 5t4

This would mean that the Hilbert series of this algebra is equal to (1−t)−5 up to degree 4. But then it follows from the fact that this algebra has as a quoti¨enta graded Cliﬀord algebra that it should have Hilbert series (1 − t)−5, which is not the case. The same argument also applies when the Koszul dual of C(A : B : C) determines 10 points, because now its Hilbert series should be 1 + t = 1 + 5t + 15t2 + 35t3 + 65t4 + 85t5 + ... 1 − 4t + 5t2 and this is clearly not (1 − t)−5.

3.5 Arbitrary p prime

2πi In this section, ω = e p with p ≥ 5 prime. The moduli space is given by p−1 P 2 and again there is an action of PSL2(p) on this space.

A classification of all regular Hp-Clifford algebras is not attainable at the moment, but one can find all algebras isomorphic to the algebra C−1[x0, ... , xp−1].

3.5.1 The quantum spaces

The following theorem will be proved here.

p−1 3.5.1 Theorem. There are exactly p + 1 points in P 2 for which the corresponding algebra C(a1 : ... : a p−1 : a0) is isomorphic to the quantum 2 ring C−1[x0, ... , xp−1] and they form an orbit under the PSL2(Fp)-action.

Before one proves this, some considerations must be made. First of all, p−1 the elements of P with a nontrivial stabilizer in Hp must be found. This is equivalent to ﬁnding the eigenvectors of all the elements of Hp. Since a central element has every vector as eigenvector (and works therefore as the identity on Pp−1), it is suﬃcient to consider elements of the form

79 k l e1 e2. Since e1e2 = ze2e1 with z = [e1, e2] central, one can take powers k l m r −1 t of (e1 e2) until something of the form z e1 e2 appears, unless k = 0. Disregarding the part zr , it suffices to determine the eigenvectors for the −1 t elements e1 e2, t = 0, ... , p − 1 and the element e2. Of course, points belonging to the same orbit have the same stabilizer (since the Hp-action p−1 on P is actually an action of Zp × Zp and this group is commutative), so it is sufficient to give one representative of such an orbit. This means there is a total of p + 1 orbits in Pp−1 with the property that such an orbit consists of p elements instead of p2 elements. −1 k 3.5.2 Lemma. The fixed points of e1 e2 , k = 0, ... , p − 1 are given by the Hp-orbit of the following element:

k 3k k p(p−1) (1 : ω : ω : ... : ω 2 ),

k i(i+1) which has as its ith co¨ordinate ω 2 , starting from 0. The ﬁxed points of e2 are given by the Hp-orbit of (0 : 0 : ... : 0 : 1). These are all the points with a nontrivial stabilizer.

Proof. Identify the image of e1 and e2 in Zp × Zp with the generators 2 p−1 of Hp. Since the order of Zp × Zp is p , every point in P has as its stabilizer the entire group, the trivial group, or a cyclic group of order p p. The entire group is impossible, since the action of Hp on V1 = C was irreducible. Every cyclic group of order p in Zp × Zp has a generator −1 k of the form e2 or e1 e2 , k = 0, ... , p − 1. Since the eigenvalues of the matrices of the representation are all distinct, there are exactly p points that are ﬁxed by any such element. So if one checks that the element −1 k claimed by the theorem indeed is ﬁxed by e1 e2 , then the lemma has been proved (the claim for the group generated by e2 is trivial). By a calculation, one gets

i(i−1) i(i+1) p(p−1) −1 k k k k e e (1 : ... : ω 2 : ω 2 : ... : ω 2 ) 1 2 i(i−1) i(i+1) p(p−1) −1 k +(i−1)k k +ik k +k(p−1) =e (1 : ... : ω 2 : ω 2 : ... : ω 2 ) 1 p(p−1) i(i−1) i(i+1) (p−2)(p−1) k +k(p−1) k +(i−1)k k +ik k +k(p−2) =(ω 2 : ... : ω 2 : ω 2 : ... : ω 2 ) | {z } | {z } i i+1

k p(p−1) +k(p−1) −k Now, ω 2 = ω , so each coordinate can be multiplied with ωk and this gives

k i(i+1) k (i+2)(i+1) k p(p−1) =(1 : ... : ω 2 : ω 2 : ... : ω 2 ). | {z } | {z } i i+1

0 −1 Calculating what the action of U = ∈ PSL ( ) on (0 : ... : 1 0 2 Fp p−1 0 : 1) ∈ P 2 in the moduli space exactly does, one sees that U switches

80 the point (0 : 0 : ... : 1) with (2 : ... : 2 : 1). Next, the action of VU ﬁxes the point (0 : 0 : ... : 0 : 1), but doesn’t ﬁx the point (2 : ... : 2 : 1). Since the order of VU is prime, this means that the VU-orbit of (2 : ... : 2 : 1) consists of p elements. Combining these 2 observations, the following lemma appears:

3.5.3 Lemma. The number of elements in the PSL2(Fp)-orbit of (0 : ... : 0 : 1) consists of at least p + 1 elements.

Proof. [Theorem 3.5.1] The quantum space C(0 : 0 : ... : 0 : 1) has the following property: there are exactly p points in Pp−1 for which the associated quadratic form has rank 1 and these p points form an orbit under the Heisenberg action. So every algebra determined by a point in p−1 our moduli space P 2 and isomorphic to the quantum space must have the same property. In Lemma 3.5.2 it was proved that there are p + 1 diﬀerent orbits consisting of p points. Let ρ = ω2. One may assume that a0 = 1, because otherwise the algebra C(a1 : ... : a p−1 : a0) is not a 2 domain and therefore it can not be isomorphic to C(0 : 0 : ... : 0 : 1). So if the algebra C(a1, ... , a p−1 ) is isomorphic to C(0 : ... : 0 : 0 : 1) = 2 C(0, ... , 0), the matrix

2 2 2  p−1 p−1  2x0 a1x p+1 a2x1 ··· a x p+ ··· a1x p+(p−1) 2 2 2 2 2  2 2 2 2  a1x p+1 2x1 a1x p+3 ········· a2x0  2 2  . . .  . . ..

2 2 has to have rank 1 when (x0 , ... , xp−1) is equal to one of the points found 2 in 3.5.2 (with ω changed by ρ since z acts on the representation Cx0 + 2 ... + xp−1 as ρIp). These conditions completely determine a1, ... , a p−1 C 2 and therefore there can only be maximal p + 1 algebras isomorphic to C(0, 0, ... , 0). But Lemma 3.5.3 gives us that there are minimally p + 1 points, so there are exactly p + 1 points. Therefore, the PSL2(Fp)-orbit p−1 of (0 : 0 : ... : 0 : 1) ∈ P 2 gives all the algebras isomorphic to the quantum space and this orbit consists of p + 1 points.

The algebras isomorphic to C−1[x0, ... , xp−1] can also be characterized by the following rule.

3.5.4 Theorem. The only Hp-Cliﬀord algebras with non-trivial representations of dimension 1 are the algebras isomorphic to the quantum polynomial ring C−1[x0, ... , xp−1].

Proof. It was just shown that the algebras isomorphic to the quantum algebra are given by the PSL2(Fp)-orbit of the point (0 : 0 : ... : 0 : 1).

81 Apart from this point, the other points in this orbit are given by the 1 0 action of the element on the point (2 : ... : 2 : 1). These other 1 1 p−1 S 2 points are elements of the open subset D(a1 ··· a p−1 ) ⊂ D(ai ) and 2 i=1 there are exactly p of them. It is enough to prove that the number of p−1 S 2 points in i=1 D(ai ) for which there exists a 1-dimensional non-trivial representation is equal to p. Suppose that ai 6= 0 and that there exists a non-trivial 1-dimensional representation. Using the Heisenberg action, ∗ assume that x0 is not sent to 0. Using the C -action, also assume that the image of x0 is 1. Let yi ∈ C be the image of xi in C under this representation. Then the following holds:

2 2yi y−i = ai , 2yi+k y−i+k = ai yk

As ai 6= 0, necessarily yi 6= 0. The following formula holds:

k (2) k ai yi yki = k , (3.6) 2(2) which is trivially true for k = 0, 1 and which will now be proved by induction. Assume that it is true for k, then

y(k+1)i = yi+ki a y 2 = i ki 2y(k−1)i a ak(k−1)y 2k 2−k(k−1) = i i i 2 (k−1) k−1 2 k−1 −( 2 ) ai yi 2 k+1 ( 2 ) k+1 ai yi = k+1 2( 2 )

p (2) p ai yi In particular, for k = p it follows that y0 = p = 1. For k = p + 1, it 2(2) p+1 ( 2 ) p+1 ai yi follows that yi = p+1 . This implies 2( 2 ) (p) (p+1) p 2 2 2 2 yi = p = p+1 (2) ( 2 ) ai ai p p So ai = 2 holds. It follows that yi is a pth root of unity. From equation 3.6 follows that yki is uniquely determined for all k, but as i 6= 0, this

82 means that all yj are uniquely determined by yi . But the aj are determined p−1 S 2 by 2yj y−j = aj . So the number of points in i=1 D(ai ) with non-trivial 1-dimensional representations is less than or equal to p. As there are certainly p points in this set, the theorem has been proved.

Chapter 4 n-dimensional Sklyanin algebras

The Sklyanin algebras are perhaps the most famous noncommutative graded algebras. Ever since the general construction in [45] using θ- functions and [63] using line bundles on elliptic curves, it has become clear that these algebras are very important in noncommutative algebraic geometry. However, there are still some mysteries regarding Sklyanin algebras in the case that the associated point on the elliptic curve is a torsion point. In this chapter this problem will be discussed in the case that the dimension is odd and τ is of order two.

Additionally, it will be shown how the relations of 4-dimensional Sklyanin algebras can be constructed using the representation theory of H4.

4.1 Deﬁnition and basic properties

In this section, [46] is used for deﬁning the n-dimensional Sklyanin algebras with n ≥ 4. Let E = C/Γ with Γ = Z ⊕ Zη, =(η) > 0 a lattice in C. Let L be a line bundle of degree n on E and let n = dim H0(L). H0(L) is identiﬁed with the holomorphic functions on C satisfying

f (z + 1) = f (z), f (z + η) = −e2πinz f (z).

85 t Let t = v2(n), so n = 2 b, gcd(b, 2) = 1. One can choose a basis 0 θ0, ... , θn−1 of H (L) such that for 0 ≤ j ≤ n − 1

1 2πI j θ (z + ) = e n θ (z), j n j η −2πI z− πI + n−1 πI η θ (z + ) = e 2t n θ (z). j n j+1

n−1 These functions deﬁne an embedding of E in P by τ 7→ [θ0(τ): ... : n(n−3) θn−1(τ)]. This way, E becomes embedded as an intersection of 2 quadrics, whose equations form a representation of the Heisenberg group Hn. n−1 Let the coordinate functions of P be y0, ... , yn−1. In the case that n is odd, the elements of order 2 in τ correspond to the intersection of E n−1 with the linear subspace V(yi − y−i , 1 ≤ i ≤ 2 ), indices taken in Zn, which follows from AppendixC.

4.1.1 Deﬁnition. The n-dimensional Sklyanin algebra Qn(E, τ) for E an elliptic curve and a point τ ∈ E is deﬁned as the algebra with n generators x0, ... , xn−1 in degree 1 and as relations the quadratic subspace Rτ generated by

n−1 X 1 ∀0 ≤ i < j ≤ n − 1 : v = x x . i,j θ (−τ)θ (τ) j−r i+r r=0 j−i−r r

The indices are taken in Zn.

These relations are quadratic and therefore Qn(E, τ) is a graded algebra. Some properties of these algebras (most of them are proved in [63] while others follow immediately from the description of these algebras given in [45]):

• the Heisenberg group Hn (as deﬁned in the AppendixA) acts on these algebras such that the degree 1 part is isomorphic to V1 with 2πI ω = e n a primitive nth root of unity, • they are noetherian domains,

• the elliptic curve E is embedded in Proj(Qn(E, τ)) as the point modules of Qn(E, τ). The associated automorphism on E is summation with (n − 2)τ. Therefore, there exists an epimorphism

Qn(E, τ) / / O(n−2)τ (E)

n−1 with O(n−2)τ (E) the twisted coordinate ring of E embedded in P with the automorphism on E deﬁned by summation with (n − 2)τ,

86 −n • the Hilbert series of Qn(E, τ) is (1 − t) .

• If τ → 0 ∈ E, then Qn(E, τ) → C[x0, ... , xn−1].

In addition, ⊕0≤i

4.1.2 Theorem. For every n-dimensional Sklyanin Qn(E, τ), there is the isomorphism of Hn-algebras ∼ Qn(E, τ) =Hn C[V1].

4.2 Order 2 Sklyanin algebras

In [63], Tate and Van den Bergh proved that when τ is a torsion point of E, then Qn(E, τ) is a ﬁnite module over its center for every dimension n. However, an explicit description of the center is only known for n = 3, 4 as proved in [59]. However, if n is odd and τ is a torsion point of order two, the center can be described and the PI-degree can be calculated.

Let Qn(E, τ) be a n-dimensional Sklyanin algebra with n odd and τ a non- trivial 2-torsion point. By the rules for the decompositions of tensor prod- ucts of Hn-representations (see AppendixA), it follows that the relations n ∼ n of Qn(E, τ) form a 2 -dimensional Hn-subrepresentation of V1⊗V1 = V2 n−1 ∼ 2 and consequently, Rτ = V2 . The following holds

n−1 n − 1 Emb (V 2 , V n) =∼ Grass , n . Hn 2 2 2

n−1 As such, there are -representatives {v1,n−1, ... , v n−1 n+1 } ⊂ V1 ⊗ V1 2 2 , 2 such that n−1 M2 Rτ = (CHp)vi,−i . i=1

4.2.1 Theorem. If τ ∈ E is a point of order 2 and n is odd, then Qn(E, τ) is a graded Cliﬀord algebra with relations

2 p−1 a0{xm+k , x−m+k } = amxk , 1 ≤ m ≤ 2 , 0 ≤ k ≤ n − 1, n−1 al {xm+k , x−m+k } = am{xl+k , x−l+k }, 1 ≤ m < l ≤ 2 , 0 ≤ k ≤ n − 1

n−1 for some [a1 : ... : a n−1 : a0] ∈ 2 . 2 P

87 This theorem follows directly from the following proposition. 4.2.2 Proposition. If τ is a non-trivial 2-torsion point of an elliptic curve E, we have for the relations vi,−i ∈ Rτ

n−1 n − 1 M2 ∀1 ≤ i ≤ : v ∈ ( {x , x }) ⊕ x 2. 2 i,−i C k −k C 0 k=1

Proof. In order to prove the statement, it is sufficient to show that for the elements vi,−i ∈ Rτ , the monomials xk x−k and x−k xk have the same coefficient if τ is a 2-torsion point. These elements are equal to X 1 vi,−i = x−i−r xi+r θr+2i (τ)θr (τ) r∈Zn 1 Let k ∈ n, then the coefficient of xk x−k is , while the co- Z θi−k (τ)θ−i−k (τ) 1 efficient of x−k xk is . By Lemma C.2.3, these two coefficients θk+i (τ)θk−i (τ) are equal to each other, so the proposition has been proved.

4.2.3 Corollary. The center of Qn(E, τ) for n odd and τ of order 2 is generated by n elements u0, ... , un−1 of degree 2 and 1 element cn of 2 degree n, with one relation of the form φ(u0, ... , un−1) − cn with φ a Hn-invariant polynomial of degree n.

Proof. The description of the center follows directly from the theory of Cliﬀord algebras as explained in for example [11], [33] and [36].

The fact that φ is Hn-invariant follows from the fact that the space of Artin-Schelter regular Cliﬀord algebras on which Hn acts is an open subset n−1 U of P 2 (the proof of Theorem 3.4.1 for n = 5 works for general odd n), such that the algebra C−1[x0, ... , xn−1] corresponds to a point of U. Qn−1 For this algebra, it is easy to see that cn = i=0 xi is Hn-invariant. By a deformation argument, it follows that the central element of degree cn for any Hn-Artin-Schelter regular algebra is Hn-invariant.

4.2.1 5-dimensional Sklyanin algebras associated to 2- torsion points

When the global dimension of the Sklyanin algebras is 5, the equations for the Sklyanin algebras Q5,1(E, τ) with τ ∈ E of order 2 can be explicitly found. The relations can be written as

( 2 {x1+k , x4+k } = axk , 0 ≤ k ≤ 4, 2 (4.1) {x2+k , x3+k } = bxk , 0 ≤ k ≤ 4,

88 with associated quadratic form

 2 2 2 2 2 2x0 bx3 ax1 ax4 bx2 2 2 2 2 2 bx3 2x1 bx4 ax2 ax0   2 2 2 2 2 Ma,b = ax1 bx4 2x2 bx0 ax3   2 2 2 2 2 ax4 ax2 bx0 2x3 bx1  2 2 2 2 2 bx2 ax0 ax3 bx1 2x4

2 2 2 2 2 over the polynomial ring C[x0 , x1 , x2 , x3 , x4 ]. Let C(a, b) be the corresponding algebra as in Chapter3.

4.2.4 Theorem. If the Clifford algebra C(a, b) determines a Sklyanin algebra Q5(E, τ) with τ of order 2, then the point (a, b) lies on the affine curve 0 3 3 5 5 2 2 2 C = V(−a b + a + b + 2a b − 8ab) ⊂ A(a,b). 0 The elliptic curve E = E/hτi is given by the curve Ct defined by the relations 2 2 txi + t xi+1xi−1 − xi+2xi−2, 0 ≤ i ≤ 4 with (ab − 4)a t = . a3 − 2b2

Proof. As Sklyanin algebras are AS regular, Proposition B.4.5 applies. In particular, if C(a, b) is a Sklyanin algebra, then the 3 × 3-minors of Ma,b 0 0 need to vanish on an elliptic curve E . As E has to be an H5-variety, it follows that the 3 × 3-minors of Ma,b need to vanish on a variety of the form 1 C = V(z2 + tz z − z z |0 ≤ i ≤ 4) ⊂ 4 t i i+1 i+4 t i+2 i+3 P by [29]. Using for example Macaulay2, the following 3 equations need to be ful- ﬁlled in order for the 3 × 3-minors to vanish on Ct for some t ∈ C  a2t2 − b2 + 4t = 0,  a2t + 2bt2 − ab = 0, a3t − a2b − 2b2t + 4a = 0.

Eliminating t from the ﬁrst and the third equation and eliminating t from the second and third equation leads to ( a(−a3b3 + a5 + b5 + 2a2b2 − 8ab) = 0, b(−a3b3 + a5 + b5 + 2a2b2 − 8ab) = 0.

89 As the Sklyanin algebras at points of order 2 form a 1-dimensional family of noncommutative algebras, it follows that

−a3b3 + a5 + b5 + 2a2b2 − 8ab = 0.

In addition, one deduces from the third equation that (ab − 4)a t = . a3 − 2b2

Some remarks about C 0 and its projective closure C in P2:

• the projective curve C has six singularities: the PSL2(F5)-orbit of the point [0 : 0 : 1], whose points give the algebras isomorphic to the quantum polynomial ring C−1[x0, x1, x2, x3, x4]. One of course expects these six points to be singular if one looks at the degeneration of the point variety: for a 5-dimensional Sklyanin algebra Q5(E, τ), the point variety is equal to the elliptic curve 4 E. In addition, a family of elliptic curves embedded in P as H5- varieties degenerates to a cycle of ﬁve lines. Consequently, if C was smooth, one expects the quantum polynomial ring C−1[x0, x1, x2, x3, x4] to have as point variety one cycle of ﬁve lines. However, a quantum Pn has at least in its point scheme the full graph on n + 1 points, as will be explained in Chapter5. • There are 12 points on C, which are smooth points, but the corresponding algebras are all isomorphic to the algebra C(1 : 0 : 0) (that is, they lie in the same PSL2(F5)-orbit as the point [1 : 0 : 0]). These 12 points form the intersection of C with the curve V(AB +C 2), which parametrizes the Koszul dual of the graded coordinate rings of elliptic curves with level 5 structure (see Chapter 3). However, these 12 points do not give the Koszul dual of graded coordinate rings of elliptic curves, but of a cycle of 5 lines.

If A = Q5(E, τ) is a graded Cliﬀord algebra with quadratic form Q over 2 2 2 2 2 the polynomial ring C[x0 , x1 , x2 , x3 , x4 ], then the center Z(A) is gener- 2 ated by the 5 elements xi , i = 0, ... , 4 of degree 2 and the square root of the determinant of Q, call this element c5 of degree 5. These six elements satisfy one relation of degree 10

2 2 2 2 2 2 φ(x0 , x1 , x2 , x3 , x4 ) = det Q = c5 , with φ an H5-invariant polynomial of degree 5. As A is a free mod- 5 2 2 2 2 2 ule of rank 2 over the polynomial ring C[x0 , x1 , x2 , x3 , x4 ], it follows

90 5−1 2 that the PI-degree of A is equal to 2 2 = 2 = 4, that is, generically the simple representations are 4-dimensional. The dimension of a simple representation lying above a point m ∈ Max(Z(A)) is determined by the rank of the quadratic form Q after specialization with respect 5 2 2 2 2 2 to m. Let S = Max(Z(A)), A = Max(C[x0 , x1 , x2 , x3 , x4 ]) and denote ψ for the double cover ψ : S / / A5 coming from the inclusion 2 2 2 2 2 C[x0 , x1 , x2 , x3 , x4 ] / Z(A) . This cover is ramiﬁed over V(det Q). 2 Set deg(xi ) = 2 for i = 0, ... , 4. 4.2.5 Theorem. Above any point of S lies a 4-dimensional simple representation of A with exception of the cone above the elliptic curve E 0. For points on the cone above the elliptic curve E 0 there exists a unique 2-dimensional simple representation, except for the trivial representation 2 2 2 2 2 lying above the maximal ideal (x0 , x1 , x2 , x3 , x4 , c5).

Proof. The fact that on the cone above E 0 there are only simple representations of dimension ≤ 2 follows from the fact that these representations are representations of the twisted coordinate ring Oτ (E), which is of PI- degree 2. Also, as none of the Sklyanin algebras are isomorphic to the quantum space, all these representations are necessarily of dimension 2 2 2 2 2 2 following Theorem 3.5.4. Let I ⊂ C[x0 , x1 , x2 , x3 , x4 ] be the ideal associated to the cone above E 0, which is generated by 5 elements of degree 4. 2 2 2 2 2 Let Jk be the ideal of C[x0 , x1 , x2 , x3 , x4 ] generated by all the k×k minors of the quadratic form Q. Using Macaulay2, one checks that (J3)6 = I6, that is, the degree 6 elements in J3 and I are the same. For J4, one checks that all the 4 × 4-minors of Q are the generators for I 2.

For the open set D(J4) the rank of Q is 4 or 5, in both cases the corresponding simple representation is 4-dimensional.

One even has the following description of φ.

4.2.6 Proposition. Let Sec2E ⊂ P4 be the secant variety of an elliptic curve E embedded in P4, that is, the Zariski-closure of all lines in P4 that intersect E in 2 points. Then the variety V(φ) is given by

2 0 4 V(φ) = Sec E ⊂ 2 2 . P[x0 :...:x1 ]

Proof. According to [29, Proposition VIII.2.5], if E is an elliptic curve 2 1 given as the intersection of 5 quadrics Qi = zi +tzi+1zi+4− t z2+i zi+3, 0 ≤ i ≤ 4 in P4, then the deﬁning equation for Sec2E is given by

∂Q 4 Sec2E = V(det i ) ∂zj i,j=0

91 (ab−4)a 0 Using the value of t = a3−2b2 from Theorem 4.2.4 for Ct = E = E/hτi, one checks that 4 ∂Qi V(det ) = V(det(Ma,b)), ∂zj i,j=0

2 where zj = xj , 0 ≤ j ≤ 4, proving the ﬁrst claim. Let

I = (Q0, Q1, Q2, Q3, Q4)

2 2 2 2 2 be the corresponding ideal in C[x0 , x1 , x2 , x3 , x4 ]. Then one checks that 2 0 I4 has the same generators as I . As it is clearly true that E ⊂ X3 ⊂ 0 X4 = E , the second claim has been proved.

The (cone over) the secant variety is singular along (the cone over) E 0, which is equal to the ramiﬁcation locus of A.

4.2.2 From trepn(A) to Proj(A)

From the previous section follows that there are 3 diﬀerent types of non- trivial simple representations:

• representations of dimension 4 where c5 does not act trivially,

• representations of dimension 4 where c5 does act trivially and • representations of dimension 2.

These representations are determined by the rank of the quadratic form Ma,b as explained in Subsections B.4.2 and B.4.1 of AppendixB. Let 4 Y = P 2 2 and deﬁne the following stratiﬁcation of Y [x0 ,...,x4 ]

Yk = {p ∈ Y : rank Ma,b(P) = k}, k = 1, ... , 5 where Ma,b(P) is the matrix in M5(C[t]) with t of degree 2 one gets after taking the quotient of Ma,b by the graded ideal determined by P. From Theorem 4.2.5 it follows that

• Y5 = Y \ V(φ),

0 • Y4 = V(φ) \ E ,

0 • Y2 = E .

Using these facts, the fat points and point modules can be determined for these algebras using Proposition B.4.5.

92 4.2.7 Theorem. The fat points and point modules of Proj(A) are determined by:

• for each point on Proj(Z(A))\V(φ), there exists one corresponding fat point module of multiplicity 4 in Proj(A),

• for each point on V(φ) \ E 0, there are 2 corresponding fat point modules of multiplicity 2 and

• for each point on E 0, there are 2 point modules in Proj(A). The point modules of A are given by E and the map between E and E 0 is the natural isogeny E / E 0 .

Proof. All this follows from Proposition B.4.5 (or for the original reference, [36, Section 4]) applied to this special case.

4.3 Relations for 4-dimensional Sklyanin algebras

In one of the first papers about n-dimensional Sklyanin algebras by Odeskii and Feigin [45] for general n, the special case n = 4 was mentioned as motivating example. Later on, these algebras were studied by Smith in [54] and [55] and by Smith and Stafford in [58], amongst others. As in the 3-dimensional case, these algebras are AS-regular, generated in degree 1 and Koszul with Hilbert series (1 − t)−4. In addition, these algebras contain in their center a 2-dimensional polynomial ring C[Ω1, Ω2], with deg(Ω1) = deg(Ω2) = 2, who generically generate the center. In [55], the Sklyanin algebras where defined as the algebra Chx0, x1, x2, x3i/(R) with defining relations

[x0, x1] = α1{x2, x3}, {x0, x1} = [x2, x3],

[x0, x2] = α2{x3, x1}, {x0, x2} = [x3, x1],

[x0, x3] = α3{x1, x2}, {x0, x3} = [x1, x2].

The point (α1, α2, α3) lies on the aﬃne surface V(α1 + α2 + α3 + 3 α1α2α3) ⊂ A , but not every point on this surface deﬁnes a Sklyanin algebra. Let Aτ (E) be such a 4-dimensional Sklyanin algebra. There are some issues one can have with these relations:

• the action of H4 on these algebras is not obvious, although it is stated in [45] that this action does exist, with the degree 1 part isomorphic to the Schr¨odingerrepresentation of H4,

93 • if τ → 0 ∈ E, then the Sklyanin algebras should degenerate to the polynomial ring in 4 variables, but this is not obvious (although it is explained in [58, page 264]), and • what is the logical compactiﬁcation of this surface in order to study degenerate 4-dimensional Sklyanin algebras? Does it live in P3, P1× P2,(P1)3, ... This section will give a description of the 4-dimensional Sklyanin algebras as H4-deformations of the polynomial ring in 4 variables, in addition the generators of the center will be given. Crucial will be the observation that Aτ (E)/(Ω1, Ω2) will be the twisted coordinate ring Oτ (E), whose point modules are parametrized by E itself.

4.3.1 Decomposition of H4-modules

Recall from AppendixA that the simple representations of H4 can be partitioned in 3 sets by their dimensions:

• 16 1-dimensional characters χa,b deﬁned by a b χa,b(e1) = I , χa,b(e2) = I for (a, b) ∈ Z4 × Z4, ∼ • 4 2-dimensional simple representations Wa,b,(a, b) ∈ Z2 × Z2 = Z4/Z2 × Z4/Z2, W0,0 coming from the quotient 2 2 2 ∼ H4/(e1 , e2 ,[e1, e2] ) = D4

and Wa,b = W0,0 ⊗ χa,b for some lift of (a, b) to an element of Z4 × Z4, ∗ • 2 4-dimensional representations, V = V1 and V = V3.

Let V = V1 = Cx0 + Cx1 + Cx2 + Cx3 be the 4-dimensional Schr¨odinger representation associated to I . Up to basechange, the elements e1 and e2 act on V by the matrices 0 1 0 0 1 0 0 0  0 0 1 0 0 I 0 0  ρ(e1) =   , ρ(e2) =   . 0 0 0 1 0 0 −1 0  1 0 0 0 0 0 0 −I

Let v00 = x0 + x2, v10 = x0 − x2, v01 = x1 + x3, v11 = x1 − x3. Using these vectors as a basis, the action of e1 and e2 is deﬁned by 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 ψ(e1) =   , ψ(e2) =   . 1 0 0 0 0 0 0 I  0 −1 0 0 0 0 I 0

94 The upside of using this basis is that vij is the 1-dimensional representa- 2 2 ∼ 2 i 2 j tion of he1 , e2 i = Z2 × Z2 such that e1 · vij = (−1) vij , e2 · vij = (−1) vij . Using ψ as representation, for any elliptic curve E there exists a λ ∈ C such that E can be embedded in P3 by the equations

( 2 2 2 2 (v00 + v10) − λ(v01 − v11) = 0 2 2 2 2 (4.2) (v01 + v11) − λ(v00 − v10) = 0

The (closure of the) union of these curves cuts out the Fermat quartic surface deﬁned by the equation

4 4 4 4 F4 = V(v00 − v10 − v01 + v11).

The following decompositions are easy to calculate

∼ 2 2 2 2 V ⊗ V = W0,0 ⊕ W1,0 ⊕ W0,1 ⊕ W1,1, ∼ V ∧ V = W1,0 ⊕ W0,1 ⊕ W1,1.

Following Chapter1, it follows that

4.3.1 Proposition. The H4-deformations of the polynomial ring C[V ] 1 3 up to degree 2 are parametrized by the projective variety V2 = (P ) .

Following [45], there exists a subvariety in V2 with an open subset corresponding to the 4-dimensional Sklyanin algebras. An explicit description of the above decompositions is given by

∼ 2 2 2 2 ∼ 2 2 2 2 W0,0 = C(v00 + v10) ⊕ C(v01 + v11) = C(v01 − v11) ⊕ C(v00 − v10), ∼ ∼ W1,0 = C{v00, v10} ⊕ C{v01, v11} = C[v11, v01] ⊕ C[v00, v10], ∼ ∼ W0,1 = C{v00, v01} ⊕ C{v10, v11} = C[v10, v11] ⊕ C[v00, v01], ∼ ∼ W1,1 = C{v00, v11} ⊕ C{v10, v01} = C[v01, v10] ⊕ C[v11, v00].

From this, it follows that the H4-deformations of C[V ] are deﬁned by ( B {v , v } = A [v , v ], 10 00 10 10 01 11 (4.3a) B10{v01, v11} = A10[v10, v00],

( B {v , v } = A [v , v ], 01 00 01 01 11 10 (4.3b) B01{v11, v10} = A01[v01, v00], ( B {v , v } = A [v , v ], 11 00 11 11 01 10 (4.3c) B11{v10, v01} = A11[v11, v00].

95 Aij Let λij = . If one would like to calculate the H4-deformations up to Bij degree 2 of the graded coordinate ring of an elliptic curve E embedded in P3, one adds the relations

( 2 2 2 2 v00 + v10 − λ00(v01 − v11) = 0, 2 2 2 2 (4.4) v01 + v11 − λ00(v00 − v10) = 0.

This implies that the H4-deformations of O(E) up to degree 2 correspond to points of (P1)4.

In particular, for each H4-elliptic curve lying on F4, there are 4 maps

fij E / P1 , 0 ≤ i, j ≤ 1 deﬁned by assigning to the point τ ∈ E the component of its relations isomorphic to Wij as H4-representation. 4.3.2 Theorem. The variety parametrizing Sklyanin algebras intersects the aﬃne variety determined by the equation

2 2 2 2 2 2 λ10λ01 − λ10λ11 − λ01λ11 + 1 = 0

Aij in an open subset. Writing λij = , this is equivalent to the Sklyanin Bij algebras corresponding to points on the surface in (P1)3 determined by the equation

2 2 2 2 2 2 2 2 2 2 2 2 A10A01B11 − A10A11B01 − A01A11B10 + B10B01B11 = 0.

Proof. For a 4-dimensional Sklyanin algebra, the point variety can be computed by taking the multi-linearisation of the relations, taking the corresponding matrix 6 × 4-matrix M and taking the 4 × 4-minors of M as the defining ideal of the point scheme (for the point variety one has to reduce this ideal). Considering the relations of 4.3, this puts conditions on (λ10, λ01, λ11) as these 4×4-minors have to vanish on an elliptic curve of the form of equation 4.2. One then verifies that this happens precisely if the triple (λ10, λ01, λ11) fulfils the relation of the theorem.

Some special points on this projective surface:

• (∞, ∞, ∞) corresponds to the commutative polynomial ring.

• (∞, 0, 0),(0, 0, ∞) and (0, ∞, 0) are 4-dimensional Sklyanin algebras associated to points of order 2, which are also quantum algebras.

96 For the elliptic curve deﬁned by equation 4.2 it follows that −1 − λ λ λ = 01 11 λ10(λ01 − λ11) and for the 2-dimensional vector space that generically generates the center, −λ10(λ01 + λ11) λ00 = λ01λ11 + 1 for the basis given by the elements of equation 4.4. 4.3.3 Remark. For n = 4 the relations for the Sklyanin algebras were determined by using the fact that the extra relations one needs to go from the Sklyanin algebra to a twisted coordinate ring were not isomorphic as H4-representation to a subrepresentation of V ∧ V . In higher dimensions however, this does not work: already for n = 5, the necessary ∼ 5 ∼ 2 decompositions are given by V1 ⊗ V1 = V2 and V1 ∧ V1 = V2 .

Chapter 5

Quantum polynomial rings

With exception of the Sklyanin algebras, the quantum polynomial rings are the most famous noncommutative graded algebras. In particular, they are multigraded, which means that the n-dimensional torus (C∗)n acts on those quantum polynomial rings of global dimension n for each n ≥ 1. This chapter discusses the possible sets of point varieties that can occur for such algebras for quantum polynomial rings of low global dimension (see Section B.2 for the definition of point modules and an introduction into noncommutative algebraic geometry in general). The first goal is to show how these algebras fit in this thesis. In this case, the degree 1 part of C[V ] will not be simple, but each subrepresentation will be multiplicity-free. However, there will be an open subset of V2 corresponding to (C∗)n-deformations of C[V ], with V the canonical n- dimensional representation of (C∗)n. The second goal is computational and shows the degeneration graphs for possible point varieties. A check to see if a linear subspace belongs to the point variety of a quantum polynomial ring is given, which in low dimensions allows us to calculate the possible point varieties. Some results of this section were already discussed in [65], but the proofs here are very different. Nothing but graded ring theory and a bit of Geometric Invariant Theory is used.

5.1 Tn+1-deformations

Quantum polynomial algebras form a class of G-algebras for G = Tn+1. n n Let V = ⊕i=0χei = ⊕i=0Cxi .

99 5.1.1 Proposition. The Tn+1-deformations up to degree 2 of the polynomial ring C[V ] are parametrized by the projective variety 1 (n+1) ∗ (n+1) (P ) 2 , with the open subset (C ) 2 corresponding to the quantum algebras Chx0, ... , xni/(xi xj − qij xj xi , 0 ≤ i < j ≤ n).

Proof. It is easy to see that

n M V ⊗ V ∼ ⊕ χ2 ⊕ χ , = 0≤i

n+1 ∼ 1 ( 2 ) From this it follows that EmbTn+1 (V ∧ V , V ⊗ V ) = (P ) . As a copy of χei +ej , i < j in V ⊗ V corresponds to a relation of the form ∗ (n+1) bij xi xj + aij xj xi , it is clear that the open subset (C ) 2 corresponds to the quantum algebras Chx0, ... , xni/(xi xj − qij xj xi , 0 ≤ i < j ≤ n), with −aij qij = . bij

To a quantum algebra A = Chx0, ... , xni/(xi xj − qij xj xi , 0 ≤ i < j ≤ n) −1 will be associated the matrix Q = (qij )i,j with qii = 1 and qji = qij . Let φ be a diagonal automorphism of Pn. Choose a diagonal matrix 0 φ = diag(a0, ... , an) as a representative of φ in GLn+1(C) and the corresponding algebra automorphism of some quantum algebra A by xi 7→ ai xi . 0 Then the algebra Aφ has relations

−1 ∀0 ≤ i < j ≤ n : xi ∗φ0 xj = aj xi xj = aj qij xj xi = aj ai qij xj ∗φ0 xi .

0 So Aφ is also a quantum algebra. In particular, there is an action of ∗ n+1 ∗ (n+1) (C ) on (C ) 2 deﬁned by −1 (ai )0≤i≤n · (qij )0≤i

100 5.1.2 Theorem. • The point modules of a quantum algebra A are unions of coordinate subspaces P(i1, ... , ik ) = V(uj : j 6= il , 1 ≤ ∗ l ≤ k), with uj = xj , 0 ≤ j ≤ n. • The point variety pts(A) contains at least the full graphs on n points deﬁned by V(ui uj uk : 0 ≤ i < j < k ≤ n).

Proof. As (C∗)n+1 acts on the algebra A, it also acts on the variety parametrizing the point modules. So pts(A) is a union of (C∗)n+1-orbits of Pn, which is always a union of coordinate subspaces.

For the second claim, let Bij = A/(xk : i 6= k 6= j). Then Bij is a 2- dimensional quantum algebra, which is always a twist of the commutative 2-dimensional polynomial ring and therefore has P1 as point variety (see example 1.3.3 or for more general results concerning twists, see [68]).

However, if one is really interested in the unions of coordinate subspaces that can occur, Theorem 5.1.2 is not suﬃcient. Using the discussion from Section B.2 and the fact that each xi is a normalizing element, this theorem can be improved. 5.1.3 Theorem. With notations as above, the following holds:

1. pts(A) = V((qij qjk − qik )ui uj uk , 0 ≤ i < j < k ≤ n).

2. P(i0, ... , ik ) is an irreducible component of pts(A) if and only if the principal k + 1 × k + 1 minor of Q   1 qi0i1 ... qi0ik

qi1i0 1 ... qi1ik  Q(i , ... , i ) =   0 k  . . .. .   . . . . 

qik i0 qik i1 ... 1

is maximal among principal Q-minors such that rk Q(i0, ... , ik ) = 1.

3. pts(A) = V(ui uj uk ; 0 ≤ i < j < k ≤ n, P(i, j, k) 6⊂ pts(A)). In particular, the point variety of A is determined by the P2 = P(u, v, w) it contains.

Proof. 1. Because each variable xi is a normalizing element in A one can consider the graded localization at the homogeneous Ore set 2 {1, xi , xi , ...}. As this localization has an invertible element of degree one it is a strongly graded ring, see [44, §1.4], and therefore is a skew Laurent extension

−1 −1 A[xi ] = Bi [xi , xi ; σ]

101 −1 where Bi is the degree zero part of A[xi ] and where σ is the automorphism on Bi given by conjugation with xi . −1 The algebra Bi is generated by the n elements vj = xj xi and −1 −1 from the commutation relations xj xi = qij xi xj it follows that the commutation relations for Bi are

−1 −1 vj vk = qij xi xj xk xi −1 −1 = qij qjk xi xk xj xi −1 −1 −1 = qij qjk qik xl xi xj xi −1 = qij qjk qik vk vj .

That is, Bi is again a quantum polynomial algebra, this time on n variables vj with corresponding n × n matrix R = (rjk )j,k with entries −1 rjk = qij qjk qik

One-dimensional representations of Bi correspond to points (aj )j ∈ n A (via the morphism vj 7→ aj ) if they satisfy all the deﬁning relations vj vk = rjk vk vj of Bi , that is,

\ ∗ ∗ (aj )j ∈ V((1 − rjk )vj vk ) (5.1) j6=i6=k

n n Observe that this affine space A can be identified with D(ui ) in P ∗ −1 with affine coordinates vj = uj ui . That is, the projective closure of rep1(Bi ), the affine variety of all one-dimensional representations n of Bi , can be identified with the following subvariety of P \ rep1(Bi ) = V((qik − qij qjk )uj uk ). j6=i6=k

Let A = Chx0, x1, x2i/(xi xj − qij xj xi , 0 ≤ i, j ≤ 2) be a quantum polynomial algebra in 3 variables. Then pts(A) is determined (see [6]) by the determinant of the following matrix   −q01u1 u0 0  0 −q12u2 u1 −q02u2 0 u0 which is equal to (q01q12 − q02)u0u1u2. This proves the claim for n = 2. Let A now be a quantum polynomial algebra in n + 1 variables. If P is a point module of A, then each of the variables xi (being normalizing elements) either acts as zero on P or as a non-zero

102 divisor. At least one of the xi must act as a non-zero divisor (oth- −1 erwise P ' C = A/(x0, ... , xn)), but then the localization P[xi ] −1 is a graded module over the strongly graded ring Bi [xi , xi ; σ] and −1 hence is fully determined by its part of degree zero (P[xi ])0, see [44, §1.3] or [7, Proposition 7.5], which is a one-dimensional representation of Bi and so P determines a unique point of rep1(Bi ) described above. Hence, there is the decomposition

pts(A) = rep1(Bi ) t pts(A/(xi )). (5.2)

A/(xi ) is a quantum polynomial algebra in n variables. Hence by induction, \ pts(A/(xi )) = V((qjl − qjk qkl )uj uk ul ) ∩ V(ui ). j6=i,k6=i,l6=i

But then

pts(A) = rep1(Bi ) ∪ pts(A/(xi )) \ = V((qik − qij qjk )uj uk )∪ j6=i6=k \ V((qjl − qjk qkl )uj uk ul ) ∩ V(ui ) j6=i,k6=i,l6=i \ = V((qik − qij qjk )ui uj uk ) 0≤i

The last equality follows from the following lemma.

5.1.4 Lemma. Fix 0 ≤ j < k < l ≤ n. If there exists an i such that  q − q q = 0,  ik ij jk qil − qij qjl = 0,  qil − qik qkl = 0, then qjl − qjk qkl = 0.

Proof. Easy calculation.

From the lemma it follows that if uj uk ul belongs to the deﬁning ideal of the variety pts(A/(xi )), then necessarily for each i either uj uk , uj ul or uk ul belongs to the deﬁning ideal of rep1(Bi ). In particular, it follows that pts(A) = Pn if and only if for all distinct i, j, k the relation −1 qjk = qik qij

103 holds. But then, all 2 × 2 minors of Q have determinant zero as

−1 −1 qju qjv qiuqij qiv qij = −1 −1 qlu qlv qiuqil qiv qil and the same applies for 2 × 2 minors involving the i-th row or column, so Q must have rank one.

2. Observe that P(i0, ... , ik ) = V(uj1 , ... , ujn−k ) if the decomposition

{0, 1, ... , n} = {i0, ... , ik } t {j1, ... , jn−k }

holds. Therefore, P(i0, ... , ik ) ⊂ pts(A) if and only if A P(i0, ... , ik ) = pts(A) with A = (xj1 , ... , xjn−k )

and as A is again a quantum polynomial algebra with corresponding matrix with coeﬃcients Q(i0, ... , ik ) it follows from the remark above that rk Q(i0, ... , ik ) = 1.

3. Recall that P(u, v, w) ⊂ pts(A) if and only if Q(u, v, w) has rank one, which is equivalent to quw = quv qvw . Then the claim follows from the previous 2 points.

5.1.5 Corollary. For generic choices of the matrix Q = (qij )i,j with corresponding algebra AQ , the point variety will be the full graph on n + 1 points.

Proof. The only thing one needs to check is whether there is a quantum algebra with only the full graph on n + 1 points as point variety, as it was already shown that ∀0 ≤ i < j ≤ n : P(i, j) ⊂ pts(A) for any n + 1-dimensional quantum algebra A. Take qij = −1 if i 6= j, then for all 0 ≤ i < j < k ≤ n, it is easy to see that qik − qij qjk = −2 6= 0, so the dimension of pts(AQ ) is 1.

5.2 Possible conﬁgurations

The possible configurations of linear subspaces that arise as point varieties of quantum algebras have to be defined by cubic polynomials. However, not all ideals defined by cubic monomials can occur, as shown in next example.

104 5.2.1 Example. In P3 only two of the P2’s (out of four in total) can arise in a proper subvariety pts(A) (P3. For example, take  1 a b x  a−1 1 a−1b c  Q =   b−1 ab−1 1 ab−1c x −1 c−1 a−1bc−1 1 then, for generic a, b, c, x we have

pts(A) = P(0, 1, 2) ∪ P(1, 2, 3) ∪ P(0, 3) However, if one includes another P2, for example, P(0, 1, 3), then one needs the relation x = ac in which case Q becomes of rank one, whence pts(A) = P3. This is a consequence of Lemma 5.1.4. A combinatorial description of all possible conﬁgurations in low dimensions will be presented. Let C be a collection of P2 = P(i, j, k) contained in Pn. Then C is adequate if the following condition is satisﬁed ∀ 0 ≤ i ≤ n, ∀ P(j, k, l) ∈ C, ∃ {u, v} ⊂ {j, k, l} : P(i, u, v) ∈ C Adequacy gives a necessary condition on the collection of P2’s not contained in the point variety of a quantum polynomial algebra. 5.2.2 Proposition. If A is a quantum polynomial algebra, then

CA = {P(i, j, k) | P(i, j, k) 6⊂ pts(A)} is an adequate collection.

Proof. This is a direct consequence of Lemma 5.1.4.

n+1 The collection of all coordinates (qij )i

In example 5.2.1 we have CA = {P(0, 1, 3), P(0, 2, 3)} and T is the complement of (C∗)4 (with coordinates a, b, c, x) by the sub-torus (C∗)3 deﬁned by x = ac, describing quantum polynomial algebras with point 3 variety P . Here, CA is adequate, but for example C = {P(0, 1, 3)} is not. In fact, for n = 3 it is easy to check that all collections are adequate apart from the singletons, so there are exactly 12 adequate collections. A collection C of P2’s in Pn is called dense if there exist 0 ≤ i < j ≤ n such that # {P(i, j, k) ∈ C} ≥ n − 2 where k 6= i, j. For small n, adequate collections are always dense.

105 5.2.3 Proposition. For n ≤ 4 all adequate collections are dense unless C = ∅.

Proof. For n = 2, the proof is trivial. For n = 3, it is easily seen that C = ∅ is the only non-dense collection. Assume now that n = 4 and that C is a non-dense collection. Then it holds for all 0 ≤ i < j ≤ 4 that

# {P(i, j, k) ∈ C} = 0, 1. If this quantity is always equal to 0 then C = ∅, which is adequate. Hence, assume that one of these quantities is equal to 1. Up to permutation by S5, assume that P(0, 1, 2) ∈ C. Then the only possible P(i, j, k) belonging to C is P(i, 3, 4) with i either equal to 0, 1 or 2. Again up to permutation, assume i = 0. But neither the collection {P(0, 1, 2)} nor {P(0, 1, 2), {P(0, 3, 4)}} is adequate (in both cases, take i = 3 and P(0, 1, 2)).

The possible conﬁgurations in small dimensions can be characterized.

5.2.4 Theorem. Assume n ≤ 5 and let C be an adequate and dense 2 n collection of P ’s in P with variables ui for 0 ≤ i ≤ n. Then,

V(ui uj uk | P(i, j, k) ∈ C) is the point variety pts(A) of a quantum polynomial algebra A with C = CA.

Proof. By the Sn+1-action, assume by denseness that P(0, n) is contained in at least n − 2 of P(0, i, n) ∈ C. One can write C as a disjoint union C1 t C2 t C3 t C4 with  C1 = {P(p, q, r) ∈ C | p, q, r ∈/ {0, n}},  C2 = {P(0, p, q) ∈ C | p, q 6= 0, n}, C = { (p, q, n) ∈ C | p, q 6= 0, n},  3 P  C4 = {P(0, p, n) ∈ C | p ∈/ {0, n}}.

Note that #C4 ≥ n −2. By adequacy of C, it follows that C1 is adequate in the variables ui for 1 ≤ i ≤ n − 1, C1 t C2 is adequate in the variables ui with 0 ≤ i ≤ n − 1 and C1 t C3 is adequate in the variables ui with 1 ≤ i ≤ n. Hence, by applying the induction hypothesis twice (which is possible by Proposition 5.2.2), a ﬁrst time with generic values for C1 t C2 and afterwards with speciﬁc values for C1 t C3, and evaluating the generic values

106 accordingly, one obtains a matrix with non-zero entries   1 q01 ... q0n−1 x −1  q 1 ... q1n−1 q1n   01   ......  Q =  . . . . .   −1 −1  q0n−1 q1n−1 ... 1 qn−1n −1 −1 −1 x q1n ... qn−1n 1 such that for all principal 3 × 3 minors Q(i, j, k) with {0, n} 6⊂ {i, j, k},

rk Q(i, j, k) = 1 if and only if P(i, j, k) ∈/ C1 t C2 t C3.

But then, the same condition is satisﬁed for all the matrices   1 q01 ... q0n−1 x −1  q 1 ... q1n−1 λq1n   01   ......  Qλ =  . . . . .   −1 −1  q0n−1 q1n−1 ... 1 λqn−1n −1 −1 −1 −1 −1 x λ q1n . . . λ qn−1n 1

∗ with λ ∈ C . If #C4 = n − 1, a generic value of x will ensure that rk Q(0, j, n) > 1 for all 1 ≤ j ≤ n − 1. If #C4 = n − 2 let i be the unique entry 1 ≤ i ≤ n − 1 such that P(0, i, n) ∈/ C, then the rank-one condition on   1 q0i x −1 −1 −1 Q(0, i, n) = q0i 1 λqin implies λ = q0i qin x −1 −1 −1 x λ qin 1 and for generic x one can ensure that for all other 1 ≤ j 6= i ≤ n − 1 one has rk Q(0, j, n) > 1.

One can verify that, up to the S6-action on the variables ui , there are exactly two adequate collections for n = 5 that are not dense, which are:

A = {P(0, 2, 4), P(0, 2, 5), P(0, 3, 4), P(0, 3, 5),

P(1, 2, 4), P(1, 2, 5), P(1, 3, 4), P(1, 3, 5)} and

B = {P(0, 1, 3), P(0, 1, 5), P(0, 2, 4), P(0, 4, 5), P(0, 2, 3), P(1, 2, 4),

P(1, 2, 5), P(1, 3, 4), P(2, 3, 5), P(3, 4, 5)}.

107 A is realisable as CA for a quantum polynomial algebra A with matrix  1 1 1 1 x x  1 1 1 1 x x    1 1 1 1 1 1    1 1 1 1 1 1   x −1 x −1 1 1 1 1 x −1 x −1 1 1 1 1 and has as point variety P(0, 1, 2, 3)∪P(0, 1, 4, 5)∪P(2, 3, 4, 5) for generic values of x. 0 B is a CA0 for the quantum algebra A with deﬁning matrix  1 −1 1 1 −1 1 −1 1 −1 1 1 1    1 −1 1 −1 1 1    1 1 −1 1 −1 1   −1 1 1 −1 1 1 1 1 1 1 1 1

The point variety of this algebra is

P(0, 1, 2) ∪ P(1, 2, 3) ∪ P(2, 3, 4) ∪ P(0, 3, 4) ∪ P(0, 1, 4)∪

P(0, 2, 5) ∪ P(1, 3, 5) ∪ P(2, 4, 5) ∪ P(0, 3, 5) ∪ P(1, 4, 5) This shows that denseness is too strong a condition for C to be realised as CA for some quantum polynomial algebra A. However, these results may imply that adequacy is a suﬃcient condition. In particular, all 175 S6- equivalence classes of adequate collections in dimension 5 can be realised as the collection of P2’s not contained in the point variety of a quantum polynomial algebra on 6 variables.

5.3 Degeneration graphs

n+1 ∗ ( 2 ) n+1 Let T2,n+1 = (C ) be the 2 -dimensional torus parametrizing quantum polynomial algebras as before with coordinate functions (qij )i

108 From the above description of point varieties of quantum polynomial algebras, one sees that this degeneration graph corresponds to degenerations of quantum polynomial algebras to other quantum polynomial algebras with a larger point module variety. Some considerations must be made in the calculations of these graphs:

n+1 • Let T3,n+1 be the 3 -dimensional torus with corresponding coordinate functions (bijk )i

• The nodes in our graphs are possible subtori up to Sn+1-action on the variables of the quantum polynomial algebras.

For n = 2, 3, 4, the complete degeneration graphs have been calculated using these methods.

5.3.1 Quantum P2’s

This case is classical [7]: the point variety is either P2 or the union of the 3 coordinate P1’s [6].

5.3.2 Quantum P3’s

The degeneration graph is given in figure 5.1. One can easily check by hand that there are 12 adequate collections and they fall into 4 S4-orbits. The label for a configuration corresponds to the dimension of the loci (in T2,n+1) parametrising these configurations. The type of a configuration describes how many Pk ’s there are as irreducible components in the point variety. The commutative situation where the point variety is the whole of P3 therefore is labeled by 0 and has type (1, 0, 0), whereas the most generic situation (labeled by 3) corresponds to 6 P1’s whose type is denoted by (0, 0, 6). In this case the degeneration graph is totally ordered, with example 5.2.1 corresponding to the configuration with label 1.

109 0

label type 1 0 (1, 0, 0) 1 (0, 2, 1) 2 (0, 1, 3) 2 3 (0, 0, 6)

Figure 5.1: Degeneration graph for quantum P3’s

5.3.3 Quantum P4’s The degeneration graph is given in figure 5.2. There are in total 314 adequate collections, falling into 16 S5-orbits. This time, the degeneration graph no longer is totally ordered. For exam- 2 ple, take the configurations 4a and 4b. These differ by how the two P ’s intersect: in an ambient P4 this happens in either a point or a line. Via similar arguments it is possible to describe each of these configurations.

Observe that 3a and 3c have the same type, but they are not the same 2 1 configuration: 3c corresponds to three P ’s intersecting in a common P , 2 2 whereas orbit 3a has two P ’s intersecting only in a point and a third P intersecting the first in two different P1’s.

5.3.4 Quantum P5’s For n = 5, there is a new phenomenon. 5.3.1 Theorem. There are at least two end points in the degeneration graph for quantum polynomial algebras in 6 variables.

Proof. An end point in the graph corresponds to a n-dimensional family of quantum polynomial algebras. Let C be the collection

{P(0, 1, 2), P(1, 2, 3), P(2, 3, 4), P(0, 3, 4), P(0, 1, 4),

P(0, 2, 5), P(1, 3, 5), P(2, 4, 5), P(0, 3, 5), P(1, 4, 5)}. Then the complement of C is adequate. An algebra A0 with exactly the union of these P2’s in its point variety was already constructed. It is

110 0 label type 0 (1, 0, 0, 0) 1a 1b 1c 1a (0, 1, 2, 0) 1b (0, 0, 5, 0) 1c (0, 2, 0, 1) 2a 2b 2c 2d 2a (0, 0, 4, 0) 2b (0, 0, 4, 2) 2c (0, 1, 1, 2) 3a 3b 3c 3d 2d (0, 0, 4, 1)

3a (0, 0, 3, 3) 3b (0, 1, 0, 4) 4a 4b 3c (0, 0, 3, 3) 3d (0, 0, 3, 2)

4a (0, 0, 2, 5) 5 4b (0, 0, 2, 4) 5 (0, 0, 1, 7) 6 (0, 0, 0, 10) 6

Figure 5.2: Degeneration graph for quantum P4’s enough to show that the family of quantum polynomial algebras with these P2 in its point variety is 5-dimensional. Using the action of K, assume that for all 0 ≤ i ≤ 4 it holds that qi5 = 1. If one can now show that there are a ﬁnite number of solutions, then the theorem has been proved as all degrees of freedom have been used up. It follows from the second row of P2’s in the point variety that

q02 = q13 = q24 = q03 = q14 = 1.

Using the ﬁrst four P2’s, one get the conditions

−1 q01 = q23 = a, q12 = q34 = q04 = a .

Now, P(0, 1, 4) belongs to the point variety if and only if a = a−1 or equivalently, a = ±1. The case a = 1 leads to the commutative polynomial ring, while a = −1 gives an quantum polynomial ring with exactly these 10 P2’s in its point variety.

111

Appendices

113

Appendix A

The ﬁnite Heisenberg groups

The motivating example for this thesis were the 3-dimensional Sklyanin algebras, which are closely related to the ﬁnite Heisenberg group of order 27. In general, the n-dimensional Sklyanin algebras are Hn-deformations of the polynomial ring in n variables. Therefore, it is useful to discuss the simple representations of the ﬁnite Heisenberg groups of order n3, n ≥ 2. This chapter gives a conceptual way to describe this representations, for a more constructive approach see for example [27].

A.1 Deﬁnition

A.1.1 Definition. The finite Heisenberg group of order n3 for n ∈ N\{0} is the group defined by generators and relations n n n he1, e2 : e1[e1, e2] = [e1, e2]e1, e2[e1, e2] = [e1, e2]e2, e1 = e2 = [e1, e2] = 1i.

It is a central extension of Zn × Zn with Zn

0 / Zn / Hn / Zn × Zn / 0 . (A.1)

The elements e1 and e2 will be called the canonical generators of Hn.

A.2 Representation theory

m To discuss the representation theory of Hn, one may assume that n = p for some m ∈ N≥0 and some prime p as a consequence of the following lemma.

115 ∼ A.2.1 Lemma. If n = qr and gcd(q, r) = 1, then Hn = Hq × Hr .

Proof. Let e1,i , e2,i be the canonical generators of Hi . Deﬁne the group morphism f : Hq × Hr / Hn , by the rule

r r f (e1,q, 0) = e1,n, f (e2,q, 0) = e2,n, q q f (0, e1,r ) = e1,n, f (0, e2,r ) = e2,n.

r q r q As the set {e1,n, e1,n, e2,n, e2,n} is a generating set of Hn, f is surjective. This implies that f is an isomorphism, as both groups have the same cardinality.

2πI Assume that n = pm is a power of a prime p and let ω = e pm be a primitive pmth root of unity. From the short exact sequence A.1, it 2m follows that Hpm has p 1-dimensional representations, labelled χa,b with (a, b) ∈ Zp2m × Zp2m , deﬁned by

a b χa,b(e1) = ω , χa,b(e2) = ω .

m ∗ m m−1 In addition, Hpm has φ(p ) = |Zpm | = p − p representations of m ∗ dimension p . These are defined as follows: for k ∈ Zpm , let Vk = pm−1 ⊕i=0 Cxi,k and define the action of Hpm on Vk as ik e1 · xi,k = xi−1,k , e2 · xi,k = ω xi,k , with i ∈ Zpm . Then a quick calculation shows that [e1, e2] acts as multiplication by ωk . This is called the Schrödingerrepresentation associated to ωk .

A.2.2 Lemma. The representation Vk is a simple representation of Hpm .

m Proof. Let χVk be the corresponding character of Hp and let h−, −iHpm be the classical inner product on character functions of Hpm . One has ( a b c 0 if a 6= 0 or b 6= 0, χV (e e [e1, e2] ) = k 1 2 pmωkc else.

Then it follows that hχVk , χVk iHpm = 1.

If m0 < m, then there exists a surjective group morphism

Hpm / / Hpm0 , consequently each simple representation of Hpm0 becomes a simple rep- 0 pm−m m0 resentation of Hpm . If ρ = ω , then ρ is a primitive p -root of

116 ∗ m0 unity. For l ∈ 0 , let W be the associated p -dimensional simple Zpm l representation of Hpm coming from Hpm0 . Recall that if ψ is a 1-dimensional representation of a group G and V is a simple representation of G, then ψ ⊗ G is also a simple representation of G. Consequently, for G = Hpm the tensor product χa,b ⊗ Wl is also a m0 simple p -dimensional representation of Hpm . ∼ A.2.3 Lemma. If (a, b) ∈ Zpm × Zpm , then χa,b ⊗ Wl = Wl if and only m−m0 m−m0 if (a, b) ∈ p Zpm × p Zpm .

Proof. This follows directly from the eigenvalue decomposition of the action of e1 and e2 on χa,b ⊗ Wl .

m0 0 Summarizing, there are for each k ∈ Zpm of order p with 0 ≤ m ≤ m 0 0 exactly p2(m−m ) simple representations of dimension pm . Let I be this set of representations of Hpm . A.2.4 Theorem. The constructed representations are all simple representations of Hpm .

Proof. One calculates

X 2 2m X m0 m0+1 2(m−m0) 2m0 (dim V ) = p + (|p Zpm | − |p Zpm |)p p V ∈I 0≤m0≤m−1 X 0 0 = p2m + (pm−m − pm−m −1)p2m 0≤m0≤m−1 = p3m.

117

Appendix B

Noncommutative algebraic geometry

This chapter is a summary of [52, Section 4]. Readers who are familiar with noncommutative geometry developed in [7] and [6] can skip this part without any problems.

B.1 Introduction

The basic problem of noncommutative algebraic geometry is to associate ‘classical’ commutative objects to noncommutative algebras, for example, to PI-algebras of degree n one associates PGLn(C)-schemes. In the case that the noncommutative algebra is graded, it is a natural question to ask if there is somehow a projective scheme/variety associated to these algebras. Assume that R is a connected, noetherian, graded algebra, ﬁnitely generated in degree one. B.1.1 Deﬁnition. The category R − gr is the category with objects the

ﬁnitely generated, graded R-modules M = ⊕n∈ZMn, RmMn ⊂ Mm+n and homomorphisms

HomR−gr(M, N) = {f ∈ HomR (M, N): ∀n ∈ Z : f (Mn) ⊂ Nn}. B.1.2 Definition. Let R be a Z-graded algebra. For a graded R-module M, define M≥n = ⊕m≥nMm. Then the category qgr(R) is defined as the category with the same objects as R − gr and homomorphisms

Homqgr(R)(M, N) = lim HomR−gr(M≥n, N) n→∞ with the direct limit taken using the inclusion M≥n+1 ,→ M≥n.

119 From the above deﬁnition it follows that there is a natural projection map π from R −gr to R −qgr, R − gr / R − qgr . The category qgr(R) is a quotient of R − gr, with respect to the category of torsion modules (that is, the full subcategory of (direct limits of) ﬁnite dimensional modules). Sometimes one writes Proj(R) for qgr(R). This follows from the fact that, in the case that R is commutative, qgr(R) is equivalent to the category of coherent sheaves on X = Proj(R) (the usual projective scheme), with R itself corresponding to the structure sheaf OX . So in a way, the qgr-construction is a generalisation of the classical commutative case.

B.2 Simple objects in qgr(R)

It is a natural question to ﬁnd the simple objects in qgr(R). One needs the notion of a 1-critical module.

B.2.1 Deﬁnition. An element π(M) ∈ qgr(R) is called 1-critical if π(M) =6∼ 0 and for every graded submodule 0 6= N ⊂ M, M/N is ﬁ- nite dimensional as a C-vector space.

From this deﬁnition, it becomes clear that 1-critical modules correspond to the simple objects in qgr(R). A special case of 1-critical modules is given by the point modules.

B.2.2 Deﬁnition. A point module P is a cyclic graded R-module with −1 Hilbert series HP (t) = (1 − t) .

B.2.3 Remark. It makes sense to classify point modules for any graded algebra A, not just noetherian rings. In [57] for example, the point modules of Chx, y, zi/(x 2, y 2, z2) were classiﬁed, although this ring is far from noetherian (see [66, Proposition 1.2]).

In the commutative case, the point modules correspond to the points of the projective scheme Proj(R) and they are all the simple objects in the category qgr(R). However, there may be other simple objects in qgr(R) in the noncommutative case.

B.2.4 Deﬁnition. A fat point (module) F of R of multiplicity e ∈ N is a 1-critical module with Hilbert series e(1 − t)−1.

A point module is consequently a fat point module of multiplicity 1. The annihilator of F , Ann(F ) = {r ∈ R : ry = 0∀y ∈ F } is a graded prime ideal (see [7, Proposition 2.30,(iv)]).

120 Assume now that x ∈ R is a regular, homogeneous normalizing element and F is a fat point module of multiplicity e. Then

R Tor1 (R/(x), F ) = {y ∈ F : xy = 0}

R is a graded submodule of F and therefore Tor1 (R/(x), F ) = 0 or F . If R −1 Tor1 (R/(x), F ) = 0, then one can localize F at x and one gets a R[x ]- −1 −1 module F [x ] such that ∀i ∈ N : F [x ]i is e-dimensional. As F was 1- critical, F [x −1] is a simple object in R[x −1]−gr. Recall from [43, Theorem 3.1.1] that for a graded ring A there is a equivalence between categories

(−)0 A − gr / A0 − mod if and only if A is a strongly graded ring (recall that A is strongly graded if and only if ∀n, m ∈ Z : AnAm = An+m). From this it follows that a fat point of multiplicity e not annihilated by −1 x corresponds to an e-dimensional simple representation of (R[x ])0.

If R has normalizing elements x1, ... , xn such that R/(x1, ... , xn) is a ﬁnite dimensional algebra, then all fat points can be recovered this way.

B.3 Connection with representation theory

Following [54], there is a connection between simple representations of R and simple objects in qgr(R). Assume in this section that R is (a quotient of) a prime, noetherian algebra, ﬁnite over its center. There exists an autoequivalence on R − gr and by extension on qgr(R), the shift functor M 7→ M[1] = ⊕i∈ZM[1]i with M[1]i = Mi+1. From this it follows that fat points of multiplicity e are mapped to fat points of the same multiplicity. Let F be a fat point and let k ∈ N \{0} be the smallest integer so that F =∼ F [k] in qgr(R). Then this fat point induces a C∗-family of ke-dimensional simple representations [36, Proposition 6, Corollary 1]. Conversely, if S is a simple, non-trivial m-dimensional representation of R, then there exists a fat point module F of multiplicity e such that F =∼ F [k] m for k = e such that S is a quotient of F . One has Ann(S)g = Ann(F ) (for I an ideal of a graded ring R, Ig is the largest homogeneous ideal contained in I ).

The connection with representation varieties is the following: let repm(R) be the variety parametrizing m-dimensional representations of R. There is the usual action of PGLm(C) on this variety by basechange. However, by ∗ the Z-gradation of R, C acts on repm(R). As these 2 actions commute, ∗ PGLm(C) × C acts on repm(R) by the rule

−1 −1 (g, λ) · (A1, ... , Ad ) = (λgA1g , ... , λgAd g ).

121 Assume now that S ∈ repm(R) is simple, S 6= 0 if m = 0. Then one calculates the stabilizer of S with respect to this action. As S is simple, this stabilizer is ﬁnite and isomorphic to Ze (see [14, Lemma 4]). Then ∗ m the PGLm(C)×C -orbit of S corresponds to e fat points of multiplicity e and these fat points are shift equivalent (for more information, see [37, Section 6]).

B.4 An example: graded Cliﬀord algebras

This section will deal with the particular case of graded Cliﬀord algebras needed for this thesis, but this is not the most general deﬁnition, for a general description, see for example [11]. For more information, see [36, Section 4] or [15].

B.4.1 Deﬁnition. Let R = C[y1, ... , yn] be a polynomial ring in n variables, graded such that deg yi = 2, 1 ≤ i ≤ n and let M be a symmetric matrix with entries in R2, det(M) 6= 0. Then the graded Cliﬀord algebra A(M) associated to M is the algebra generated by x1, ... , xn, y1 ... , yn with relations

xi xj + xj xi = Mij ,[xi , yj ] = 0, [yi , yj ] = 0, 1 ≤ i, j ≤ n (B.1) and deg(xi ) = 1, 1 ≤ i ≤ n.

B.4.2 Proposition. A(M) is a free module of rank 2n over the ring C[y1, ... , yn].

The center of A(M) depends on the parity of n:

• If n is even, then Z(A(M)) = C[y1, ... , yn], a polynomial ring in n variables.

• If n is odd, then Z(A(M)) = C[y1, ... , yn, g], with g a central element of degree n fulﬁlling the relation g 2 = det(M).

It is possible that A(M) is generated by x1, ... , xn alone. B.4.3 Example. Take A = Chx, yi/(xy + yx). The center of A is generated by x 2 and y 2 and A is a free module of rank 4 over C[x 2, y 2]. A is a Cliﬀord algebra over C[x 2, y 2] with associated symmetric matrix

2x 2 0 . 0 2y 2

122 B.4.1 Representations of graded Cliﬀord algebras

In order to describe the representation theory of A(M), recall the deﬁni- tion of the PI degree of an algebra ﬁnite over its center.

B.4.4 Deﬁnition. Let A be a ﬁnite module over its center Z(A), with Z(A) a normal domain. Then the PI degree of A is equal to

φ a = max{m ∈ N : ∃ A / Mm(C) simple}.

The set of points m ∈ Max(Z(A)) that fulfil the condition that there exists ∼ a maximal ideal M ∈ Max(A) such that M∩Z(A) = m and A/M = Ma(C) is called the Azumaya locus of A. Returning to the the special case of graded Clifford algebras, let m be a maximal ideal of C[y1, ... , yn], let n be odd and assume that det(M) is not a square in C[y1, ... , yn]. It is easy to see (using the theory of Clifford algebras over C, see for example [16] and [36]) that the dimension of simple representations depends on the rank of M after taking the quotient with respect to m. Let Ym be the corresponding symmetric matrix in Mn(C).

n−1 • If rank Ym = n, then there are 2 simple 2 2 -dimensional representations. These 2 representations are separated in the center by the double cover

Max(Z(A)) / / Max(C[y1, ... , yn])

coming from the inclusion C[y1, ... , yn] / Z(A).

n−1 • If rank Ym = n − 1, then there is 1 simple 2 2 -dimensional representation lying over m.

k−1 • If rank Ym = k, k odd, then there are 2 2 2 -dimensional simple representations lying over m.

k • If rank Ym = k, k even, then there is 1 2 2 -dimensional simple representation lying over m.

B.4.2 The Proj of graded Cliﬀord algebras

For a graded Cliﬀord algebra A(M) with M the corresponding symmetric matrix in Mn(C[y1, ... , yn]), the classiﬁcation of the fat points and point modules is determined by [36, Proposition 9] on the condition that A(M)

123 is a domain and generated in degree 1. For a maximal graded prime ideal p of C[y1, ... , yn], let M(p) be the symmetric matrix one gets after specialization with respect to p.

n−1 B.4.5 Proposition. Let Y = Proj(C[y1, ... , yn]) = P and let

Yk = {p ∈ Y : rank M(p) = k} ⊂ Y .

Then for each p ∈ Y , there exists a unique graded prime ideal P of A(M) −1 such that A(p) = A(M)/P ⊗C[y] C[y, y ] has the following structure.

• If p ∈ Yk with k odd, then there exists an isomorphism

∼ −1 A(p) = M k−1 (C[x, x ])(0, 0, ... , 0) 2 2 | {z } k−1 2 2

and deg(x) = 1. This implies that there is one fat point of multi- k−1 plicity 2 2 .

• If p ∈ Yk with k even, the isomorphism becomes

∼ −1 A(p) = M k (C[y, y ])(0, 0, ... , 0, 1, 1, ... , 1) 2 2 | {z } | {z } k −1 k −1 2 2 2 2

and deg(y) = 2. This implies that there are two fat points of k −1 multiplicity 2 2 .

From this it follows that the points p ∈ Pn−1 for which there are point Sk modules in Proj(A) are determined by Y2. Set Xk = i=0 Yi , then this proposition shows that the point modules of Proj(A) determine a 2-to-1 cover of X2, with ramiﬁcation over X1.

B.4.6 Example. Take A = Chx1, ... , xni/({xi , xj } : 1 ≤ i < j ≤ n). The following is easy to check:

2 2 • if n is even, then Z(A) = C[x1 , ... , xn ], or

2 2 Qn • if n is odd, then Z(A) = C[x1 , ... , xn , i=1 xi ].

∼ 2 Qn In the last case one has Z(A) = C[u1, ... , un, g]/(g − i=1 ui ). 2 2 The associated symmetric matrix is clearly diag(2x1 , ... , 2xn ). Using the fact that the point modules are parametrized by a double cover of X2 with ramiﬁcation over X1, the point modules of A are determined by the full graph on n points, with ramiﬁcation over the intersection points.

124 B.5 Koszul algebras

Koszul algebras are a class of quadratic algebras that have good homological properties. For more information, please see [25], [40] or [49]. If R is a connected, quadratic algebra, finitely generated in degree 1, one would like that the trivial module C has a linear resolution for homological reasons. From this follows the definition of a Koszul algebra. B.5.1 Definition. Let R = T (V )/I be a quadratic algebra with defining ! ∗ relations I2. The Koszul dual is the quadratic algebra R = T (V )/J, ∗ ∗ with J = (J2) and J2 defined as the subspace of V ⊗ V such that

∀w ∈ J2, ∀v ∈ I2 : w(v) = 0.

If R is quadratic, one has the associated complex K(R, C)

n−1 dn d d2 d1 d0 K ! ∗ K K ! ∗ K K / R ⊗ (R )n−1 / ... / R ⊗ (R )1 / R / C / 0

! ∗ n−2 ⊗j ⊗n−2−j with (R )n = ∩j=0 V ⊗ I2 ⊗ V and diﬀerential

n dK (r ⊗ (v1 ⊗ v2 ⊗ ... ⊗ vn)) = rv1 ⊗ (v2 ⊗ ... ⊗ vn), n ≥ 1, 0 dK (r) = r.

B.5.2 Deﬁnition. A quadratic algebra R is a Koszul algebra if and only if K(R, C) is exact.

The standard properties of Koszul algebras needed here are that there is a relation between the Hilbert series of R and R!, given by

HA(t)HA! (−t) = 1 and that R is Koszul if and only if R! is Koszul. B.5.3 Example. The polynomial ring C[V ] for V a ﬁnite dimensional vector space is a Koszul algebra, with Koszul dual ∧V ∗, the wedge algebra over V ∗.

125

Appendix C

Theta functions

As the coeﬃcients of the relations for Sklyanin algebras depend on theta functions, it might prove useful to have an explicit description of the theta functions of order n on an elliptic curve E = C/(Z + Zη) with =(η) > 0. In addition, some additional results are shown in the speciﬁc case that n is odd and τ is of order 2.

C.1 Solution of the functional equations

Let E = C/(Z + Zη) be an elliptic curve with =(η) > 0. Recall from [45, Section 1] that the theta functions of order n with respect to the lattice Z + Zη were deﬁned by the following functional equations 1 α ∀0 ≤ α ≤ n − 1 :θ z + = exp 2πI θ (z), (C.1) α n n α η πI (n − 1)πI η ∀0 ≤ α ≤ n − 1 :θ z + = exp −2πI z − + θ (z). α n 2P n α+1 (C.2)

The value P is equal to the 2-valuation of P in these equations.

One can ﬁnd an explicit solution {θ0(z), ... , θn−1(z)} of these functional equations. Let Θ(z) = exp(2πI z).

m mη C.1.1 Proposition. For m ∈ Z, let am = Θ 2P+1 + 2n (m − n) . Then a solution of the equations C.1 and C.2 is given by X ∀0 ≤ α ≤ n − 1 : θα(z) = aα+knΘ ((α + kn)z). k∈Z

127 Proof. The ﬁrst set of equations C.1 are trivially fulﬁlled. For the second set, one calculates that η X m mη η θ z + = Θ + (m − n) Θ m(z + ) α n 2P+1 2n n m≡α mod n X m mη = Θ + (m − n + 2) Θ (mz) 2P+1 2n m≡α mod n and

πI (n − 1)πI η exp −2πI z − + θ (z) 2P n α+1 X 1 (n − 1)η m + 1 (m + 1)η = Θ − + Θ + (m + 1 − n) 2P+1 2n 2P+1 2n m≡α mod n Θ (mz).

For the latter equation, one calculates the coeﬃcients of Θ(mz) to be

1 (n − 1)η m + 1 (m + 1)η Θ − + Θ + (m + 1 − n) 2P+1 2n 2P+1 2n m η =Θ + (n − 1 + (m + 1)(m + 1 − n) 2P+1 2n m mη =Θ + (m − n + 2) . 2P+1 2n

C.2 n is odd

Theorem 4.2.1 deals with Sklyanin algebras of prime global dimension and points of order 2. As such, it is useful to calculate θα(z) and θα(−z) for z ∈ C such that its image in C/(Z + Zη) is of order 2 for n odd. C.2.1 Corollary. If n is odd, it follows that

∀0 ≤ α ≤ n − 1, ∀z ∈ C : θα(−z) = −Θ(−nz)θ−α(z).

Proof. Under this condition, the following holds:

X m mη θ (−z) = Θ + (m − n) Θ(m(−z)) α 2 2n m≡α mod n X −m + n mη = −Θ(−nz)Θ + (m − n) 2 2n m≡α mod n Θ((−m + n)z).

128 If m0 = n − m, then 0 0 X m (n − m )η 0 0 = −Θ(−nz) Θ + (−m ) Θ(m z) 2 2n m0≡−α mod n

= −Θ(−nz)θ−α(z).

C.2.2 Corollary. The equations for the Sklyanin algebra Qn(E, τ) for n odd can be written in the simpler form X 1 ∀0 ≤ i < j ≤ n − 1 : vi,j = xj−r xi+r = 0. θr−j+i (τ)θr (τ) r∈Zn

The following lemma is used in Chapter4.

1 η 1+η C.2.3 Lemma. Let τ ∈ { 2 , 2 , 2 } and n odd. Then the following equalities hold:

∀0 ≤ α ≤ n − 1 : θα(τ) = θ−α(τ).

1 Proof. By calculation, for τ = 2 it holds that 1 X m mη m θ = Θ + (m − n) Θ α 2 2 2n 2 m≡α mod n X mη = Θ (m − n) 2n m≡α mod n 1 X m mη m θ = Θ + (m − n) Θ −α 2 2 2n 2 m≡−α mod n X mη = Θ (m + n) 2n m≡α mod n m=l+n X lη = Θ (l − n) . 2n l≡α mod n

η For τ = 2 , one calculates that η X m mη mη θ = Θ + (m − n) Θ α 2 2 2n 2 m≡α mod n 2 X m m η = (−1) Θ 2n m≡α mod n 2 X m m η η = (−1) Θ = θ . 2n −α 2 m≡−α mod n

129 1+η Lastly, for τ = 2 one ﬁnds

1 + η X m mη m(1 + η) θ = Θ + (m − n) Θ α 2 2 2n 2 m≡α mod n X m2η 1 + η = Θ = θ . 2n −α 2 m≡α mod n

130 Bibliography

[1] Tarig Abdelgadir, Shinnosuke Okawa, and Kazushi Ueda, Com- pact moduli of noncommutative projective planes, arXiv preprint arXiv:1411.7770 (2014). [2] Michael Artin, On Azumaya algebras and ﬁnite dimensional representations of rings, Journal of Algebra 11 (1969), no. 4, 532–563. [3] , Geometry of quantum planes, Contemp. Math 124 (1992), 1–15. [4] Michael Artin and William Schelter, Graded algebras of global dimension 3, Advances in Mathematics 66 (1987), no. 2, 171–216. [5] Michael Artin, William Schelter, and John Tate, The centers of 3-dimensional Sklyanin algebras, Barsotti Symposium in Algebraic Geometry: Perspectives in Mathematics, vol. 15, Academic Press, 2014, p. 1. [6] Michael Artin, John Tate, and Michel Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Springer, 1990, pp. 33–85. [7] , Modules over regular algebras of dimension 3, Inventiones mathematicae 106 (1991), no. 1, 335–388. [8] Michael Artin and Michel Van den Bergh, Twisted homogeneous coordinate rings, Journal of Algebra 133 (1990), no. 2, 249–271. [9] Michael Artin and James J. Zhang, Abstract Hilbert schemes, Alge- bras and representation theory 4 (2001), no. 4, 305–394. [10] Maurice Auslander and Oscar Goldman, Maximal orders, Transac- tions of the American Mathematical Society 97 (1960), no. 1, 1–24. [11] Hyman Bass, Cliﬀord algebras and spinor norms over a commutative ring, American Journal of Mathematics 96 (1974), no. 1, 156–206.

131 [12] Pieter Belmans, Kevin De Laet, and Lieven Le Bruyn, The point variety of quantum polynomial rings, Journal of Algebra 463 (2016), 10–22.

[13] Raf Bocklandt, Travis Schedler, and Michael Wemyss, Superpoten- tials and higher order derivations, Journal of pure and applied algebra 214 (2010), no. 9, 1501–1522.

[14] Raf Bocklandt and Stijn Symens, The local structure of graded representations, Communications in Algebra 34 (2006), no. 12, 4401– 4426.

[15] Thomas Cassidy and Michaela Vancliﬀ, Generalizations of graded Cliﬀord algebras and of complete intersections, Journal of the Lon- don Mathematical Society 81 (2010), no. 1, 91–112.

[16] Roy Chisholm, John Stephen, and Alan K. Common, Cliﬀord algebras and their applications in mathematical physics, vol. 183, Springer Science & Business Media, 2012.

[17] Aldo Conca, Emanuela De Negri, and Maria Evelina Rossi, Koszul algebras and regularity, Commutative algebra, Springer, 2013, pp. 285–315.

[18] Kevin De Laet, Character series and quaternionic Sklyanin algebras, arXiv preprint arXiv:1412.7001 (2014).

[19] , Graded Cliﬀord algebras of prime global dimension with an action of Hp, Communications in Algebra 43 (2015), no. 10, 4258– 4282.

[20] , Constructing G-algebras, arXiv preprint arXiv:1605.09265 (2016).

[21] , Quotients of degenerate Sklyanin algebras, Journal of Al- gebra and Its Applications (2016).

[22] Kevin De Laet and Lieven Le Bruyn, The geometry of representations of 3-dimensional Sklyanin algebras, Algebras and Representa- tion Theory 18 (2015), no. 3, 761–776.

[23] Tom Fisher, Pfaﬃan presentations of elliptic normal curves, Trans- actions of the American Mathematical Society 362 (2009), no. 5, 2525.

[24] Edward Formanek, Noncommutative invariant theory, Contemp. Math., Group actions on rings, Amer. Math. Soc., Providence, RI 43 (1985), 87–119.

132 [25] Ralph Fröberg, Koszul algebras - Advances in commutative ring theory (fez, 1997), 337–350, Lecture Notes in Pure and Appl. Math 205 (1999). [26] William Fulton and Joe Harris, Representation theory, vol. 129, Springer Science & Business Media, 1991. [27] Johannes Grassberger and GüntherHörmann, A note on representations of the finite Heisenberg group and sums of greatest common divisors., Discrete Mathematics & Theoretical Computer Science 4 (2001), no. 2, 91–100. [28] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathe- matics, vol. 52, Springer Science & Business Media, 1977. [29] Klaus Hulek, Projective geometry of elliptic curves, Société mathématiquede France, 1986. [30] Naihuan Jing and James J. Zhang, On the trace of graded automorphisms, Journal of Algebra 189 (1997), no. 2, 353–376. [31] George Kempf, Projective coordinate rings of abelian varieties, Alge- braic analysis, geometry, and number theory (Baltimore, MD, 1988), 1989, pp. 225–235. [32] Ulrich Kraehmer, Notes on Koszul algebras. [33] Tsit-Yuen Lam, Introduction to quadratic forms over fields, vol. 67, American Mathematical Soc., 2005. [34] , A first course in noncommutative rings, vol. 131, Springer Science & Business Media, 2013. [35] Lieven Le Bruyn, Sklyanin algebras and their symbols, K-theory 8 (1994), no. 1, 3–17. [36] , Central singularities of quantum spaces, Journal of Algebra 177 (1995), no. 1, 142–153. [37] , Generating graded central simple algebras, Linear algebra and its applications 268 (1998), 323–344. [38] , Noncommutative geometry and Cayley-smooth orders, CRC Press, 2007. [39] Domingo Luna, Slices étales, Mémoiresde la SociétéMathématique de France 33 (1973), 81–105. [40] Yuri I. Manin, Some remarks on Koszul algebras and quantum groups, Annales de l’institut Fourier, vol. 37, 1987, pp. 191–205.

133 [41] John C. McConnell, James Christopher Robson, and Lance W. Small, Noncommutative noetherian rings, vol. 30, American Mathematical Soc., 2001.

[42] Susan Montgomery, Group actions on rings, vol. 43, American Math- ematical Soc., 1985.

[43] Constantin Nastasescu and Freddy Van Oystaeyen, Methods of graded rings, Springer Science & Business Media, 2004.

[44] , Graded ring theory, Elsevier, 2011.

[45] Alexander Vladimirovich Odesskii and Boris Lvovich Feigin, Elliptic Sklyanin algebras, Funkt. Anal. i Priloz 23 (1989), 45–54.

[46] , Constructions of Sklyanin elliptic algebras and quantum R- matrices, Functional Analysis and Its Applications 27 (1993), no. 1, 31–38.

[47] Christopher Phan, The Yoneda algebra of a graded Ore extension, Communications in Algebra 40 (2012), no. 3, 834–844.

[48] Alexander Polishchuk, On the Koszul Property of the Homogeneous Coordinate Ring of a Curve, Journal of Algebra 1 (1995), no. 178, 122–135.

[49] Leonid Positselski and Alexander Vishik, Koszul duality and Galois cohomology, arXiv preprint alg-geom/9507010 (1995).

[50] Claudio Procesi, A formal inverse to the Cayley-Hamilton theorem, Journal of Algebra 107 (1987), no. 1, 63–74.

[51] , Lie Groups: An Approach through Invariants and Repre- sentations, Springer Science & Business Media, 2007.

[52] Daniel Rogalski, An introduction to Noncommutative Projective Ge- ometry, arXiv preprint arXiv:1403.3065 (2014).

[53] Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Graduate Texts in Mathematics, no. 106, Springer New York, 2009.

[54] S. Paul Smith, The 4-dimensional Sklyanin algebra at points of ﬁnite order, preprint (1992).

[55] , The four-dimensional Sklyanin algebras, K-theory 8 (1994), no. 1, 65–80.

[56] , Point modules over Sklyanin algebras, Mathematische Zeitschrift 215 (1994), no. 1, 169–177.

134 [57] , Degenerate 3-dimensional Sklyanin algebras are monomial algebras, Journal of Algebra 358 (2012), 74–86.

[58] S. Paul Smith and J. Toby Staﬀord, Regularity of the four dimensional Sklyanin algebra, Compositio Mathematica 83 (1992), no. 3, 259–289. [59] S. Paul Smith and John Tate, The center of the 3-dimensional and 4-dimensional Sklyanin algebras, K-theory 8 (1994), no. 1, 19–63.

[60] J. Toby Staﬀord, Auslander-regular algebras and maximal orders, Journal of the London Mathematical Society 50 (1994), no. 2, 276– 292. [61] , Noncommutative projective geometry, Algebra- Representation Theory (2001), 415–417.

[62] J. Toby Staﬀord and James J. Zhang, Homological properties of (graded) Noetherian PI rings, Journal of Algebra 168 (1994), no. 3, 988–1026. [63] John Tate and Michel Van den Bergh, Homological properties of Sklyanin algebras, Inventiones mathematicae 124 (1996), no. 1-3, 619–648. [64] Craig A. Tracy, Embedded elliptic curves and the Yang-Baxter equations, Physica D: Nonlinear Phenomena 16 (1985), no. 2, 203–220. [65] Jorge Vitoria, Equivalences for noncommutative projective spaces, arXiv preprint arXiv:1001.4400 (2010). [66] Chelsea Walton, Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings, Journal of Algebra 322 (2009), no. 7, 2508–2527.

[67] , Representation theory of three-dimensional Sklyanin algebras, Nuclear Physics B 860 (2012), no. 1, 167–185. [68] James J. Zhang, Twisted graded algebras and equivalences of graded categories, Proceedings of the London Mathematical Society 3 (1996), no. 2, 281–311.

135