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 noncom- mutative 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 noncommuta- tive 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 theo- rems 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 gen- erated 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 2 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 sub- spaces, with the generic point corresponding to the full graph on n points. v 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, 1 ................. 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 ix 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 xi 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 associ¨eren. Hierbij worden er vaak extra voor- waarden op de beoogde algebra's gelegd, doordat de meeste algebra¨ısche technieken niet op elke niet-commutatieve algebra van toepassing zijn. Deze voorwaarden kunnen vari¨erenvan 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 alge- bra'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 mas- terthesis: 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 infor- matie over 3-dimensionale Sklyanin algebra's te bekomen? Of in een meer algemene context, kan de Hn-actie op de Sklyanin al- gebra'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 2 N gegeven en bestuderen we een aantal voorbeelden. Ook een aantal constructies om G-deformaties te vinden worden behandeld, namelijk twisten en Ore extensies.