Susan J. Sierra
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On Ideal Classes of Three Dimensional Sklyanin Algebras
On ideal classes of three dimensional Sklyanin algebras 4 expos´esau seminaire d'alg`ebre Universit´eJean Monnet, Saint-Etienne, France by Koen De Naeghel University of Hasselt, Belgium May - June 2005 Abstract These notes reflect a series of four lectures that I have given at the university of Jean Monnet in Saint-Etienne, France. The purpose of these talks was to outline the paper [13] in more detail than a regular talk. Of course it was not my intention to prove every theorem in full detail. Rather, the idea was to show the path towards the results. This means I had to make a selection which proofs to discuss - and which not. I hope that the reader agrees in my selection. I also wanted to stress the computational methods on Hilbert series, Euler forms and Grothendieck groups. I am greatful for the pre-doctoral position offered by the European research training network Liegrits. I would like to thank Roland Berger and Nicolas Marconnet for their kind inviation and the joint research we have done in Saint-Etienne. Also, I want to express my sincere thanks to my thesis advisor Michel Van den Bergh for introducing me to the subject and for sharing with me his mathematical ideas. Contents 1 Introduction and motivation 3 1.1 Hilbert schemes on affine planes . 3 1.1.1 The commutative polynomial algebra k[x; y] . 3 1.1.2 The first Weyl algebra . 4 1.2 Hilbert scheme of projective planes . 6 1.2.1 The commutative polynomial algebra k[x; y; z] . -
To the Sklyanin Algebra
transactions of the american mathematical society Volume 341, Number 2, February 1994 REGULARITY OF ALGEBRAS RELATED TO THE SKLYANINALGEBRA J. T. STAFFORD Abstract. This paper continues the research of [SS] by finding further ex- amples of (Artin-Schelter) regular rings of dimension four. Unlike the three- dimensional case studied in [ATV1, 2], these examples show that the four- dimensional regular rings are not uniquely determined by their associated geo- metric data. Indeed, we find a one-parameter family of regular algebras associ- ated to this data. Introduction 0.1. Fix once and for all an algebraically closed field k of characteristic not equal to 2. Throughout this paper, a graded algebra A will mean a (connected) N-graded algebra, generated in degree 1; thus A = 0,>o^i> where A$ = k is central, dim kAi < oo for all /', and A is generated as an algebra by A\. Following [AS], a graded ring A is called regular {of dimension d) if (i) A has finite global homological dimension, gldim A = d ; (ii) A has finite Gelfand-Kirillov dimension, GKdim A < oo ; that is, there exists p £ R such that dim An < np for all « . (iii) A is Gorenstein; that is, ExtA{k, A) = Sdq.k . 0.2. The papers [ATV1, 2] classify all regular graded rings of dimension 3 and, in doing so, introduce some novel geometric techniques into the study of noncommutative ring theory. Those techniques were shown to have an analogue in dimension 4 in [SS] and this paper continues the research of [SS]. To describe these techniques and the questions that motivated this paper, assume that A is either one of the nontrivial examples considered in [ATV1 ] (specifically, assume that A is one of the three-generated, type A examples of [AS]), or that A is a Sklyanin algebra, as considered in [SS] and defined in (0.3) below. -
Representation Theory of Three-Dimensional Sklyanin Algebras
REPRESENTATION THEORY OF THREE-DIMENSIONAL SKLYANIN ALGEBRAS CHELSEA WALTON Abstract. We determine the dimensions of the irreducible representations of the Sklyanin algebras with global dimension 3. This contributes to the study of marginal deformations of the N=4 super Yang-Mills theory in four dimensions in supersymmetric string theory. Namely, the classification of such representations is equivalent to determining the vacua of the aforementioned deformed theories. We also provide the polynomial identity degree for the Sklyanin algebras that are module finite over their center. The Calabi-Yau geometry of these algebras is also discussed. 1. Introduction During the last several years, much attention has been paid to supersymmetric gauge theories, and the mathematical results in the work are prompted by these physical principles. In particular, we provide representation-theoretic, geometric, and algebraic results about so- called three-dimensional Sklyanin algebras (Definition 1.1), all of which are motivated by Leigh-Strassler’s seminal work on supersymmetric quantum field theories [22]. Here, it was demonstrated that marginal deformations of the N = 4 super Yang Mills (SYM) theory in four dimensions yield a class of N =1conformalfieldtheoriesthatareconformallyinvariant. A few years later, Berenstein-Jejjala-Leigh continues this study by considering deformations of the N = 4 SYM theory in an attempt to classify their moduli space of vacua [9]. Although the full classification of vacua was not achieved, the authors introduce a noncommutative geometric framework to study vacua remarkably without the use of C⇤-algebras. The work in this article begins with the observation that, since an N = 4 SYM theory can be described in terms of an N = 1 SYM theory equipped with a superpotential, the problem of classifying vacua reduces to computing matrix-valued solutions to F -flatness. -
The Center of the 3-Dimensional and 4-Dimensional Sklyanin Algebras
K-Theory 8: 19-63, 1994, t9 © 1994 Kluwer Academic Publishers. Printed in the Netherlands. The Center of the 3-Dimensional and 4-Dimensional Sklyanin Algebras S. P. SMITH Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A. e-mail: [email protected] and J. TATE Department of Mathematics, University of Texas, Austin, TX 78712, U.S.A. e-mail: [email protected] (Received: April, 1993) Abstract. Let A = A(E, z) denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curve E and a point z 6 E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension of A. It is known that A is a finite module over its center if and only if z is of finite order. Generators and defining relations for the center Z(A) are given. If S = Proj(Z(A)) and d is the sheaf of d~s-algebras defined by d(Scr)) = A[f- 1]o then the center .~e of d is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result that Sloe Y' ~ p2. However, for the 4-dimensional Sklyanin algebra there is not such a simple result: although Slm: ~e is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus of d. Key words. Sklyanin algebras, center. 1. Introduction Let k be an algebraically closed field, E an elliptic curve over k, and z e E. -
Interactions Between Noncommutative Algebra and Algebraic Geometry
Interactions between noncommutative algebra and algebraic geometry Michael Artin (Massachusetts Institute of Technology), Colin Ingalls (University of New Brunswick), Lance Small (University of California, San Diego), James J. Zhang (University of Washington) October 26 - October 31, 2008 Overview of the Field The root of noncommutativealgebra goes back to 1843 when Hamilton discovered the quaternions. The sub- ject of abstract algebra has been developed since the early twentieth century by Wedderburn, Artin, Brauer, Noether, and later by Amitsur, Jacobson, Kaplansky, Goldie, Herstein and many others. New research top- ics in noncommutative algebra and its interplay with other fields such as Lie theory, geometry and physics has emerged in recent years. The structure of noncommutative algebras has been understood by the use of algebraic, combinatorial, geometric and homological means. Noncommutative algebraic geometry is an interdisciplinary subject that arises from the interaction be- tween noncommutative algebra and algebraic geometry. Originated by Artin, Schelter, Tate and Van den Bergh [6, 9, 10], noncommutative projective geometry has grown from the basic principle that the global techniques of classical projective algebraic geometry furnish powerful methods and intuition for the study of noncommutative graded algebras. Specifically, the category of graded modules modulo those of finite length over a noetherian connected graded algebra form an appropriate analogue for the category of coher- ent sheaves on a projective variety [12]. This technique has been particularly successful in understanding important classes of noncommutative algebras. Over the last fifteen years, noncommutative projective curves have been ”understood” by Artin and Stafford [7, 8] in terms of graded algebras of Gelfand-Kirillov dimension two and by Reiten and Van den Bergh [40] in terms of proper categories of global dimension two. -
Regularity of the Four Dimensional Sklyanin Algebra Compositio Mathematica, Tome 83, No 3 (1992), P
COMPOSITIO MATHEMATICA S. P. SMITH J. T. STAFFORD Regularity of the four dimensional Sklyanin algebra Compositio Mathematica, tome 83, no 3 (1992), p. 259-289 <http://www.numdam.org/item?id=CM_1992__83_3_259_0> © Foundation Compositio Mathematica, 1992, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 83: 259-289,259 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands. Regularity of the four dimensional Sklyanin algebra S. P. SMITH1 and J. T. STAFFORD2 ’Department of Mathematics, University of Washington, Seattle, WA 98195; 2Department of Mathematics, University of Michigan, Ann Arbor, MI 48109 Received 4 September 1990; accepted 2 May 1991 Abstract. The notation of a (non-commutative) regular, graded algebra is introduced in [AS]. The results of that paper, combined with those in [ATV1, 2], give a complete description of the regular graded rings of global dimension three. This paper considers certain algebras that were defined by Sklyanin [Sk 1, 2] in connection with his work on the Quantum Inverse Scattering Method. We prove that these Sklyanin algebras are regular graded algebras of global dimension four, and are Noetherian domains. Moreover, we show that much of the machinery developed in [ATV] to deal with 3-dimensional algebras has an analogue in dimension four. -
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[Math.QA] 2 Oct 2001 Hr H Rces[ Brackets the Where Ue-Ler Tutr Nλ on Structure Super-Algebra Faypisnmnfl.Nml,Teeaecodnts( Coordinates Are There [19]
ITEP-TH-42/01 hep-th/0110032 POLYNOMILAL POISSON ALGEBRAS WITH REGULAR STRUCTURE OF SYMPLECTIC LEAVES ALEXANDRE ODESSKII AND VLADIMIR RUBTSOV Abstract. We study polynomial Poisson algebras with some regularity con- ditions. Linear (Lie-Berezin-Kirillov) structures on dual spaces of semi-simple Lie algebras, quadratic Sklyanin elliptic algebras of [3],[4] as well as polyno- mial algebras recently described by Bondal-Dubrovin-Ugaglia ([7],[11]) belong to this class. We establish some simple determinantal relations between the brackets and Casimirs in this algebras. These relations imply in particular that for Sklyanin elliptic algebras the sum of Casimir degrees coincides with the dimension of the algebra. We are discussing some interesting examples of these algebras and in particular we show that some of them arise naturally in Hamiltonian integrable systems. Among these examples is a new class of two- body integrable systems admitting an elliptic dependence both on coordinates and momenta. 1. Introduction We shall understand under polynomial Poisson structures those ones whose brackets are polynomial in terms of local coordinates on underlying Poisson man- ifold. The typical example of a such structure is the famous Sklyanin algebra. Remind that a Poisson structure on a manifold M(for instance it does not play an important role is it smooth or algebraic) is given by a bivector antisymmetric tensor field π ∈ Λ2(T M) defining on the corresponded algebra of functions on M a structure of (infinite dimensional) Lie algebra by means of the Poisson brackets {f,g} = hπ, df ∧ dgi. arXiv:math/0110032v1 [math.QA] 2 Oct 2001 The Jacobi identity for this brackets is equivalent to an analogue of (classical) Yang-Baxter equation namely to the ”Poisson Master Equation”: [π, π] = 0, where the brackets [, ]:Λp(T M) × Λq(T M) 7→ Λp+q−1(T M) are the only Lie super-algebra structure on Λ.(T M) given by the so-called Schouten brackets. -
An Introduction to Noncommutative Projective Algebraic Geometry
AN INTRODUCTION TO NONCOMMUTATIVE PROJECTIVE ALGEBRAIC GEOMETRY DANIEL ROGALSKI Contents 1. Lecture 1: Review of basic background and the Diamond Lemma 2 2. Lecture 2: Artin-Schelter regular algebras 14 3. Lecture 3: Point modules 24 4. Lecture 4: Noncommutative projective schemes 31 5. Lecture 5: Classification of noncommutative curves and surfaces 40 6. Solutions to exercises 46 References 53 These notes are a significantly expanded version of the author's lectures at the graduate workshop \Noncommutative algebraic geometry" held at the Mathematical Sciences Research Institute in June 2012. The main point of entry to the subject we chose was the idea of an Artin-Schelter regular algebra. The introduction of such algebras by Artin and Schelter motivated many of the later developments in the subject. Regular algebras are sufficiently rigid to admit classification in dimension at most 3, yet this classification is non-trivial and uses many interesting techniques. There are also many open questions about regular algebras, including the classification in dimension 4. Intuitively, regular algebras with quadratic relations can be thought of as the coordinate rings of noncommutative projective spaces; thus, they provide examples of the simplest, most fundamental noncommutative projective varieties. In addition, regular algebras provide some down-to-earth examples of Calabi-Yau algebras. This is a class of algebras defined by Ginzburg more recently, which is related to several of the other lecture courses given at the workshop. Our first lecture reviews some important background and introduces noncommutative Gr¨obner bases. We also include as part of Exercise set 1 a few exercises using the computer algebra system GAP. -
Bridges Between Noncommutative Algebra and Algebraic Geometry: 16W5088
Bridges Between Noncommutative Algebra and Algebraic Geometry: 16w5088 Michael Artin (Massachusetts Institute of Technology), Jason Bell (University of Waterloo), Colin Ingalls (University of New Brunswick), Lance Small (University of California, San Diego), James Zhang (University of Washington) Sept. 11, 2016 – Sept. 16, 2016 1 Overview of the Field Noncommutative algebra is a rich and diverse field that has influences rooted in algebraic and differential geometry, representation theory, algebraic combinatorics, mathematical physics, and other areas. Rapid developments in noncommutative algebra, especially in noncommutative algebraic geometry, influence many other mathematical disciplines. We briefly give an overview of some of the main trends that shape our field at this time, with an emphasis on the areas represented during the workshop. 1. Noncommutative surfaces. Since curves, having one parameter, are close to being commutative, (or finitely generated modules over their centers), as shown in [5, 6], it is natural to study noncommutative surfaces next. In algebraic geometry there is a classification of surfaces with the most crucial results being Castelnouvo’s and Enriques’ which characterize rational and ruled surfaces respectively. Artin’s conjecture predicts that noncommutative surfaces are either rational, ruled or module finite over their centers [2]. The study of noncommutative rational surfaces produced seminal work in this area [7, 8], as well as new techniques and objects such as twisted homogeneous coordinate rings. Recently, there have been remarkable developments in the theory of noncommutative surfaces (and 3-folds) that are finite modules over their centers and these results extend several commutative programs to the noncom- mutative setting. 1 2 2. Birationally commutative surfaces and birationally Sklyanin surfaces In [31, 48, 49, 50] Keeler, Rogalski, Sierra and Stafford have defined and studied a family of noncommutative surfaces that are birationally commutative, some of which are na¨ıve blowups of commutative surfaces. -
The Interplay of Algebra and Geometry in the Setting of Regular Algebras
Commutative Algebra and Noncommutative Algebraic Geometry, I MSRI Publications Volume 67, 2015 The interplay of algebra and geometry in the setting of regular algebras MICHAELA VANCLIFF This article aims to motivate and describe the geometric techniques introduced by M. Artin, J. Tate and M. Van den Bergh in the 1980s at a level accessible to graduate students. Additionally, some advances in the subject since the early 1990s are discussed, including a recent generalization of complete intersection to the noncommutative setting, and the notion of graded skew Clifford algebra and its application to classifying quadratic regular algebras of global dimension at most three. The article concludes by listing some open problems. Introduction Many noncommutative algebraists in the 1980s were aware of the successful marriage of algebra and algebraic geometry in the commutative setting and wished to duplicate that relationship in the noncommutative setting. One such line of study was the search for a subclass of noncommutative algebras that “behave” enough like polynomial rings that a geometric theory could be developed for them. One proposal for such a class of algebras are the regular algebras, introduced in [Artin and Schelter 1987], that were investigated using new geometric techniques in the pivotal papers of M. Artin, J. Tate and M. Van den Bergh [Artin et al. 1990; 1991]. About the same time, advances in quantum mechanics in the 20th century had produced many new noncommutative algebras on which traditional techniques had only yielded limited success, so a need had arisen to find new techniques to study such algebras (see [Reshetikhin et al. 1989; Kapustin et al. -
Shifted Poisson Structures and Moduli Spaces of Complexes
Shifted Poisson structures and moduli spaces of complexes Zheng Hua∗1 and Alexander Polishchuk y2 1Department of Mathematics, the University of Hong Kong, Hong Kong SAR, China 2University of Oregon and National Research University Higher School of Economics Abstract In this paper we study the moduli stack of complexes of vector bun- dles (with chain isomorphisms) over a smooth projective variety X via derived algebraic geometry. We prove that if X is a Calabi-Yau variety of dimension d then this moduli stack has a (1−d)-shifted Poisson structure. In the case d = 1, we construct a natural foliation of the moduli stack by 0-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projective planes studied by Nevins and Stafford, and the Pois- son structures on the moduli spaces of stable triples over an elliptic curves considered by one of us. We also relate the latter Poisson structures to the semi-classical limits of the elliptic Sklyanin algebras studied by Feigin and Odesskii. ∗[email protected] [email protected] 1 1 Introduction The framework of derived algebraic geometry, including the theory of derived Artin stacks (see [9], [32]), has become an important tool in understanding moduli spaces of vector bundles and their generalizations. In this paper we study the moduli spaces of bounded complexes of vector bundles in this context. Our main inspiration comes from the work of Pantev-To¨en-Vaqui´e-Vezzosi [22], where the classical symplectic structure on the moduli spaces of sheaves over K3-surfaces discovered by Mukai is \explained" and generalized using shifted symplectic structures. -
On Algebraic Values of Function Exp (2Ni X+ Log Log Y)
On algebraic values of function exp (2πi x + loglog y) Igor Nikolaev Abstract It is proved that for all but a finite set of the square-free integers d the value of transcendental function exp (2πi x + log log y) is an algebraic number for the algebraic arguments x and y lying in a real quadratic field of discriminant d. Such a value generates the Hilbert class field of the imaginary quadratic field of discriminant d. − Key words and phrases: real multiplication; Sklyanin algebra MSC: 11J81 (transcendence theory); 46L85 (noncommutative topol- ogy) 1 Introduction It is an old problem to determine if given irrational value of a transcendental function is algebraic or transcendental for certain algebraic arguments; the algebraic values are particularly remarkable and worthy of thorough investi- arXiv:1210.4747v9 [math.NT] 26 Oct 2017 gation, see [Hilbert 1902] [4], p. 456. Only few general results are known, see e.g. [Baker 1975] [1]. We shall mention the famous Gelfond-Schneider Theo- rem saying that eβ log α is a transcendental number, whenever α 0, 1 is an algebraic and β an irrational algebraic number. In contrast, Klein’s6∈ { invariant} j(τ) is known to take algebraic values whenever τ H := x+iy C y > 0 is an imaginary quadratic number. ∈ { ∈ | } The aim of our note is a result on the algebraic values of transcendental function (x, y) := e2πi x + log log y <x< , 1 <y< (1) J { |−∞ ∞ ∞} 1 for the arguments x and y in a real quadratic field; the function (x, y) can be viewed as an analog of Klein’s invariant j(τ), hence the notation.J Namely, let k = Q(√d) be a real quadratic field and Rf = Z+fOk be an order of conductor f 1 in the field k; let h = Cl (R ) be the class number of R and denote by ≥ | f | f Z + Zθ 1 i h the set of pairwise non-isomorphic pseudo-lattices in k { i | ≤ ≤ } having the same endomorphism ring Rf, see [Manin 2004] [5], Lemma 1.1.1.