# New Trends in Noncommutative Algebra and Algebraic Geometry (12w5049)

Arriving in Banff, Alberta Sunday, October 28 and departing Friday November 2, 2012

## Organizers

Michael Artin (MIT)

Jason Bell (Simon Fraser University)

Colin Ingalls (University of New Brunswick)

Lance Small (University of California, San Diego)

James Zhang (University of Washington)

## Objectives

This workshop would provide a useful follow-up to the 5-day BIRS workshop "Interactions Between Noncommutative Algebra and Algebraic Geometry (08w5072)'' from October 26-31, 2008; the 5-day Manchester workshop on "Noncommutative algebraic geometry'', the University of Manchester, UK, from August 3-8, 2009; the RIMS workshop on "Noncommutative Algebraic Geometry and Related Topics'', the University of Kyoto, Japan, from August 24-28, 2009; the Oberwolfach workshop on "Interactions between Algebraic Geometry and Noncommutative Algebra'', Oberwolfach, Germany, from May 9-15, 2010, and the recent conference "New Trends in Noncommutative Algebra'' in Seattle, USA, from August 9-13, 2010. The main theme of this workshop is the interplay between noncommutative algebra and algebraic geometry. An important goal is to bring together researchers who use geometric and/or homological methods in different areas of noncommutative algebra (including representation theory of algebras) and algebraic geometers who have an interest in non-commutative phenomena, in order to encourage their interaction and collaboration. Given the fast pace of research in this area, and the growing number of mathematicians working in it, we expect the proposed workshop to be very different from these earlier meetings.

We now describe some of the topics that will be discussed at the workshop.

(a) Combinatorial and Geometric Structure of Quantum Groups

Quantum groups form a class of Hopf algebras which involve a parameter $q$ with the property that when $q$ is set to be equal to $1$, the algebra becomes a classical object. Recently, there has been a lot of work on the study of the prime and primitive spectra of these objects, coming from three different approaches: geometric, algebraic, and combinatorial. Goodearl and Letzter showed that in many cases, quantum groups have a natural torus action which partitions the prime spectrum into a finite number of strata, with each stratum homeomorphic to the prime spectrum of a Laurent polynomial ring. In the case that one is working with a quantized enveloping algebra, one can use semiclassical limits to endow the underlying Lie algebra with a Poisson structure. Using this approach, Brown, Goodearl, and Yakimov showed that for a quantization of the coordinate ring of $mtimes n$ matrices, the number of orbits of symplectic leaves under a natural torus action is finite. Furthermore, the number of orbits was shown to be precisely the number of torus-invariant prime ideals in the quantized ring. Goodearl, Launois, and Lenagan have given a natural bijection between the torus-invariant prime ideals in a quantization of the coordinate ring of $mtimes n$ matrices and so-called totally nonnegative cells in ordinary matrices, which play an important role in algebraic combinatorics. Yakimov has also recently given explicit generating sets for prime ideals in a class of quantizations of enveloping algebras of semisimple Lie algebras. These results give a great deal of insight into the topological structure of the prime and primitive spectra of quantum groups, but there is still much progress to be made towards a full understanding of these objects.

(b) Noncommutative Crepant Resolutions

The research area of noncommutative crepant resolutions gives an important interplay between resolutions of singularities and noncommutative algebras. Noncommutative algebras can be used to build resolutions as moduli spaces and exhibit derived equivalences between them. They are also important in applications of algebraic geometry to string theory. The area began with Van den Bergh's definition and his new approach to Bridgeland's proof of Orlov's conjecture that crepant resolutions of terminal Goreinstein singularities are derived equivalent. Recent results in this area include work of Iyama and Weymss which build resolutions of commutative singularities from noncommutative algebras. Kawamata is also studying components of the derived category of terminal singularities which have fractional Calabi-Yau components. Yasuda has defined a Frobenius morphism on noncommutative blowups which he uses to obtain good properties of singularities.

(c) Artin-Schelter Regular Algebras

Artin-Schelter regular (or regular for short) algebras of dimension three were classified by Artin, Tate, Schelter and Van den Bergh, and the resulting algebras have appeared in many contexts. There have been many recent developments in the problem of classification of regular algebras of dimension four, or quantizations of projective three space. Certain classes of algebras have been classified by Lu, Palmieri, Wu and Zhang by using a new technique involving the A-infinity Koszul Dual. Explicit computations of this fairly abstract structure have yielded interesting new objects and solved certain parts of the classification problem. Double Ore extensions were understood in the work of Zhang and Zhang. Also, ongoing projects of Vancliff and Rogalski are aimed at the study of potentially the largest class of regular algebras of dimension four and their geometric properties. Recenly, Rogalski and Sierra have given a family of four-dimensional graded algebras that are birationally commutative, noetherian, and are not Artin-Schelter regular. These examples are surprising, as it had been conjectured that four-dimensional, birationally commutative graded rings should never be noetherian.

(d) Noncommutative Surfaces

There has been much recent development in the theory of non-commutative surfaces that are finite modules over their centers, by Artin, Chan, de Jong, and Ingalls. Many results from the theory of surfaces have now been shown to have more general versions in this setting. The birational theory of existence and uniqueness of minimal models was extended by Chan and Ingalls, using ideas of Mori's minimal model program. The study of moduli of vector bundles on surfaces has been extended by Artin and de Jong, yielding results such as a generalization of Bogomolov's inequality and de Jong's exponent equals index theorem. Current work on the explicit construction of del Pezzo and ruled models will yield workable interesting examples. This area has applications to higher dimensional algebraic geometry, in particular the study of threefold conic bundles. An important open problem in threefold geometry is Iskovskih's conjecture on the rationality of conic bundles. Noncommutative surfaces are providing new methods and results on the study of this difficult conjecture. In particular, Hacking has exhibited local models for terminal conic bundles whose effective threshold is two. Corti suggested the study of these conic bundles as part of a program to attach Iskovskih's conjecture. On the other hand, Keeler, Rogalski, Sierra and Stafford have defined and studied a family of noncommutative surfaces that are birationally commutative, some of which are na{"i}ve blow-ups of commutative surfaces. These surfaces behave pathologically in some ways; for example, while they are often noetherian, they do not retain the noetherian property when one tensors with certain commutative noetherian rings. Rogalski and Stafford have recently shown that all birationally commutative surfaces that are generated in degree $1$ are either part of this new family of algebras or are twisted homogeneous coordinate rings.

(e) Infinite-Dimensional Division Algebras

The study of infinite-dimensional division algebras is much less developed than the study of their finite-dimensional counterparts. These algebras arise naturally as the quotient division rings of Ore domains like the Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, and many quantum groups. Resco computed transcendence degrees of the maximal subfields of the Weyl division rings and his approach has recently been extended in work of Yekutieli and Zhang. Their work lays the foundations for an appropriately "noncommutative" transcendence degree. Moreover, they raise interesting questions as to when tensor products of division rings with themselves remain noetherian and when extending the base field preserves "noetherianity." Other questions of interest focus on the subfield structure of various division rings and the existence of free subalgebras on two generators; for example, are the maximal subfields of the division ring of quotients of the quantum plane purely transcendental? This is emph{not} the case for the Weyl division rings as a famous example of Dixmier shows. Another interesting question has to do with the minimal number of generators for a division algebra. For finite-dimensional algebras this number has been described by Reichstein, building on earlier work of Procesi. The infinite-dimensional case is still open. Finally, there is the question of whether a division algebra which is neither locally PI nor algebraic over its centre must contain a free algebra on two generators. Such a result would say that noncommutative localization is much more pathological than its commutative counterpart. Recently, Bell and Rogalski have shown that this is the case for quotient division algebras of skew polynomial rings over a field and Bell has shown that this is the case for any quotient division ring of a domain of Gelfand-Kirillov dimension strictly less than $3$ over an uncountable field.

(f) Algebras of Low Gelfand-Kirillov Dimension

In recent years there has been much interest in obtaining a noncommutative analogue of the birational classification of projective surfaces. In the noncommutative setting, Gelfand-Kirillov dimension plays the role of Krull dimension and consequently this project involves the study of graded domains of Gelfand-Kirillov dimension $3$. Artin and Stafford gave a concrete description of graded domains of Gelfand-Kirillov dimension $2$ and Smoktunowicz showed that there do not exist graded domains whose dimension is strictly between two and three. Recently Rogalski and Stafford and Sierra have given a complete classification of birationally commutative graded domains of Gelfand-Kirillov dimension $3$. There remains, however, a large gap before a full birational classification can be realized. To properly complete this picture, it is necessary to understand ungraded domains of Gelfand-Kirillov dimension two. Bell has recently shown that complex domains of Gelfand-Kirillov dimension two that do not satisfy a polynomial identity have only finitely many height one primes of infinite codimension and that a division subalgebra of the ring of quotients is either a field or is large in the sense that the ring of quotients is a finite module over the subalgebra.

(g) Other Possible Topics

Cherednik algebras and the more general symplectic reflection algebras $H_{t,c}=H_{t,c}(W)$ (which include certain deformed preprojective algebras) are deformations of skew group rings $mathbb{O}(V)ast W$ of polynomial rings by a finite group $W$. As such, they can also be regarded as another facet of quantum algebra; certainly they have many of the same basic properties as quantum coordinate rings, and many of the same techniques apply. Although they only first appeared in print in 2002 in a seminal paper of Etingof and Ginzburg, these algebras have already been used to answer conjectures in a wide range of subjects and are related to many others. For example, they have been used by Gordon to answer a combinatorial conjecture of Haiman, by Berest, Etingof and Ginzburg to answer questions about rings of quasi-invariants that arise in integrable systems, and (combining work of Ginzburg, Gordon, and Kaledin) the (non)existence of symplectic (or crepant) resolutions. There are an enormous number of open problems concerned with these algebras; in particular little is understood about their representation theory outside type $A$ and, as one might expect from the range of subjects mentioned above, their interaction with other areas of mathematics is flourishing. To give three out of many such examples, they are closely related to Hilbert schemes and Calogero-Moser spaces, to $q$-Schur algebras and Hecke algebras and, through recent work of Ginzburg, Gordon and Stafford, to Haiman's $n!$ Theorem. One can expect further applications in all these areas.

Derived Categories are becoming a fundamental invariant in algebraic geometry

There are many examples of rational varieties which share their derived category with representations of a noncommutative finite-dimensional algebra. This is true for many toric varieties. Also, semiorthogonal decompositions of derived categories of varieties may have components which are naturally equivalent to the triangulated category of modules over a noncommutative variety. This is a natural way to study orthogonal components that arise, for example, as the left perpendicular of a Mori fiber space contraction. Hence, once one studies the structure of derived categories of commutative varieties, noncommutative structures appear naturally. Bridgeland and Roquier have recently done pioneering work with derived categories and have defined subtle invariants of triangulated categories. Bridgeland's stability manifolds have rich structure and yields connections with mirror symmetry. Roquier's dimension is not easy to compute and is related to the difficult problem of showing a tilting object generates the derived category. An analysis and understanding of these new invariants will yield invaluable information about triangulated categories.

An exciting new development in the theory of representations of finite-dimensional algebras is the study of cluster categories started by Buan, Marsh, Reineke, Reiten and Todorov. The combinatorics of clusters is shown to be tightly related to tilting objects in a triangulated category which is a Calabi-Yau quotient (orbit category) of the derived category of the path algebra of a quiver. Clusters, developed by Fomin and Zelevinsky, are a rapidly developing area of algebraic combinatorics. Clusters are appearing in many areas and may eventually provide enrichment to many objects classified by ADE Dynkin diagrams. There have been many questions in algebra motivated by the study of cluster categories, and the study of representations of finite dimensional algebras has provided results in cluster combinatorics. This new interaction is currently very active and more results in this area are sure to follow.

We now describe some of the topics that will be discussed at the workshop.

(a) Combinatorial and Geometric Structure of Quantum Groups

Quantum groups form a class of Hopf algebras which involve a parameter $q$ with the property that when $q$ is set to be equal to $1$, the algebra becomes a classical object. Recently, there has been a lot of work on the study of the prime and primitive spectra of these objects, coming from three different approaches: geometric, algebraic, and combinatorial. Goodearl and Letzter showed that in many cases, quantum groups have a natural torus action which partitions the prime spectrum into a finite number of strata, with each stratum homeomorphic to the prime spectrum of a Laurent polynomial ring. In the case that one is working with a quantized enveloping algebra, one can use semiclassical limits to endow the underlying Lie algebra with a Poisson structure. Using this approach, Brown, Goodearl, and Yakimov showed that for a quantization of the coordinate ring of $mtimes n$ matrices, the number of orbits of symplectic leaves under a natural torus action is finite. Furthermore, the number of orbits was shown to be precisely the number of torus-invariant prime ideals in the quantized ring. Goodearl, Launois, and Lenagan have given a natural bijection between the torus-invariant prime ideals in a quantization of the coordinate ring of $mtimes n$ matrices and so-called totally nonnegative cells in ordinary matrices, which play an important role in algebraic combinatorics. Yakimov has also recently given explicit generating sets for prime ideals in a class of quantizations of enveloping algebras of semisimple Lie algebras. These results give a great deal of insight into the topological structure of the prime and primitive spectra of quantum groups, but there is still much progress to be made towards a full understanding of these objects.

(b) Noncommutative Crepant Resolutions

The research area of noncommutative crepant resolutions gives an important interplay between resolutions of singularities and noncommutative algebras. Noncommutative algebras can be used to build resolutions as moduli spaces and exhibit derived equivalences between them. They are also important in applications of algebraic geometry to string theory. The area began with Van den Bergh's definition and his new approach to Bridgeland's proof of Orlov's conjecture that crepant resolutions of terminal Goreinstein singularities are derived equivalent. Recent results in this area include work of Iyama and Weymss which build resolutions of commutative singularities from noncommutative algebras. Kawamata is also studying components of the derived category of terminal singularities which have fractional Calabi-Yau components. Yasuda has defined a Frobenius morphism on noncommutative blowups which he uses to obtain good properties of singularities.

(c) Artin-Schelter Regular Algebras

Artin-Schelter regular (or regular for short) algebras of dimension three were classified by Artin, Tate, Schelter and Van den Bergh, and the resulting algebras have appeared in many contexts. There have been many recent developments in the problem of classification of regular algebras of dimension four, or quantizations of projective three space. Certain classes of algebras have been classified by Lu, Palmieri, Wu and Zhang by using a new technique involving the A-infinity Koszul Dual. Explicit computations of this fairly abstract structure have yielded interesting new objects and solved certain parts of the classification problem. Double Ore extensions were understood in the work of Zhang and Zhang. Also, ongoing projects of Vancliff and Rogalski are aimed at the study of potentially the largest class of regular algebras of dimension four and their geometric properties. Recenly, Rogalski and Sierra have given a family of four-dimensional graded algebras that are birationally commutative, noetherian, and are not Artin-Schelter regular. These examples are surprising, as it had been conjectured that four-dimensional, birationally commutative graded rings should never be noetherian.

(d) Noncommutative Surfaces

There has been much recent development in the theory of non-commutative surfaces that are finite modules over their centers, by Artin, Chan, de Jong, and Ingalls. Many results from the theory of surfaces have now been shown to have more general versions in this setting. The birational theory of existence and uniqueness of minimal models was extended by Chan and Ingalls, using ideas of Mori's minimal model program. The study of moduli of vector bundles on surfaces has been extended by Artin and de Jong, yielding results such as a generalization of Bogomolov's inequality and de Jong's exponent equals index theorem. Current work on the explicit construction of del Pezzo and ruled models will yield workable interesting examples. This area has applications to higher dimensional algebraic geometry, in particular the study of threefold conic bundles. An important open problem in threefold geometry is Iskovskih's conjecture on the rationality of conic bundles. Noncommutative surfaces are providing new methods and results on the study of this difficult conjecture. In particular, Hacking has exhibited local models for terminal conic bundles whose effective threshold is two. Corti suggested the study of these conic bundles as part of a program to attach Iskovskih's conjecture. On the other hand, Keeler, Rogalski, Sierra and Stafford have defined and studied a family of noncommutative surfaces that are birationally commutative, some of which are na{"i}ve blow-ups of commutative surfaces. These surfaces behave pathologically in some ways; for example, while they are often noetherian, they do not retain the noetherian property when one tensors with certain commutative noetherian rings. Rogalski and Stafford have recently shown that all birationally commutative surfaces that are generated in degree $1$ are either part of this new family of algebras or are twisted homogeneous coordinate rings.

(e) Infinite-Dimensional Division Algebras

The study of infinite-dimensional division algebras is much less developed than the study of their finite-dimensional counterparts. These algebras arise naturally as the quotient division rings of Ore domains like the Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, and many quantum groups. Resco computed transcendence degrees of the maximal subfields of the Weyl division rings and his approach has recently been extended in work of Yekutieli and Zhang. Their work lays the foundations for an appropriately "noncommutative" transcendence degree. Moreover, they raise interesting questions as to when tensor products of division rings with themselves remain noetherian and when extending the base field preserves "noetherianity." Other questions of interest focus on the subfield structure of various division rings and the existence of free subalgebras on two generators; for example, are the maximal subfields of the division ring of quotients of the quantum plane purely transcendental? This is emph{not} the case for the Weyl division rings as a famous example of Dixmier shows. Another interesting question has to do with the minimal number of generators for a division algebra. For finite-dimensional algebras this number has been described by Reichstein, building on earlier work of Procesi. The infinite-dimensional case is still open. Finally, there is the question of whether a division algebra which is neither locally PI nor algebraic over its centre must contain a free algebra on two generators. Such a result would say that noncommutative localization is much more pathological than its commutative counterpart. Recently, Bell and Rogalski have shown that this is the case for quotient division algebras of skew polynomial rings over a field and Bell has shown that this is the case for any quotient division ring of a domain of Gelfand-Kirillov dimension strictly less than $3$ over an uncountable field.

(f) Algebras of Low Gelfand-Kirillov Dimension

In recent years there has been much interest in obtaining a noncommutative analogue of the birational classification of projective surfaces. In the noncommutative setting, Gelfand-Kirillov dimension plays the role of Krull dimension and consequently this project involves the study of graded domains of Gelfand-Kirillov dimension $3$. Artin and Stafford gave a concrete description of graded domains of Gelfand-Kirillov dimension $2$ and Smoktunowicz showed that there do not exist graded domains whose dimension is strictly between two and three. Recently Rogalski and Stafford and Sierra have given a complete classification of birationally commutative graded domains of Gelfand-Kirillov dimension $3$. There remains, however, a large gap before a full birational classification can be realized. To properly complete this picture, it is necessary to understand ungraded domains of Gelfand-Kirillov dimension two. Bell has recently shown that complex domains of Gelfand-Kirillov dimension two that do not satisfy a polynomial identity have only finitely many height one primes of infinite codimension and that a division subalgebra of the ring of quotients is either a field or is large in the sense that the ring of quotients is a finite module over the subalgebra.

(g) Other Possible Topics

Cherednik algebras and the more general symplectic reflection algebras $H_{t,c}=H_{t,c}(W)$ (which include certain deformed preprojective algebras) are deformations of skew group rings $mathbb{O}(V)ast W$ of polynomial rings by a finite group $W$. As such, they can also be regarded as another facet of quantum algebra; certainly they have many of the same basic properties as quantum coordinate rings, and many of the same techniques apply. Although they only first appeared in print in 2002 in a seminal paper of Etingof and Ginzburg, these algebras have already been used to answer conjectures in a wide range of subjects and are related to many others. For example, they have been used by Gordon to answer a combinatorial conjecture of Haiman, by Berest, Etingof and Ginzburg to answer questions about rings of quasi-invariants that arise in integrable systems, and (combining work of Ginzburg, Gordon, and Kaledin) the (non)existence of symplectic (or crepant) resolutions. There are an enormous number of open problems concerned with these algebras; in particular little is understood about their representation theory outside type $A$ and, as one might expect from the range of subjects mentioned above, their interaction with other areas of mathematics is flourishing. To give three out of many such examples, they are closely related to Hilbert schemes and Calogero-Moser spaces, to $q$-Schur algebras and Hecke algebras and, through recent work of Ginzburg, Gordon and Stafford, to Haiman's $n!$ Theorem. One can expect further applications in all these areas.

Derived Categories are becoming a fundamental invariant in algebraic geometry

There are many examples of rational varieties which share their derived category with representations of a noncommutative finite-dimensional algebra. This is true for many toric varieties. Also, semiorthogonal decompositions of derived categories of varieties may have components which are naturally equivalent to the triangulated category of modules over a noncommutative variety. This is a natural way to study orthogonal components that arise, for example, as the left perpendicular of a Mori fiber space contraction. Hence, once one studies the structure of derived categories of commutative varieties, noncommutative structures appear naturally. Bridgeland and Roquier have recently done pioneering work with derived categories and have defined subtle invariants of triangulated categories. Bridgeland's stability manifolds have rich structure and yields connections with mirror symmetry. Roquier's dimension is not easy to compute and is related to the difficult problem of showing a tilting object generates the derived category. An analysis and understanding of these new invariants will yield invaluable information about triangulated categories.

An exciting new development in the theory of representations of finite-dimensional algebras is the study of cluster categories started by Buan, Marsh, Reineke, Reiten and Todorov. The combinatorics of clusters is shown to be tightly related to tilting objects in a triangulated category which is a Calabi-Yau quotient (orbit category) of the derived category of the path algebra of a quiver. Clusters, developed by Fomin and Zelevinsky, are a rapidly developing area of algebraic combinatorics. Clusters are appearing in many areas and may eventually provide enrichment to many objects classified by ADE Dynkin diagrams. There have been many questions in algebra motivated by the study of cluster categories, and the study of representations of finite dimensional algebras has provided results in cluster combinatorics. This new interaction is currently very active and more results in this area are sure to follow.