Support varieties for transporter category algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let G be a finite group. Over any finite G-poset \({\mathcal{P}}\) we may define a transporter category as the corresponding Grothendieck construction. Transporter categories are generalizations of subgroups of G, and we shall demonstrate the finite generation of their cohomology. We record a generalized Frobenius reciprocity and use it to examine some important quotient categories of transporter categories, customarily called local categories.

New approaches to finite generation of cohomology rings

Varieties of algebras

A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.

A variety of algebras is specified by a collection of symbols of ground operations and a collection of identities. Similarly, one may speak of syntactic specification of other algebraic theories. In this chapter we point out another approach to description of algebraic theories, grounded on categorial ideas. What plays the role of the corresponding syntax is a certain category T, called an algebraic theory. Algebras of this theory are treated as functors from T into the category of sets.

This survey article is an expanded version of a series of lectures given at the conference on Advances in Group Theory and its Applications which was held in Porto Cesareo in June of 2011. We are concerned with representations of finite group schemes, a class of objects that generalizes the more familiar finite groups. In the last 30 years, this discipline has enjoyed considerable attention. One reason is the application of geometric techniques that originate in Quillen’s fundamental work concerning the spectrum of the cohomology ring v[25, 26]. The subsequent developments pertaining to cohomological support varieties and representation-theoretic support spaces have resulted in many interesting applications. Here we will focus on those aspects of the theory that are motivated by the problem of classifying indecomposable modules. Since the determination of the simple modules is often already difficult enough, one can in general not hope to solve this problem in a naive sense. However, the classification problem has resulted in an important subdivision of the category of algebras, which will be our general theme. The algebras we shall be interested in are the so-called cocommutative Hopf algebras, which are natural generalizations of group algebras of finite groups. The module categories of these algebras are richer than those of arbitrary algebras: • They afford tensor products which occasionally allow the transfer of information between various blocks of the algebra. • Their cohomology rings are finitely generated, making geometric methods amenable to application. The purpose of these notes is to illustrate how a combination of these features with methods from the abstract representation theory of algebras and quivers provides insight into classical questions.

After the seminal paper [BPZ] of Belavin, Polyakov and Zamolodchikov, conformal field theory has become by now a large field with many remarkable ramifications to other fields of mathematics and physics. A rigorous mathematical definition of the “chiral part” of a conformal field theory, called a vertex (= chiral) algebra, was proposed by Borcherds [Bo] more than ten years ago and continued in [DL], [FHL], [FLM], [K], [L] and in numerous other works. However, until now a classification of vertex algebras, similar, for example, to the classification of finite-dimensional Lie algebras, seems to be far away. In the present paper we give a solution to the special case of this problem when the chiral algebra is generated by a finite number of quantum fields, closed under the operator product expansion (in the sense that only derivatives of the generating fields may occur). In this situation the adequate tool is the notion of a conformal algebra [K] which, to some extent, is related to a chiral algebra in the same way a Lie algebra is related to its universal enveloping algebra. At the same time, the theory of conformal algebras sheds a new light on the problem of classification of infinite-dimensional Lie algebras. About thirty years ago one of the authors posed (and partially solved) the problem of classification of simple Z-graded Lie algebras of finite Gelfand-Kirillov dimension [K1]. This problem was completely solved by Mathieu [M1]-[M3] in a remarkable tour de force. The point of view of the present paper is that the condition of locality (which is the most basic axiom of quantum field theory) along with a finiteness condition, are more natural conditions, which are also much easier to handle. In this paper we develop a structure theory of finite rank conformal algebras. Applications of this theory are two-fold. On the one hand, the conformal algebra structure is an axiomatic description [K] of the operator product expansion (OPE) of chiral fields in a conformal field theory [BPZ]. Hence the theory of finite conformal algebras provides a classification of finite systems of fields closed under the OPE. On the other hand, the category of finite conformal algebras is (more or less) equivalent to the category of infinite-dimensional Lie algebras spanned by Fourier coefficients of a finite number of pairwise local fields (or rather formal distributions)

On Gorenstein algebras of finite Cohen-Macaulay type: Dimer tree algebras and their skew group algebras

Homotopical uniqueness of classifying spaces

One of the cornerstones of the representation theory of Hopf algebras and finite tensor categories is the theory of support varieties. Balmer introduced tensor triangular geometry for symmetric monoidal triangulated categories, which united various support variety theories coming from disparate areas such as homotopy theory, algebraic geometry, and representation theory. In this thesis a noncommutative version will be introduced and developed. We show that this noncommutative analogue of Balmer's theory can be determined in many concrete situations via the theory of abstract support data, and can be used to classify thick tensor ideals. We prove an analogue of prime ideal contraction, connecting the Balmer spectrum of a stable category of a finite tensor category with the stable category of its Drinfeld center. We classify the Balmer spectra for various examples arising in representation theory, such as Drinfeld doubles of cosemisimple Hopf algebras, the smash coproducts studied by Benson and Witherspoon, and the small quantum Borels. Lastly, we leverage the theory to prove the tensor product property for cohomological support varieties in a family of small quantum Borels, a conjecture of Negron and Pevtsova.

Let Λ be a finite dimensional algebra with an action by a finite group G and A:=Λ*G the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair induced by the skew group algebra extension Λ⊂A induces a poset isomorphism between the poset of G-stable support τ-tilting modules over Λ and that of (modG)-stable support τ-tilting modules over A. We also establish a similar poset isomorphism between posets of appropriate classes of silting complexes over Λ and A. These two results generalize and unify the preceding results by Zhang–Huang, Breaz–Marcus–Modoi and the second and the third authors. Moreover, we give a practical condition under which τ-tilting finiteness and silting discreteness of Λ are inherited by A. As applications we study τ-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support τ-tilting modules and of silting complexes over preprojective algebra Π(𝕃 n ) of type 𝕃 n .

Preprojective algebra structure on skew-group algebras

Skew group algebras in the representation theory of artin algebras

The aim of this paper is two fold. We study first finite groups G of automorphisms of the homogenized Weyl algebra Bn, the skew group algebra Bn ∗ G, the ring of invariants BG n , and the relations of these algebras with the Weyl algebra An, with the skew group algebra An ∗G, and with the ring of invariants An . Of particular interest is the case n=1. On the other hand, we consider the invariant ring C[X]G of the polynomial ring C[X] in n generators, where G is a finite subgroup of Gl(n,C) such that any element in G different from the identity does not have one as an eigenvalue. We study the relations between the category of finitely generated modules over C[X]G and the corresponding category over the skew group algebra C[X]∗G. We obtain a generalization of known results for n=2 and G a finite subgroup of Sl(2,C ). In the last part of the paper we extend the results for the polynomial algebra C[X] to the homogenized Weyl algebra Bn Mathematics Subject Classification: Primary 16S30, 16P40; Secondary 16G70, 18E30