Results 1  10
of
13
On injective modules and support varieties for the small quantum group
 International Mathematics Research Notices
"... ar ..."
Examples of support varieties for Hopf algebras with noncommutative tensor products
"... ar ..."
(Show Context)
SUPPORT VARIETIES AND REPRESENTATION TYPE OF SELFINJECTIVE ALGEBRAS
"... Abstract. We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg [7]: If a finite dimensional selfinjective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg [7]: If a finite dimensional selfinjective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed in [23]. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small halfquantum groups, and Nichols (quantum symmetric) algebras.
Research Program
, 2012
"... I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and groupgraded algebras. My research program involves collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past and ong ..."
Abstract
 Add to MetaCart
(Show Context)
I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and groupgraded algebras. My research program involves collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past and ongoing research projects, which fall loosely into three categories: Hochschild cohomology and deformations. A large part of my research program involves Hochschild cohomology and deformations of algebras. Hochschild cohomology is important in deformation theory, since a deformation of an algebra is infinitesimally a Hochschild 2cocycle, and obstructions to lifting 2cocycles to deformations live in degree 3 cohomology. My work in deformation theory began with [9], a paper I wrote with Căldăraru and Giaquinto. We were inspired by some examples of Vafa and Witten, which are deformations of certain skew group algebras arising from orbifolds. Specifically, these are the skew group algebras S(V) ⋊ G (and twisted versions), where G is a finite group acting by graded automorphisms on the symmetric algebra S(V) of a vector space V (i.e. a polynomial
Cohomology of Hopf Algebras
, 2015
"... Group algebras are Hopf algebras, and their Hopf structure plays crucial roles in representation theory and cohomology of groups. A Hopf algebra is an algebra A (say over a field k) that has a comultiplication ( ∆ : A → A ⊗k A) generalizing the diagonal map on group elements, an augmentation (ε: A → ..."
Abstract
 Add to MetaCart
(Show Context)
Group algebras are Hopf algebras, and their Hopf structure plays crucial roles in representation theory and cohomology of groups. A Hopf algebra is an algebra A (say over a field k) that has a comultiplication ( ∆ : A → A ⊗k A) generalizing the diagonal map on group elements, an augmentation (ε: A → k) generalizing the augmentation on a group algebra, and an antipode (S: A → A) generalizing the inverse map on group elements. Hopf algebras of interest include group algebras, universal enveloping algebras of Lie algebras, restricted enveloping algebras, quantum groups, coordinate rings of groups, and more. The category of modules of a Hopf algebra A is a tensor category with unit object and duals; this extra structure on the category arises from the comultiplication, augmentation, and antipode. Denote the unit object (that is, trivial module) by k. The cohomology of A is H∗(A): = Ext∗A(k, k). The cohomology of A generally has some of the same properties as group cohomology: It is an algebra under a cup product arising from the tensor product of resolutions, equivalently Yoneda composition. It is graded commutative. If M is an Amodule, then Ext∗A(M,M) is an H ∗(A)module by tensoring with M. These are some of the ingredients required for a support variety theory of modules. My research involves the structure of Hopf algebra cohomology, support variety theory, and applications for various types of finite dimensional Hopf algebras. The following conjecture motivates some of my work. Conjecture (Etingof and Ostrik 2004 [3]) If A is a finite dimensional Hopf algebra, then H∗(A) is finitely generated. In fact, Etingof and Ostrik conjectured more generally that the cohomology of a finite tensor category is finitely generated. This conjecture harkens back to finite generation of group cohomology proved by Golod [9], Venkov [15], and Evens [4] over 50 years ago. Friedlander and Parshall [5] in 1986 proved finite generation of cohomology of restricted enveloping algebras. Ginzburg and Kumar [8] in 1993 proved finite generation of cohomology of small quantum groups (in characteristic 0) under some restrictions on the parameters; these restrictions were removed by Bendel, Nakano, Parshall, and
ERRATUM TO “SUPPORT VARIETIES AND REPRESENTATION TYPE OF SMALL QUANTUM GROUPS”
"... ar ..."
(Show Context)