Karin Erdmann

Radical embeddings and representation dimension

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.

SUPPORT VARIETIES FOR SELFINJECTIVE ALGEBRAS

Support varieties for any finite dimensional algebra over a field were introduced in (20) using graded subalgebras of the Hochschild cohomol- ogy. We mainly study these varieties for selfinjective algebras under appropri- ate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In par- ticular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true.

On Auslander-Reiten components of blocks and self-injective biserial algebras

We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here

On Auslander-Reiten components for group algebras

Abstract Let Λ be the group algebra of a finite group over a field, and let Θ be a component of the stable Auslander-Reiten quiver of Λ. We show that if the block containing the modules in Θ is not of finite or tame representation type then Θ has tree class A ∞ .

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs An (n > 1), D n (n > 4), EG, E 7 , Eg and L n (n ≥ 1) and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift v -1 S of every non-projective simple module S is isomorphic to its third syzygy Ω 3 S.

Representation type of Hecke algebras of type

In this paper we provide a complete classification of the representation type for the blocks for the Hecke algebra of type A, stated in terms of combinatorical data. The computation of the complexity of Young modules is a key component in the proof of this classification result.

On Standardly Stratified Algebras

Abstract Let A be a finite dimensional algebra over an algebraically closed field k. For any fixed partial ordering of an index set,Λ say,labelling the simple A-modules L(i),there are standard modules,denoted by Δ(i),i ∈ Λ. By definition,Δ(i) is the largest quotient of the projective cover of L(i) having composition factors L(j) with j ≤ i. Denote by ℱ(Δ) the category of A-modules which have a filtration whose quotients are isomorphic to standard modules. The algebra A is said to be standardly stratified if all projective A-modules belong to ℱ(Δ). In this paper we define a “stratifying system” and we show that this produces a module Y,whose endomorphism ring A is standardly stratified. In particular,we construct stratifying systems for special biserial self-injective algebras.

Representation type of -Schur algebras

Abstract. We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics.

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.

Twisted bimodules and Hochschild cohomology for self-injective algebras of class A, II

Up to derived equivalence, the representation-finite self-injective algebras of class An are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math.11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.