by Doty, Stephen R.; Erdmann, Karin; Nakano, Daniel K.

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.

DOI: 10.1023/B:ALGE.0000019454.27331.59
Print publication date: 3/1/2004
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by Zimmermann, Alexander

In an earlier paper, Raphaël Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.

DOI: 10.1023/B:ALGE.0000019388.23407.e2
Print publication date: 3/1/2004
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by Putcha, Mohan S.

We introduce the concept of a Brauer character of a representation of a finite monoid M in characteristic p>0. When p does not divide the order of any subgroup of M, we develop a theory of p-monoid quivers. We apply our results to the full transformation semigroup.

DOI: 10.1023/B:ALGE.0000019387.07748.9b
Print publication date: 3/1/2004
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by Doty, Stephen; Giaquinto, Anthony

We give a presentation of the Schur algebras S

_Q
(2,d) by generators and relations, in fact a presentation which is compatible with Serre’s presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S

_Q
(2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant’s Z-form, and show that it has an integral basis which is a truncated version of Kostant’s basis.

DOI: 10.1023/B:ALGE.0000019386.04383.f9
Print publication date: 3/1/2004
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by Mantese, Francesca; Tonolo, Alberto

Let S be an arbitrary associative ring and
_S

W be a left S-module. Denote by R the ring End
_S

W and by Δ both the contravariant functors Hom
_S
(−,W) and Hom
_R
(−,W). A module M is reflexive if the evaluation map δ
_M
: M→Δ²
M is an isomorphism. Any direct summand of finite direct sums of copies of
_S

W and of R

_R
is reflexive. Increasing in a minimal way the classes of reflexive modules, a “cotilting condition” on finitely generated R-modules naturally arises.

DOI: 10.1023/B:ALGE.0000019385.66745.59
Print publication date: 3/1/2004
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by Gómez-Torrecillas, J.; Lobillo, F. J.

The quantized enveloping C(q)-algebra U

_q
(C) associated to a Cartan matirx C is Auslander-regular and Cohen–Macaulay. This is deduced from a general theorem, which also applies to solvable polynomial algebras. The results are obtained by constructing a new filtration keeping the properties of the associated graded algebra of a given multi-filtered algebra.

DOI: 10.1023/B:ALGE.0000019384.36800.fa
Print publication date: 3/1/2004
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by Yekutieli, Amnon

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic
_K
(A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic
_K
(A) is a locally algebraic group, and its identity component is Out⁰

_K
(A). If B is a derived Morita equivalent algebra then DPic
_K
(A)≅DPic
_K
(B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.

DOI: 10.1023/B:ALGE.0000019383.78214.31
Print publication date: 3/1/2004
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