Square-free algebras and their automorphism groups
Titel:
Square-free algebras and their automorphism groups
Auteur:
Anderson, Frank W. D'Ambrosia, Barbara K.
Verschenen in:
Communications in algebra
Paginering:
Jaargang 24 (1996) nr. 10 pagina's 3163-3191
Jaar:
1996
Inhoud:
A finite-dimensional algebra A over a field K is square-free in case for every pair e,f of primitive idempotents in AdimK(eAf) ≤ 1. For example, every incidence algebra of a finite pre-ordered set over a field is square-free. The automorphism groups of the latter have been studied by Stanley, Scharlau, Baclawski, and more recently by Coelho. In this paper we characterize all finite-dimensional square-free K-algebras A as certain semigroup algebras A ≅ KξS over a square-free semigroup S twisted by some ξ ∈Z2 (SK*), a two-dimensional cocycle of S with coefficients in the group of units A* of K. We prove that for each such A ≅ KξS, its outer automorphism group Out A is the middle term of a short exact sequence [image omitted] where H1 (SK*) is the first cohomology group of S with coefficients in K*Aut0 S is the group of “normal” automorphisms of the semigroup S, and Stabξ(Auto S) is the stabilizer in Auto S of ξ under the action of Auto S on H2 (SK* ). Finally, if ξ ≅ 1, so that A ss KS is untwisted, then the sequence splits.
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