Square-free algebras and their automorphism groups.

A finite-dimensional algebra A over a field K is square-free in case for every pair e,f of primitive idempotents in A, dim(K)(eAf) less than or equal to 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 congruent to KxiS over a square-free semigroup S twisted by some xi is an element of Z(2)(S, K*), a two-dimensional cocycle of S with coefficients in the group of units K* of K. We prove that for each such A congruent to K xi S, its outer automorphism group Out A is the middle term of a short exact sequence 1 --> H-1(S, K*) --> Out A --> Stab xi(Aut(0) S) --> 1, where H-1(S, K*) is the first cohomology group of S with coeffcients in K*, Aut(0) S is the group of ''normal'' automorphisms of the semigroup S, and Stab xi (Aut(0) S) is the stabilizer in Aut(0) S of xi under the action of Aut(0) S on H-2(S, K*). Finally, if xi = 1, so that A congruent to KS is untwisted, then the sequence splits.