BINOMIAL ALGEBRAS

Abstract

ABSTRACT Let K be a field. A split basic finite-dimensional K-algebra with left quiver Γ is binomial if it can be represented as the path algebra modulo relations of the form , where and p and q are paths in Γ. We here characterize all binomial algebras A as twisted semigroup algebras , where S is an algebra semigroup, and where is a two-dimensional cocycle of S with coefficients in the multiplicative group of units of K. Subject to certain conditions on an algebra semigroup S, we classify the twisted semigroup algebras of S up to isomorphism. Finally, subject to the same conditions on S, we show that for each binomial algebra there exists a short exact sequence where is the first cohomology group of S with coefficients in , is the group of outer automorphisms of A, is the group of semigroup automorphisms of S, and is the stabilizer in of under the natural action of on the second cohomology group, , of S. Moreover, if (so that is untwisted), then the above sequence splits, yielding .