Rigid monomial algebras

Abstract

In this paper we obtain a classification of rigid monomial algebras with directed quiver. By a monomial algebra will be meant a finite dimensional algebra over a field k of the form kQ/(Z) where kQ is the quiver algebra for a finite quiver Q and ( Z ) is a two-sided ideal generated by a minimal set of paths Z of length at least 2 (these algebras have also been called "zero relations algebras"). The problem of classifying rigid finite dimensional algebras was raised by Gerstenhaber in [11 ]. In that paper he shows that if the Hochschild cohomology in degree two H2(A, ,4) of a k-algebra A vanishes, then A is rigid. The converse of this statement is false, at least if the field is of positive characteristic (see [12]). However for monomial algebras with directed quiver the converse of Gerstenhaber's result holds as a consequence of our classification. In [10, p. 141] Gabriel wrote that "it should be one of the main tasks of associative algebra to determine for every n the number of irreducible components of the variety of algebras of dimension n". Also Kraft [17, p. 140] points out that the generic structures should be understood. Moreover the classification of rigid algebras can give lower bounds for the number of irreducible components as for example Mazzola did in [18]. Some of the techniques developed to study the representation theory of finite dimensional algebras have been proven to be useful to treat rigidity problems. In particular quivers with relations are at the origin of the definition of combinatorial invariants closely related to the Hochschild cohomology, homology and cyclic homology (see [14, 9, 4, 19, 5-7]). Moreover Happel and Schaps have highlighted recently the connections between deformation and tilting theories in [15]. In this paper we compute the dimension of H2(A,A) for A=kQ/(Z) a monomial algebra using a new quiver ~ associated to Q and Z. Some of the connected components of this "parallel quiver" ~ are relevant and we call them