Comparison morphisms between two projective resolutions of monomial algebras

Abstract

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra A : the bar resolution Bar A and Bardzell’s resolution Ap A ; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology HH ∗ ( A ) and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for A = k Q/I a monomial algebra such that dim k e i Ae j = 1 whenever there exists an arrow α : i → j ∈ Q 1 , we describe the Lie action of the Lie algebra HH 1 ( A ) on HH ∗ ( A ).