$L_\infty$-structure on Bardzell's complex for monomial algebras

REDONDO, MARÍA JULIA; ROSSI BERTONE, FIORELA

JOURNAL OF PURE AND APPLIED ALGEBRA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2021

0022-4049

Let $A$ be a monomial associative finite dimensional algebra over a field $K$ of characteristic zero. It is well known that the Hochschild cohomology of $A$ can be computed using Bardzell´s complex $B(A)$. The aim of this article is to describe an { explicit} $L_infty$-structure on $B(A)$ that induces a weak equivalence of $L_infty$-algebras between $B(A)$ and the Hochschild complex $C(A)$ of $A$. This allows us to describe the Maurer-Cartan equation in terms of elements of degree $2$ in $B(A)$. Finally, we make concrete computations when $A$ is a truncated algebra, and we prove that Bardzell´s complex for radical square zero algebras is in fact a dg-Lie algebra.