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

MARIA JULIA REDONDO

Paris (virtual)

Seminario; Paris algebra seminar; 2020

Institut de Math ́ematiques de Jussieu - Paris Rive Gauche, Sorbonne Universite Universite de Paris, Universite de Reims Champagne-Ardennes

When dealing with a monomial algebra A, Bardzell?s complex B(A) has shown to be more efficient for computing Hochschild cohomology groups of $A$ than Hochschild complex C(A). Since C(A)[1] is a dg-Lie algebra, it is natural to ask if the comparison morphisms between these complexes allow us to transfer the dg-Lie structure to B(A)[1].  This is true for radical square zero algebras, but it is not true in general for monomial algebras.In this talk I will describe an explicit L_\infty-structure on B(A) that induces a weak equivalence of L_\infty-algebras between B(A) and  C(A). This allows us to describe the Maurer-Cartan equation in terms of elements of degree 2 in B(A) and make concrete computations when A is a truncated algebra.