Bigraded differential algebra for vertex algebra complexes

Abstract

For an infinite chain bicomplex we show that the orthogonality and grading conditions provide it with the structure of a bigraded differential algebra with respect to a natural multiplication of several elements bicomplex spaces. Corresponding bigraded algebra commutation relations generate a sequence of non-vanishing cohomology invariants associated to vertex algebras. In particular, we apply this result to the bicomplex of grading-restricted vertex algebra cohomology endowed with a multiplication we introduce. We provide examples associated to various choices of vertex algebra bicomplex subspaces. The generators and commutation relations of the bigraded differential algebra form a continual Lie algebra with the root space provided by a grading-restricted vertex algebra.