Massey products and deformations

TL;DR

This paper generalizes Massey products in cohomology to study deformation theory of Lie algebras, providing new tools to understand extendibility conditions for deformations with various local bases.

Contribution

It introduces a construction of generalized Massey products depending on an arbitrary graded commutative, associative algebra, extending classical concepts to broader deformation contexts.

Findings

Generalized Massey products depend on arbitrary graded commutative, associative algebras.

Extendibility conditions for deformations are generalized using these new Massey products.

Classical and matric Massey products are special cases related to specific Lie algebras.

Abstract

The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given, depending on an arbitrary graded commutative, associative algebra. In terms of these products, the above condition of extendibility is generalized to deformations with arbitrary local bases. Dually, a construction of generalized Massey products on the cohomology of a differential graded commutative associative algebra depends on a nilpotent graded Lie algebra. For example, the classical Massey products correspond to the Lie algebra of strictly upper triangular matrices, while the matric Massey products correspond to the Lie algebra of block strictly upper triangular matrices.