Massey Products and Deformations

It is common knowledge that the construction of one-parameter deformations of various algebraic structures, like associative algebras or Lie algebras, involves certain conditions on cohomology classes, and that these conditions are usually expressed in terms of Massey products, or rather Massey powers. The cohomology classes considered are those of certain differential graded Lie algebras (DGLAs). It is also known that the Massey products arising in deformation theory are not precisely the products considered in the general theory of DGLAs (see [R2]). Actually, different natural problems of deformation theory give rise to different kinds of Massey products. The definitions of these Massey products involve certain equations whose coefficients turn out to be, quite unexpectedly, the structure constants of a graded commutative associative algebra. Thus to define Massey products in the cohomology of a differential graded Lie algebra one should begin by choosing a graded commutative associative algebra. It is interesting that, dually, different kinds of Massey products in the cohomology of a differential commutative associative algebra correspond in a similar way to Lie algebras. In particular, the classical Massey products correspond to the Lie algebra of strictly upper triangular matrices, while May’s matric Massey products [Ma] correspond to the Lie algebra of block strictly upper triangular matrices. The article is organized as follows. Section 2 contains a list of various Massey-like products which arise in the cohomology of DGLAs. Most of them are related to deformations of Lie algebras. In section 3 we give a general construction of Massey products in the cohomology of DGLAs. The main result of this section is Proposition 3.1, which shows the necessity of the associativity of the auxiliary algebra. Section 4 contains an application of the construction of Section 3 to Lie algebra deformations, and in Section 5 we consider Massey products in the cohomology of graded commutative associative differential algebras.