Drinfel'd algebra deformations, homotopy comodules and the associahedra

Abstract

The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The task is accomplished in three steps. The first step is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involve a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure on the bar resolution of a general Drinfeld algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. As an application we show that the Drinfel'd deformation of the universal enveloping algebra of a simple Lie algebra is not a jump deformation. The results of the paper were announced in the paper "Drinfel'd algebra deformations and the associahedra" (IMRN, Duke Math. Journal, 4(1994), 169-176, appeared also as preprint hep-th/9312196).