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 A A and thus finish the program which began in [M. Markl and J.D. Stasheff, Deformation theory via deviations, Journal of Algebra, 70 (1994), 122–155], [S. Shnider and S. Sternberg, The cobar construction and a restricted deformation theory for Drinfeld algebras, Journal of Algebra, 169 (1994), 343–366]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves 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 a 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 (Definition 3.3 below) on the bar resolution of a general Drinfel′d 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. The results of the paper were announced in [12].