Formal deformation theory of bialgebras

Abstract

The modern algebraic deformation theory was introduced for associative algebras by Gerstenhaber in the 1960s. The constraints for the deformation of an associative algebra can be naturally interpreted in terms of the Hochschild cochain complex. In particular, the linear term of the deformation is a cocycle in the Hochschild complex. A structure of differential graded Lie algebra on the Hochschild cochain complex was also introduced by Gerstenhaber. The solutions to the Maurer-Cartan equation of this differential graded Lie algebra determine the deformations of an associative algebra. In this sense, the deformations of associative algebras are governed by the Hochschild cochain complex and the structure of differential graded Lie algebra on it. In the 1980s, the Gerstenhaber-Schack complex was introduced by Gerstenhaber and Schack to study the deformations of bialgebras. They conjectured that there is a structure of $L_\infty$-algebra on the Gerstenhaber-Schack complex, which is related to the deformations of bialgebras. The conjecture is partially answered by Markl's work. He gives a construction of $L_\infty$-algebras on the deformation complexes of algebraic structures. When one applies the construction to bialgebras, the deformation complex is a truncation of the Gerstenhaber-Schack complex. The main purpose of this thesis is to review the construction of Markl's and then apply it to the case of associative algebras and the case of bialgebras. We study the property of the $L_\infty$-algebras obtained by Markl's construction in these two cases. The construction is based on the language of PROP. We also give an elementary introduction to the PROP theory to support the construction.