HOPF ALGEBRA OF MULTIPLE POLYLOGARITHMS AND ASSOCIATED MIXED HODGE STRUCTURES

Abstract

This thesis constructs a variation of mixed Hodge structures based on multiple polylogarithms, and attempts to build candidate complexes for computing motivic cohomology. Firstly, we consider Hopf algebras with generators representing multiple polylogarithms. By quotienting products and functional relations, we get Lie coalgebras whose Chevalley-Eilenberg complexes are conjectured to compute rational and integral motivic cohomologies. We also associate one-forms to multiple polylogarithms, which exhibit combinatorial properties that are easy to work with. Next, we introduce a variation matrix which describes a variation of mixed Hodge structures encoded by multiple polylogarithms. Its corresponding connection form is composed of the one-forms associated to the multiple polylogarithms. Lastly, to ensure the well-definedness of the Hodge structures, we must compute the monodromies of multiple polylogarithms, for which we provide an explicit formula, extending the previous work done for multiple logarithms, a subfamily of multiple polylogarithms.