On the graded quotients of the $\mathrm{SL}(m,\C)$-representation algebras of groups

TL;DR

This paper introduces new filtrations of $ ext{SL}(m, ext{C})$-representation algebras of groups, generalizing previous results for $ ext{SL}(2, ext{C})$, and connects them to Johnson homomorphisms and known cocycles.

Contribution

It develops analogs of Johnson homomorphisms for $ ext{SL}(m, ext{C})$-representation algebras and extends these to automorphism groups as crossed homomorphisms, linking to Kawazumi's and Morita's cocycles.

Findings

Extended Johnson homomorphisms to automorphism groups as crossed homomorphisms

Connected the homomorphisms to Kawazumi's and Morita's cocycles

Generalized previous $ ext{SL}(2, ext{C})$ results to $ ext{SL}(m, ext{C})$

Abstract

In this paper, we consider certain descending filtrations of the $\mathrm{SL}(m,\C)$-representation algebras of free groups and free abelian groups. By using it, we introduce analogs of the Johnson homomorphisms of the automorphism groups of free groups. We show that the first homomorphisms are extended to the automorphism groups of free groups as crossed homomorphisms. Furthermore we show that the extended crossed homomorphisms induce Kawazumi's cocycles and Morita's cocycles. This works are generalization of our previous results for the $\mathrm{SL}(2,\C)$-representation algebras.