Nontriviality results for the characteristic algebra of a DGA

TL;DR

This paper proves that for a certain class of noncommutative DGAs with an action filtration, the homology injects into the characteristic algebra, using Cohn's weak division algorithm.

Contribution

It establishes a monomorphism from homology to the characteristic algebra for filtered semifree noncommutative DGAs, a new structural insight.

Findings

Homology injects into the characteristic algebra.

The result applies to semifree noncommutative DGAs with action filtration.

Uses Cohn's weak division algorithm as a key tool.

Abstract

Assume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.