Module Structure on Lie Powers and Natural Coalgebra-Split Sub Hopf Algebras of Tensor Algebras

TL;DR

This paper explores the structure of tensor powers and tensor algebras, identifying natural coalgebra summands and decompositions of Lie powers over general linear groups, advancing understanding of algebraic module decompositions.

Contribution

It provides explicit descriptions of natural coalgebra summands of tensor algebras and new decompositions of Lie powers, extending prior structural results.

Findings

Explicit natural coalgebra summands of tensor algebras identified

Decompositions of Lie powers over general linear groups established

Enhanced understanding of module and Hopf algebra structures in tensor contexts

Abstract

In this article, we investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups.