Hopf structures in the representation theory of direct products

TL;DR

This paper explores how direct product constructions in group theory can be used to build and analyze Hopf algebras in representation theory, revealing structural properties and specific cases like noncommutative symmetric functions.

Contribution

It introduces a general framework for constructing Hopf algebras from towers of groups via direct products, including structural insights and formulas for key operations.

Findings

Character groups of the constructed Hopf algebras are characterized.

A formula for the antipode in these Hopf algebras is provided.

The noncommutative symmetric functions NSym emerge as a special case.

Abstract

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structure via the representation theory of families of algebras. This paper examines a fundamental construction in group theory, the direct product, and how it can be used to build representation theoretic Hopf algebras out of towers of groups. A key special case gives us the noncommutative symmetric functions NSym, but there are many things that we can say for the general Hopf algebras, including the structure of their character groups and a formula for the antipode.