Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

TL;DR

This paper introduces polynomial invariants for finite-dimensional semisimple and cosemisimple Hopf algebras, revealing new insights into their representation categories and distinguishing non-isomorphic categories with identical representation rings.

Contribution

The paper develops new polynomial invariants based on braiding structures, demonstrating their stability and their role as tensor invariants of the representation category.

Findings

Polynomial invariants are stable under base field extension.

Polynomial invariants distinguish non-equivalent representation categories.

Examples of Hopf algebras with isomorphic representation rings but different categories.

Abstract

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability under extension of the base field. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference of the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but representation categories are distinct.