On a general notion of a polynomial identity and codimensions

TL;DR

This paper develops a unified framework for polynomial identities and codimensions across various algebraic structures within braided monoidal categories, extending classical concepts to new contexts.

Contribution

It introduces a general notion of polynomial identities and codimensions using braided algebraic theories, applicable to structures like coalgebras, bialgebras, and Hopf algebras.

Findings

Established bases for polynomial identities in several algebraic structures

Calculated codimensions for key cases

Extended classical polynomial identity theory to braided categories

Abstract

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter-Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.