A $q$-Identity Related to a Comodule

TL;DR

This paper establishes a connection between comodule algebra structures over Taft Hopf algebras and $q$-binomial identities, providing a combinatorial proof of these identities when $q$ is a primitive $n$th root of 1.

Contribution

It reveals that certain algebraic comodule conditions are equivalent to $q$-binomial identities and offers a direct combinatorial proof for these identities.

Findings

Algebra being a comodule algebra over Taft Hopf algebra relates to $q$-binomial identities.

Provides a combinatorial proof of the identities for primitive $n$th roots of unity.

Establishes an equivalence between algebraic structures and combinatorial identities.

Abstract

In this paper we show that a certain algebra being a comodule algebra over the Taft Hopf algebra of dimension $n^2$ is equivalent to a set of identities related to the $q$-binomial coefficient, when $q$ is a primitive $n^{th}$ root of 1. We then give a direct combinatorial proof of these identities.