$\times_R$-Bialgebras associated with iterative $q$-difference rings

TL;DR

This paper demonstrates that iterative q-difference rings and modules can be understood within a broader algebraic framework involving $ imes_R$-bialgebras, unifying existing Picard-Vessiot theories.

Contribution

It extends the Picard-Vessiot theory for iterative q-difference rings by embedding it into the framework of $ imes_R$-bialgebras and module algebras over Hopf algebras.

Findings

Iterative q-difference modules are represented by modules over cocommutative $ imes_R$-bialgebras.

The framework generalizes and unifies previous Picard-Vessiot theories for iterative q-difference rings.

The approach connects iterative q-difference rings with Hopf algebra theory through $ imes_R$-bialgebras.

Abstract

Realizing the possibility suggested by Hardouin [6], we show that her own Picard-Vessiot Theory for iterative $q$-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [2], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative $q$-difference modules over an iterative $q$-difference ring $R$, by modules over a certain cocommutative $\times_R$-bialgebra. Recall that the notion of $\times_R$-bialgebras was defined by Sweedler [17], as a generalization of bialgebras.