Integral theory for left Hopf left bialgebroids

TL;DR

This paper extends integral theory to left Hopf left bialgebroids, which lack an antipode but have related elements, and applies these results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.

Contribution

It generalizes integral theory from Hopf algebroids to left Hopf left bialgebroids, broadening the scope of algebraic structures studied.

Findings

Extended integral theory to left Hopf left bialgebroids.

Applied the theory to restricted enveloping algebras.

Connected the results to recent developments in algebraic structures.

Abstract

We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G. B\"ohm who studied integral theory for Hopf algebroids. We make use of recent results about left Hopf left bialgebroids. We apply our results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.