Internal Hopf algebroid

TL;DR

This paper generalizes the concept of symmetric Hopf algebroids to an internal setting within symmetric monoidal categories, motivated by the study of Heisenberg doubles of Hopf algebras as internal structures.

Contribution

It introduces a new definition of internal symmetric Hopf algebroids applicable in categories like indproVect, expanding the framework for studying noncommutative phase spaces.

Findings

Defines internal symmetric Hopf algebroids in symmetric monoidal categories.

Provides examples involving Heisenberg doubles of Hopf algebras.

Connects the theory to noncommutative geometry and quantum algebra.

Abstract

We introduce a natural generalization of the definition of a symmetric Hopf algebroid, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. Motivation for this is the study of Heisenberg doubles of countably dimensional Hopf algebras $A$ as internal Hopf algebroids over a (noncommutative) base $A$ in the category $\mathrm{indproVect}$ of filtered cofiltered vector spaces introduced by the author. One example of such Heisenberg double is internal Hopf algebroid $U(\mathfrak{g}) \sharp U(\mathfrak{g})^*$ over universal enveloping algebra $U(\mathfrak{g})$ of a finite-dimesional Lie algebra $\mathfrak{g}$ that is a properly internalized version of a completed Hopf algebroid previously studied as a Lie algebra type noncommutative phase space.