Lie algebra type noncommutative phase spaces are Hopf algebroids

TL;DR

This paper demonstrates that noncommutative phase spaces derived from Lie algebra coordinate algebras can be structured as Hopf algebroids, providing a new algebraic framework for these physical models.

Contribution

It introduces a coproduct structure making the entire phase space algebra a Hopf algebroid over a noncommutative base, extending previous models.

Findings

Phase space algebra is a Hopf algebroid.

Provides a coproduct structure for noncommutative phase spaces.

Connects noncommutative geometry with Hopf algebroid theory.

Abstract

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.