Heisenberg double versus deformed derivatives

TL;DR

This paper demonstrates the mathematical equivalence between the Heisenberg double construction and deformed derivatives approach for noncommutative spaces based on Lie algebras, unifying two previously separate formalisms.

Contribution

It proves that the Heisenberg double and deformed derivatives formalisms are isomorphic, linking different community approaches through a parametrization of orderings.

Findings

Heisenberg double is isomorphic to a twisted smash product algebra.

The isomorphism involves a datum $$ that parametrizes orderings.

The two formalisms are mathematically equivalent for Lie algebra noncommutative spaces.

Abstract

Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphic to that product in a nontrivial way, involving a datum $\phi$ parametrizing the orderings or realizations in other approaches. This way, we show that the two different formalisms, used by different communities, for introducing the noncommutative phase space for the Lie algebra type noncommutative spaces are mathematically equivalent.