A generalization of the Jordan-Schwinger map: classical version and its q--deformation

TL;DR

This paper extends the classical Jordan--Schwinger map to all three-dimensional Lie algebras, providing explicit phase space realizations and their q-deformations, including for quantum groups like ${ m SL}_q(2, )$ and ${ m U}_q(n)$.

Contribution

It generalizes the Jordan--Schwinger map to all three-dimensional Lie algebras and their q-deformations, offering explicit phase space constructions and Poisson bracket realizations.

Findings

Explicit phase space realizations for all three-dimensional Lie algebras.

Extension of the Jordan--Schwinger map to q-deformed algebras.

Discussion of ${ m U}_q(n)$ algebra within this framework.

Abstract

For all three--dimensional Lie algebras the construction of generators in terms of functions on 4-dimensional real phase space is given with a realization of the Lie product in terms of Poisson brackets. This is the classical Jordan--Schwinger map which is also given for the deformed algebras ${\cal {SL}}_{q}(2,\R)$, ${\cal E}_{q} (2)$ and ${\cal H}_q(1)$. The ${\cal U}_{q}(n)$ algebra is discussed in the same context.