Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications

TL;DR

This paper introduces a generalized Heisenberg algebra for $rak{gl}(n)$ realizations, constructs star products and twists, and explores dualities, with applications to noncommutative space theories.

Contribution

It provides new algebraic frameworks and explicit realizations for $rak{gl}(n)$, including star products, coproducts, and twists, advancing the mathematical tools for noncommutative geometry.

Findings

Constructed linear realizations of $rak{gl}(n)$

Developed star product and twist formulations

Presented dual realizations and applications to noncommutative spaces

Abstract

We introduce the generalized Heisenberg algebra appropriate for realizations of the $\mathfrak{gl}(n)$ algebra. Linear realizations of the $\mathfrak{gl}(n)$ algebra are presented and the corresponding star product, coproduct of momenta and twist are constructed. The dual realization and dual $\mathfrak{gl}(n)$ algebra are considered. Finally, we present a general realization of the $\mathfrak{gl}(n)$ algebra, the corresponding coproduct of momenta and two classes of twists. These results can be applied to physical theories on noncommutative spaces of the $\mathfrak{gl}(n)$ type.