The Weyl realizations of Lie algebras and left-right duality

TL;DR

This paper explores dual realizations of Lie algebra-based non-commutative spaces using Weyl algebra formalism, establishing connections with star-products, shift operators, and Bernoulli number generating functions, exemplified by the eformed space.

Contribution

It introduces a novel framework linking dual realizations of Lie algebras to extension problems and shift operators, providing explicit formulas in Weyl symmetric ordering.

Findings

Derived closed-form expressions for dual realizations using Bernoulli numbers.

Linked dual realizations to extension problems involving shift operators.

Illustrated the theory with the eformed space example.

Abstract

We investigate dual realizations of non--commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra $\g$ we associate a star--product on the symmetric algebra $S(\g)$ and an ordering on the enveloping algebra $U(\g)$. Dual realizations of $\g$ are defined in terms of left--right duality of the star--products on $S(\g)$. It is shown that the dual realizations are related to an extension problem for $\g$ by shift operators whose action on $U(\g)$ describes left and right shift of the generators of $U(\g)$ in a given monomial. Using properties of the extended algebra, in the Weyl symmetric ordering we derive closed form expressions for the dual realizations of $\g$ in terms of two generating functions for the Bernoulli numbers. The theory is illustrated by considering the $\kappa$--deformed space.