The Weyl realizations of Lie algebras and left-right duality

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.