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. Published by AIP Publishing.