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 κ-deformed space.

Abstract. This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0 we construct a Lie superalgebra g = g0 ⊕ g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g) gives the commutation relations between monomials in U (g0) and one-forms. Realizations of noncommutative coordinates, one-forms and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0) = dim (g1) we also find a realization of the exterior derivative on U (g0). The realizations of these geometric objects yield a bicovariant differential calculus on U (g0) as a deformation of the standard calculus on the Euclidean space.

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