Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces

摘要

This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g(0), we construct a Lie superalgebra g = g(0) circle plus g(1) containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators T-AB whose action on the enveloping algebra U(g) gives the commutation relations between monomials in U(g(0)) 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(g(0)) = dim(g(1)), we also find a realization of the exterior derivative on U(g(0)). The realizations of these geometric objects yield a bicovariant differential calculus on U(g(0)) as a deformation of the standard calculus on the Euclidean space. Published by AIP Publishing.