Non-Gaussian Malliavin calculus on real Lie algebras

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra is extended to the affine algebra. A differential calculus and a non-commutative integration by parts are established. As an application we obtain sufficient conditions for the smoothness of Wigner-type laws of non-commutative random variables with gamma or continuous binomial marginals.