Casimir Chaos in a Boson Fock Space

We investigate Casimir processes corresponding to central elements of the universal enveloping algebra associated with a family of representations of the unitary Lie group U(n) which arises naturally in n-dimensional quantum stochastic calculus. In particular for a family of Casimir elements Gnm, whose eigenvalues in the irreducible representation with highest weight (l1 − n + 1, l2 − n + 2, …, ln) are linear combinations of elementary symmetric polynomials in (l1, …, ln), we obtain chaotic decompositions using iterated stochastic integrals. The terms of the chaotic decomposition are themselves Casimir processes corresponding to Casimir elements Znm. We find explicit formulae for the eigenvalues of Znm in an irreducible represenation. Finally, by passing to a product of Fock spaces we realise, in the context of quantum stochastic calculus, the random walk on the dual of U(n) first constructed by Biâne.