On q-Analogues of Bounded Symmetric Domains and Dolbeault Complexes

A very well known result by Harish-Chandra claims that any Hermitiansymmetric space of non-compact type admits a canonical embedding into acomplex vector space V. The image of this embedding is a bounded symmetricdomain in V. This work provides a construction of q-analogues of apolynomial algebra on V and the differential algebra of exterior forms on V.A way of producing a q-analogue of the bounded function algebra in a boundedsymmetric domain is described. All the constructions are illustrated bydetailed calculations in the case of the simplest Hermitian symmetric spaceSU (1,1)/U(1).