  A very well known result by Harish-Chandra claims that any Hermitian
symmetric space of non-compact type admits a canonical embedding into a complex
vector space $V$. The image of this embedding is a bounded symmetric domain in
$V$. This work provides a construction of q-analogues of a polynomial 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 bounded symmetric
domain is described. All the constructions are illustrated by detailed
calculations in the case of the simplest Hermitian symmetric space
$SU(1,1)/U(1)$. The development of these ideas can be found in math.QA/9803110
and math.QA/9809038 .
