  We introduce a class of non-commutative, complex, infinite-dimensional
Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic
functions which are also square integrable with respect to a heat kernel
measure $\mu$ on these groups are studied. In particular, we establish a
unitary equivalence between the square integrable holomorphic functions and a
certain completion of the universal enveloping algebra of the "Lie algebra" of
this class of groups. Using quasi-invariance of the heat kernel measure, we
also construct a skeleton map which characterizes globally defined functions
from the $L^{2}(\nu)$-closure of holomorphic polynomials by their values on the
Cameron-Martin subgroup.
