2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/170240We 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 $μ$ 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 $L^{2}(ν)$-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup.ProbabilityDifferential Geometry35K05, 43A15, 58G32text