  The parentage between Weyl pairs, generalized Pauli group and unitary group
is investigated in detail. We start from an abstract definition of the
Heisenberg-Weyl group on the field R and then switch to the discrete
Heisenberg-Weyl group or generalized Pauli group on a finite ring Z_d. The main
characteristics of the latter group, an abstract group of order d**3 noted P_d,
are given (conjugacy classes and irreducible representation classes or
equivalently Lie algebra of dimension d**3 associated with P_d). Leaving the
abstract sector, a set of Weyl pairs in dimension d is derived from a polar
decomposition of SU(2) closely connected to angular momentum theory. Then, a
realization of the generalized Pauli group P_d and the construction of
generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs.
Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra
of the Lie algebra associated with P_d. This leads to a development of the Lie
algebra of U(d) in a basis consisting of d**2 generalized Pauli matrices. In
the case where d is a power of a prime integer, the Lie algebra of SU(d) can be
decomposed into d-1 Cartan subalgebras.
