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**Variations on a theme of Heisenberg, Pauli and Weyl.**
*(English)*
Zbl 1147.81014

Summary: The parentage between Weyl pairs, the generalized Pauli group and the unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field \({\mathbb R} \) and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring \({\mathbb Z}_d \). 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 \(\text{SU}(d)\) can be decomposed into \(d - 1\) Cartan subalgebras.