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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.

MSC:

81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R15 Operator algebra methods applied to problems in quantum theory
81S10 Geometry and quantization, symplectic methods
17B81 Applications of Lie (super)algebras to physics, etc.
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