2008/07/17 by M. Kibler, M. R. Kibler · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Quantum optics and atomic interactions #quant-ph
paper · pdf · doi:10.1088/1751-8113/41/37/375302
published as Journal of Physics A Mathematical and Theoretical 41 (2008) 375302 · Dedicated to the memory of Moshé Flato on the occasion of the tenth anniversary of his death
arxiv created 2008/07/17 · openalex publication_date 2008/08/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 and then switch to the discrete Heisenberg–Weyl group or generalized Pauli group on a finite ring . 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.