Explicit Parameterizations of Ortho-Symplectic Matrices in R 4

Clementina D. Mladenova () and Ivaïlo M. Mladenov
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Clementina D. Mladenova: Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria
Ivaïlo M. Mladenov: Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria

Mathematics, 2024, vol. 12, issue 16, 1-11

Abstract: Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension.

Keywords: Cayley formula; Cayley map; group factorization; Hamiltonian matrices; Lie algebra; Lie group; orthogonal matrices; rotations; symplectic matrices; unitary matrices; vector parameterization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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