Journal
PROCEEDINGS OF THE TWENTY-SECOND INTERNATIONAL CONFERENCE ON GEOMETRY, INTEGRABILITY AND QUANTIZATION
Volume 22, Issue -, Pages 154-164Publisher
INST BIOPHYSICS & BIOMEDICAL ENGINEERING BULGARIAN ACAD SCIENCES
DOI: 10.7546/giq-22-2021-154-164
Keywords
Cayley formula; Cayley map; group composition; Hamiltonian matrices; Lie algebra; Lie group; rotations; symmetric matrices; symplectic matrices; vector parameterization
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Despite their importance, symplectic groups are not as popular as orthogonal groups possibly due to their more complex algebraic structures. However, it has been shown that in some sense symplectic groups can be represented by even dimensional symmetric matrices.
Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.
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