Cayley Map for Symplectic Groups

Abstract

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.