Relationships Between The Symplectic Groups
The relationship between the groups Sp(2n, C), Sp(2n, R) and Sp(n) is most evident at the level of their Lie algebras. It turns out that the first of these Lie algebras is a complexification of the Lie algebras of either of the latter two groups.
Stated slightly differently, the complex Lie algebra sp(2n, C) of the complex Lie group Sp(2n, C) has several different real forms:
- the compact form, sp(n), which is the Lie algebra of Sp(n),
- the algebras, sp(p, n − p), which are the Lie algebras of Sp(p, n − p), the indefinite signature equivalent to the compact form,
- the normal form (or split form), sp(2n, R), which is the Lie algebra of Sp(2n, R).
Read more about this topic: Symplectic Group
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