Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted $Sp(2 n, F)$ and $Sp(n)$ for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by $\mathrm {USp} (n)$ . Many authors prefer slightly different notations, usually differing by factors of $2$ . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group $Sp(2 n, C)$ is denoted $C n$ , and $Sp(n)$ is the compact real form of $Sp(2 n, C)$ . Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension $n$ .

For finite groups with all characteristic abelian subgroups cyclic, see group of symplectic type.

The from the Lie algebra $sp (2 n, R)$ to the group $Sp(2 n, R)$ is not surjective. However, any element of the group can be represented as the product of two exponentials.^[4] In other words,

exponential map

Hamiltonian mechanics

Metaplectic group

Orthogonal group

Paramodular group

Projective unitary group

Representations of classical Lie groups

Symplectic matrix, Symplectic vector space, Symplectic representation