For $n \in \mathbb{N}$, the symplectic group $Sp(2n, \mathbb{R})$ is one of the classical Lie groups.
It is the subgroup of the general linear group $GL(2n, \mathbb{R})$ of elements preserving the canonical symplectic form $\Omega$ on the Cartesian space $\mathbb{R}^{2n}$, that is: the group consisting of those matrices $A$ such that
The symplectic group should not be confused with the compact symplectic group $Sp(n)$, which is the maximal compact subgroup of the complex symplectic group $Sp(2n,\mathbb{C})$.
The maximal compact subgroup of the symplectic group $Sp(2n, \mathbb{R})$ is the unitary group $U(n)$.
By the above the homotopy groups of the symplectic group are those of the corresponding unitary group.
In particular rhe first homotopy group of the symplectic group is the integers
The unique connected double cover obtained from this is the metaplectic group extension $Mp(2n) \to Sp(2n, \mathbb{R})$.
A higher analog of the symplectic group in 2-plectic geometry is the exceptional Lie group G2 (see there for more details).
The term “symplectic group” was suggested in
by
The name “complex group” formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word “complex” in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective “symplectic.” Dickson calls the group the “Abelian linear group” in homage to Abel who first studied it.