See also quaternionic unitary group for the compact form.
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
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 real symplectic group $Sp(2n,\mathbb{R})$ 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 the 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.
On homotopy groups:
O. Meara’s book studies symplectic group of a finite dimensional symplectic or even alternating space (space with an alternating form, not necessarily nondegenerate) over an arbitrary field
A generalization, a symplectic group over a noncommutative algebra with involution is studied in
Last revised on September 21, 2024 at 13:41:52. See the history of this page for a list of all contributions to it.