nLab symplectic group

See also quaternionic unitary group for the compact form.

Definition

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

$$A^T \Omega A = \Omega \,.$$

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 $Sp(2n,\mathbb{C})$.

Properties

Maximal compact subgroup

The maximal compact subgroup of the symplectic group $Sp(2n, \mathbb{R})$ is the unitary group $U(n)$.

Homotopy groups

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

$$\pi_1(Sp(2n,\mathbb{R})) \simeq \mathbb{Z} \,.$$

The unique connected double cover obtained from this is the metaplectic group extension $Mp(2n) \to Sp(2n, \mathbb{R})$.

Related concepts

• conformal symplectic group

• affine symplectic group

• metaplectic group

• extended affine symplectic group

• orthosymplectic supergroup

• A higher analog of the symplectic group in 2-plectic geometry is the exceptional Lie group G₂ (see there for more details).

• quaternionic unitary group

References

The term “symplectic group” was suggested in

• Hermann Weyl, The Classical Groups: their invariants and representations (1939, p. 165)

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:

• Mamoru Mimura, Hiroshi Toda, Homotopy groups of symplectic groups, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 251-273 (Euclid:1250524819)

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

• O. T. Meara, Symplectic groups, Math. Surveys 16, Amer. Math. Soc. 1978

A generalization, a symplectic group over a noncommutative algebra with involution is studied in

• D. Alessandrini, A. Berenstein, V. Retakh, E. Rogozinnikov, A. Wienhard, Symplectic groups over noncommutative algebras Sel. Math. New Ser. 28, 82 (2022) doi