nLab symplectic group


See also quaternionic unitary group.


Group Theory

Symplectic geometry



For nn \in \mathbb{N}, the symplectic group Sp(2n,)Sp(2n, \mathbb{R}) is one of the classical Lie groups.

It is the subgroup of the general linear group GL(2n,)GL(2n, \mathbb{R}) of elements preserving the canonical symplectic form Ω\Omega on the Cartesian space 2n\mathbb{R}^{2n}, that is: the group consisting of those matrices AA such that

A TΩA=Ω. A^T \Omega A = \Omega \,.

The symplectic group should not be confused with the compact symplectic group Sp(n)Sp(n), which is the maximal compact subgroup of the complex symplectic group Sp(2n,)Sp(2n,\mathbb{C}).


Maximal compact subgroup

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

π 1(Sp(2n,)). \pi_1(Sp(2n,\mathbb{R})) \simeq \mathbb{Z} \,.

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


The term “symplectic group” was suggested in

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


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

Last revised on September 22, 2022 at 08:29:18. See the history of this page for a list of all contributions to it.