Symplectic group

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One of the classical groups, defined as the group of automorphisms of a skew-symmetric bilinear form on a left -module , where is a commutative ring (cf. Classical group). In the case when and the matrix of with respect to the canonical basis of has the form

where is the identity matrix of order , the corresponding symplectic group is called the symplectic group of variables over the ring and is denoted by or . The matrix of any automorphism in with respect to is called a symplectic matrix.

Let be a field and a non-degenerate skew-symmetric bilinear form on an -dimensional vector space over . If is even, then the symplectic group associated with is isomorphic to and is generated by all linear transformations of of the form , given by

where , . Linear transformations of the form are called symplectic transvections, or translations in the direction of the line . The centre of consists of the matrices and if , and if . The quotient group is called the projective symplectic group and is denoted by . All projective symplectic groups are simple, except

(here denotes the field of elements) and these are isomorphic to the symmetric groups , (cf. Symmetric group) and the alternating group , respectively. The order of is

The symplectic group coincides with the special linear group . If , is isomorphic to the quotient group of by its centre, where is the commutator subgroup of (index 2 in) the orthogonal group associated with a symmetric bilinear form in five variables.

Except when and , every automorphism of can be written as

where is an automorphism of the field , and is a linear transformation of the space , represented on the basis by a matrix of the form

( is a non-zero element of ).

coincides with the group of -points of the linear algebraic group defined by the equation . This algebraic group, also called a symplectic group, is a simple simply-connected linear algebraic group of type of dimension .

In the case when or , is a connected simple complex (respectively, real) Lie group. is one of the real forms of the complex symplectic group . The other real forms of this group are also sometimes called symplectic groups. These are the subgroups of , , , consisting of those elements of that preserve the Hermitian form

where for and , and otherwise. The group is a compact real form of the complex symplectic group . The symplectic group is isomorphic to the group of all linear transformations of the right vector space of dimension over the division ring of quaternions that preserve the quaternionic Hermitian form of index , that is, the form


and the bar denotes conjugation of quaternions.


[1] E. Artin, "Geometric algebra" , Interscience (1957)
[2] N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Addison-Wesley (1973) (Translated from French)
[3] J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955)
[4] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)
[5] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)


is also simply connected. But has the homotopy type of , so that . Here is the circle and is the special unitary group. The unitary symplectic group is the intersection (in ) of the unitary group and . Topologically, .

In Hamiltonian mechanics (cf. Hamilton equations) the phase space is a symplectic manifold, a manifold provided with a symplectic form (a closed differential form of degree which is non-degenerate at each point). If , the cotangent bundle of a configuration space , with local coordinates , then the symplectic form is called canonical. The flow of a Hamiltonian system leaves the symplectic form invariant. As a consequence, its tangent mapping at a fixed point belongs to the symplectic group of the tangent space.

Cf. Symplectic homogeneous space; Symplectic structure.

How to Cite This Entry:
Symplectic group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article