In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by . Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension .
The name "symplectic group" is was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex".
The metaplectic group is a double cover of the symplectic group over R; it has analogues over other local fields, finite fields, and adele rings.
The symplectic group is a classical group defined as the set of linear transformations of a -dimensional vector space over the field which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the symplectic group of an abstract symplectic vector space is denoted . Upon fixing a basis for , the symplectic group becomes the group of symplectic matrices, with entries in , under the operation of matrix multiplication. This group is denoted either or . If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then
where MT is the transpose of M. Often Ω is defined to be
where In is the identity matrix. In this case, can be expressed as those block matrices , where , satisfying the three equations:
Since all symplectic matrices have determinant , the symplectic group is a subgroup of the special linear group . When , the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that . For , there are additional conditions, i.e. is then a proper subgroup of .
Typically, the field is the field of real numbers or complex numbers . In these cases is a real or complex Lie group of real or complex dimension , respectively. These groups are connected but non-compact.
The center of consists of the matrices and as long as the characteristic of the field is not . Since the center of is discrete and its quotient modulo the center is a simple group, is considered a simple Lie group.
The real rank of the corresponding Lie algebra, and hence of the Lie group , is .
The Lie algebra of is the set
equipped with the commutator as its Lie bracket. For the standard skew-symmetric bilinear form , this Lie algebra is the set of all block matrices subject to the conditions
The symplectic group over the field of complex numbers is a non-compact, simply connected, simple Lie group.
is the complexification of the real group . is a real, non-compact, connected, simple Lie group. It has a fundamental group isomorphic to the group of integers under addition. As the real form of a simple Lie group its Lie algebra is a splittable Lie algebra.
Some further properties of :
The exponential map from the Lie algebra to the group is not surjective. However, any element of the group can be represented as the product of two exponentials. In other words,
For all in :
The matrix is positive-definite and diagonal. The set of such s forms a non-compact subgroup of whereas forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition. Further symplectic matrix properties can be found on that Wikipedia page.
As a Lie group, has a manifold structure. The manifold for is diffeomorphic to the Cartesian product of the unitary group with a vector space of dimension .
Infinitesimal generators
The members of the symplectic Lie algebra are the Hamiltonian matrices.
These are matrices, such thatwhere and are symmetric matrices. See classical group for a derivation.
Example of symplectic matrices
For , the group of matrices with determinant , the three symplectic -matrices are:
Sp(2n, R) It turns out that can have a fairly explicit description using generators. If we let denote the symmetric matrices, then is generated by whereare subgroups of pg 173pg 2.