In mathematics, the symplectic group is the group of linear transformations that preserve the geometric structure of phase space, the space of position and momentum variables used in classical mechanics. It is defined as the group of linear changes of coordinates on phase space that preserve the symplectic form. The symplectic groups are usually denoted Sp ( 2 n , F ) {\displaystyle \operatorname {Sp} (2n,\mathbb {F} )} , where n {\displaystyle n} is a positive integer and F {\displaystyle \mathbb {F} } is a field, often the real numbers or complex numbers. They are among the four families of classical groups and play a central role in symplectic geometry, Hamiltonian mechanics, and representation theory. A related but different family is the compact symplectic group, usually denoted Sp ( n ) {\displaystyle \operatorname {Sp} (n)} or U S p ( n ) {\displaystyle \mathrm {USp} (n)} .
Terminology and notation The name "symplectic" was introduced by Hermann Weyl as a replacement for older terminology such as line complex group. It was intended as a Greek-based analogue of the word "complex". The notation Sp ( 2 n , F ) {\displaystyle \operatorname {Sp} (2n,\mathbb {F} )} usually denotes the symplectic group of a 2 n {\displaystyle 2n} -dimensional symplectic vector space over a field F {\displaystyle \mathbb {F} } . A related but different family is the compact symplectic group, denoted Sp ( n ) {\displaystyle \operatorname {Sp} (n)} or U S p ( n ) {\displaystyle \mathrm {USp} (n)} , which is the compact real form of the complex symplectic group. Many authors use slightly different notations, often differing by factors of 2 {\displaystyle 2} . In Cartan's classification, the Lie algebra of Sp ( 2 n , C ) {\displaystyle \operatorname {Sp} (2n,\mathbb {C} )} has type C n {\displaystyle C_{n}} .
Sp(2n, F) The symplectic group is a classical group defined as the set of linear transformations of a 2 n {\displaystyle 2n} -dimensional vector space over the field F {\displaystyle \mathbb {F} } 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 V {\displaystyle V} is denoted Sp ( V ) {\displaystyle \operatorname {Sp} (V)} . Upon fixing a basis for V {\displaystyle V} , the symplectic form is represented by a nonsingular skew-symmetric matrix J {\displaystyle J} , and Sp ( V ) {\displaystyle \operatorname {Sp} (V)} is identified with the group of 2 n × 2 n {\displaystyle 2n\times 2n} matrices over F {\displaystyle \mathbb {F} } satisfying
{ M ∈ M 2 n × 2 n ( F ) : M T J M = J } . {\displaystyle \{M\in M_{2n\times 2n}(\mathbb {F} ):M^{\mathrm {T} }JM=J\}.}
This matrix group is denoted Sp ( 2 n , F ) {\displaystyle \operatorname {Sp} (2n,\mathbb {F} )} or Sp ( n , F ) {\displaystyle \operatorname {Sp} (n,\mathbb {F} )} , although the notation depends on the convention being used. Here M T {\displaystyle M^{T}} denotes the transpose of M {\displaystyle M} . In an arbitrary basis, the matrix J {\displaystyle J} need not have any particular form. However, one can choose a symplectic basis, in which the form is represented by the standard matrix
Ω = ( 0 I n − I n 0 ) , {\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}},}
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