Symplectic group

The Lie group of linear transformations preserving a nondegenerate alternating form on .

Symplectic group

Definition

Let $J$ be the standard symplectic matrix

J=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix},

so the bilinear form $\omega(u,v)=u^T J v$ on $\mathbb R^{2n}$ is nondegenerate and skew-symmetric. The real symplectic group is

\mathrm{Sp}(2n,\mathbb R)=\{A\in GL(2n,\mathbb R)\mid A^T J A = J\}.

This is a closed subgroup of the general linear group , hence a Lie subgroup by the closed subgroup theorem .

More generally, for any field $\mathbb F$ of characteristic $\neq 2$ , one defines $\mathrm{Sp}(2n,\mathbb F)$ as the group preserving a fixed nondegenerate alternating form; different choices are conjugate in $GL(2n,\mathbb F)$ .

Lie algebra

The Lie algebra of $\mathrm{Sp}(2n,\mathbb R)$ is the symplectic Lie algebra $\\mathfrak{sp}(2n,\\mathbb R)$ , obtained by differentiating the defining equation $A^TJA=J$ at the identity.

Context

$\mathrm{Sp}(2n,\mathbb R)$ is the basic symmetry group of linear symplectic geometry and Hamiltonian mechanics: it preserves the standard symplectic form and hence the canonical Poisson structure on $\mathbb R^{2n}$ . In representation theory, its complexification corresponds to the simple Lie algebra of type $C_n$ (compare root systems and Dynkin diagrams ).