Symplectic Lie algebra

The Lie algebra of the symplectic group: matrices satisfying $X^T J + JX = 0$ with commutator bracket.

Symplectic Lie algebra

Fix the standard symplectic matrix

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

The symplectic Lie algebra over a field $\mathbb F$ of characteristic $\neq 2$ is

\mathfrak{sp}(2n,\mathbb F)=\{X\in M_{2n}(\mathbb F)\mid X^T J + JX = 0\},

with Lie bracket given by the commutator $[X,Y]=XY-YX$ (see the Lie bracket ). It is the Lie algebra of the symplectic group $\\mathrm{Sp}(2n,\\mathbb F)$ in the sense of the Lie algebra of a Lie group .

Writing a matrix in $n\times n$ blocks

X=\begin{pmatrix}A&B\\C&D\end{pmatrix},

the condition $X^T J + JX=0$ is equivalent to

D=-A^T,\qquad B=B^T,\qquad C=C^T.

In particular, over $\mathbb R$ or $\mathbb C$ one computes

\dim \mathfrak{sp}(2n)=n^2+\frac{n(n+1)}{2}+\frac{n(n+1)}{2}=n(2n+1).

Over an algebraically closed field of characteristic $0$ , $\mathfrak{sp}(2n)$ is a simple Lie algebra (type $C_n$ ). A choice of Cartan subalgebra leads to the usual root space description (see root space decomposition ); the associated reflections generate the Weyl group . Nondegeneracy of the Killing form is a key semisimplicity test (compare Killing form nondegeneracy and semisimplicity ).