Compact symplectic Lie algebra
= Compact symplectic Lie algebra
{title2=$\mathfrak{usp}(2n)\cong\mathfrak{sp}(n)$}
= Lie algebra of the compact symplectic group
{c}
{synonym}
The <Lie algebra> of the <compact symplectic group> consists of $X^\dagger=-X$ and $XJ+JX^T=0$. Equivalently $X=\begin{pmatrix}A&B\\-\overline B&\overline A\end{pmatrix}$ with $A^\dagger=-A$ and $B^T=B$. Its real dimension is $n(2n+1)$, and its complexification is the <symplectic Lie algebra>.