Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted $Sp(2 n, F)$ and $Sp(n)$ , the latter is called the compact symplectic group. Many authors prefer slightly different notations, usually differing by factors of $2$ . The notation used here is consistent with the size of the matrices used to represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group $Sp(2 n, C)$ is denoted $C n$ , and $Sp(n)$ is the compact real form of $Sp(2 n, C)$ . 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 $n$ .

The name "symplectic group" is due to Hermann Weyl (details) as a replacement for the previous confusing names of (line) complex group and Abelian linear group, and is the Greek analog of "complex".

The symplectic group of degree $2 n$ over a field $F$ , denoted $Sp(2 n, F)$ , is the group of $2 n \times 2 n$ symplectic matrices with entries in $F$ , and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant $1$ , the symplectic group is a subgroup of the special linear group $SL(2 n, F)$ .

...
Wikipedia