Symplectic Group

 The orthogonal groups were based on a symmetric metric. Symmetric matrices Determinant of 1  An antisymmetric metric can also exist. Transpose is negative Bilinear function Antisymmetry

Dimension 2n  To have an inverse a matrix must be non-singular. 1 x 1 gives det(S) = 01 x 1 gives det(S) = 0 2 x 2 gives det(S) ≠ 02 x 2 gives det(S) ≠ 0 2 n x 2 n gives det(S) ≠ 02 n x 2 n gives det(S) ≠ 0  Use vector spaces of even dimension. Determinant squared is 1Determinant squared is 1 2 n x 2 n antisymmetric matrix2 n x 2 n antisymmetric matrix

Symplectic Metric  Antisymmetric matrices must have 0 on the diagonal.  With unity determinant there is a canonical form. 1 at J i+n,i1 at J i+n,i -1 at J i,i+n-1 at J i,i+n  This is symmetric on the minor diagonal. Symplectic symmetrySymplectic symmetry

Symplectic Group  Find elements in GL(2n) that preserve the antisymmetry. Matrix T called symplectic Means twisted  The symplectic matrices form a group Sp(2n) Sometimes Sp(n), n even Lie group identity inverse closure

Sample Elements  There are three 2x2 matrices with elements 0 or 1 that are in Sp(2).

General Form of Sp(2)  There general form of Sp(2) is the same as SL(2). Isomorphic groupsIsomorphic groups  Sp(2) is also isomorphic to SU(2) Dimension 3Dimension 3

Inverse Matrices  The inverse of a symplectic matrix is easy to compute. Use properties of J next