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Symplectic Group

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.

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Symplectic Group

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  1. Symplectic Group

  2. 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

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

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

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

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

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

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

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