symplectic group. the orthogonal groups were based on a symmetric metric. symmetric matrices...
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Symplectic GroupSymplectic Group
The orthogonal groups were The orthogonal groups were based on a symmetric based on a symmetric metric.metric.• Symmetric matricesSymmetric matrices
• Determinant of 1Determinant of 1
An antisymmetric metric can An antisymmetric metric can also exist.also exist.• Transpose is negativeTranspose is negative
• Bilinear functionBilinear function
AntisymmetryAntisymmetry
jiji
jiij vusvus
),(),( uvsvus
jiij ss SS T
),(),(),( wusvuswvus
),(),(),( vwsvusvwus
Dimension 2nDimension 2n
To have an inverse a matrix To have an inverse a matrix must be non-singular.must be non-singular.• 1 x 1 gives det(1 x 1 gives det(SS) = 0) = 0
• 2 x 2 gives det(2 x 2 gives det(SS) ) ≠≠ 0 0
• 22nn x 2 x 2nn gives det( gives det(SS) ) ≠≠ 0 0
Use vector spaces of even Use vector spaces of even dimension.dimension.• Determinant squared is 1Determinant squared is 1
• 22nn x 2 x 2nn antisymmetric matrix antisymmetric matrixki
jkijss
00 2
0
0a
a
a
0
0
0
0
ab
ac
bc
Symplectic MetricSymplectic Metric
Antisymmetric matrices must Antisymmetric matrices must have 0 on the diagonal.have 0 on the diagonal.
With unity determinant there With unity determinant there is a canonical form.is a canonical form.• 1 at 1 at JJi+n,ii+n,i
• -1 at -1 at JJi,i+ni,i+n
This is symmetric on the This is symmetric on the minor diagonal.minor diagonal.• Symplectic symmetrySymplectic symmetry
nn
nnn I
IJ
0
02
01
102J
0010
0001
1000
0100
4J
Symplectic GroupSymplectic Group
Find elements in GL(2n) that Find elements in GL(2n) that preserve the antisymmetry.preserve the antisymmetry.• Matrix Matrix T T calledcalled symplecticsymplectic
• Means twistedMeans twisted
The symplectic matrices The symplectic matrices form a group Sp(2n)form a group Sp(2n)• Sometimes Sp(n), n evenSometimes Sp(n), n even
• Lie groupLie group
cdbd
acab sTTs
SSTT T
identity
inverse
closure
cdbd
acab ss
ki
jkijss
SSBBSTT
SABABSTTTT
TTT
Sample ElementsSample Elements
There are three 2x2 matrices There are three 2x2 matrices with elements 0 or 1 that are with elements 0 or 1 that are in Sp(2).in Sp(2).
10
01
10
11
11
01
10
11
01
10
11
01JTT T
10
11
11
10JTT T
JJTT T
01
10
General Form of Sp(2)General Form of Sp(2)
There general form of Sp(2) There general form of Sp(2) is the same as SL(2).is the same as SL(2).• Isomorphic groupsIsomorphic groups
Sp(2) is also isomorphic to Sp(2) is also isomorphic to SU(2)SU(2)• Dimension 3Dimension 3
db
ca
dc
baJTT T
01
10
db
ca
cd
abJTT T
Jadbc
bcadJTT T
0
0
Inverse MatricesInverse Matrices
The inverse of a symplectic The inverse of a symplectic matrix is easy to compute.matrix is easy to compute.• Use properties of JUse properties of J
JTTJ T
11 JTTTJT T
JTJJTJ T111
JJTT T 1
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