Student Seminar: Classical and Quantum Integrable Systems

• Pseudo-orthogonal groups SO(p, q). Let g will be pseudo-Euclidean metric in
the space R n p,q with p + q = n. The group SO(p, q) is the group of real matrices
which preserve the form g:
AgA t = g , detA = 1 .
• Unitary group U(n) – the group of unitary n × n matrices:
UU † = I .
• Special unitary group SU(n) – the group of unitary n × n matrices with the
unit determinant
UU † = I , detU = 1 .
• Pseudo-unitary group U(p, q):
AgA † = g ,
where g is the pseudo-Euclidean metric. Special pseudo-unitary group requires
in addition the unit determinant detA = 1.
• Symplectic group Sp(2n, R) or Sp(2n, C) is a group or real or complex matrices
satisfying the condition
AJA t = J
where J is 2n × 2n matrix
and I is n × n unit matrix.
J =
( ) 0 I
−I 0
Question to the class: What are the eigenvalues of J? Answer:
J = diag(i, · · · i; −i, · · · , −i).
Thus, the group Sp(2n) is really different from SO(2n)!
The powerful tool in the theory of Lie groups are the Lie algebras. Let us see how
they arise by using as an example SO(3). Let A be “close” to the identity matrix
A = I + ɛa
is an orthogonal matrix A t = A −1 . Therefore,
I + ɛa t = (I + ɛa) −1 = I − ɛa + ɛ 2 a 2 + · · ·
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