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Monday, September 5, 2016

Classical Lie groups

 Lecture topic

A Lie group G is both a group and a manifold, with a smooth map G×GG, given by (g,h)gh1. The Lie algebra g of G is the tangent space TeG of G at the identity.

We distinguish between real and complex Lie groups by saying that the base manifold is either real or complex analytic, respectively.

Example:
Here are some examples of classical Lie groups and their dimension:
general linear groupn2GL(n)={n×n matrices with non-zero determinant}special linear groupn21SL(n)={MGL(n) : det(M)=1}orthogonal groupn(n1)/2O(n)={MGL(n) : MMt=I}special orthogonal groupn(n1)/2SO(n)={MO(n) : det(M)=1}unitary groupn2U(n)={MGL(n,C) : MM=I}special unitary groupn21SU(n)={MU(n) : det(M)=1}symplectic groupn(2n+1)Sp(n)={n×n matrices: ω(Mx,My)=ω(x,y)}
For the symplectic group, the skew-symmetric bilinear form ω is defined as
ω(x,y)=ni=1xiyi+nyixi+n=(0II0)xy,
where is the regular dot product (a symmetric bilinear form). Also note that the unitary group is a real Lie group - real because there is no holomorphic map G×GG as would be necessary, so we view the entries of a matrix in U(n) in terms of its real and imaginary parts. Hence the dimension indicated above is real dimension.

References: Kirillov Jr (An introduction to Lie groups and Lie algebras, Chapter 2)

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