Lecture topic
A Lie group G is both a group and a manifold, with a smooth map G×G→G, given by (g,h)↦gh−1. 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 groupn2−1SL(n)={M∈GL(n) : det(M)=1}orthogonal groupn(n−1)/2O(n)={M∈GL(n) : MMt=I}special orthogonal groupn(n−1)/2SO(n)={M∈O(n) : det(M)=1}unitary groupn2U(n)={M∈GL(n,C) : MM∗=I}special unitary groupn2−1SU(n)={M∈U(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)=n∑i=1xiyi+n−yixi+n=(0−II0)x⋅y,
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×G→G 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)
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 groupn2−1SL(n)={M∈GL(n) : det(M)=1}orthogonal groupn(n−1)/2O(n)={M∈GL(n) : MMt=I}special orthogonal groupn(n−1)/2SO(n)={M∈O(n) : det(M)=1}unitary groupn2U(n)={M∈GL(n,C) : MM∗=I}special unitary groupn2−1SU(n)={M∈U(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)=n∑i=1xiyi+n−yixi+n=(0−II0)x⋅y,
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×G→G 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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