Lie groups and Lie algebras

24 janeiro 2022, 16:30 • John Huerta

We recalled the definition of Lie group, and gave abundant examples:

  1. (ℝn, +) is an abelian Lie group.
  2. The real general linear group GLn(ℝ) is, first of all, an open set in the set of n x n real matrices, Matn(ℝ). This makes it a manifold with one chart, and it is easy to check the group operations, given by matrix multiplication and matrix inversion, are both smooth. Hence, GLn(ℝ) is a Lie group.
  3. Similarly, the complex general linear group GLn(ℂ) is a Lie group.
  4. (a) The real special linear group SLn(ℝ) is the subgroup of GLn(ℝ) consisting of matrices with determinant 1. Because the determinant is a continous function, SLn(ℝ) is a closed subgroup. Hence, SLn(ℝ) is a Lie group by Cartan's theorem. Similarly, the complex special linear group SLn(ℂ) is a Lie group.
  • (b) The real orthogonal group On(ℝ) is the subgroup of GLn(ℝ) consisting of matrices A that satisfy ATA = I. Because the map sending A to ATA is continuous, this is a closed subgroup. Hence, On(ℝ) is a Lie group by Cartan's theorem. Similarly, the complex orthogonal group On(ℂ) is a Lie group.
  • (c) The Lorentz group O(1,n) is the subgroup of GLn+1(ℝ) consisting of matrices that satisfy AThA = h, where h is the (n+1) x (n+1) diagonal matrix with diagonal [-1, 1, 1, ..., 1].
  • (d) The unitary group Un is the subgroup of GLn(ℂ) of matrices satisfying A*A = I, where A* denotes the conjugate transpose of A. The special unitary group SUn is the subgroup of GLn(ℂ) satisfying A* A = I and det(A) = 1.
  • (e) The symplectic group Spn(𝕂) is the subgroup of GL2n(𝕂) of matrices satisfying ATΩA = Ω. Here, 𝕂 denotes ℝ or ℂ and Ω denotes the 2n x 2n that looks like this (each block is n x n):
     0  -I
     I
    		 0
We also defined a Lie algebra, and gave abundant examples:

  1. ℝn with zero bracket, [v,w] = 0, is an abelian Lie algebra.
  2. The real general linear algebra gln(ℝ) is defined on the vector space Matn(ℝ), with Lie bracket given by the commutator: [X,Y] = XY - YX.
  3. The complex general linear algebra gln(ℂ) is defined on the vector space Matn(ℂ), with Lie bracket given by the commutator: [X,Y] = XY - YX.
  4. (a) The real special linear algebra sln(𝕂) is the Lie subalgebra of gln(𝕂) consisting of matrices of zero trace, for 𝕂 = ℝ or ℂ.
  • (b) The real orthogonal algebra on(𝕂) is the Lie subalgebra of gln(𝕂) consisting of matrices X that satisfy XT+ X = 0.
  • (c) The Lorentz algebra o(1,n) is the Lie subalgebra of gln+1(ℝ) consisting of matrices that satisfy XTh + hX = 0, where h is the (n+1) x (n+1) diagonal matrix with diagonal [-1, 1, 1, ..., 1].
  • (d) The unitary algebra un is the Lie subalgebra of gln(ℂ) of matrices satisfying X*+ X = 0, where X* denotes the conjugate transpose of X. The special unitary algebra sun is the Lie subalgebra of gln(ℂ) satisfying X* + X = 0 and tr(X) = 0.
  • (e) The symplectic algebra spn(𝕂) is the Lie subalgebra of gl2n(𝕂) of matrices satisfying XTΩ + ΩX = 0, where Ω is the same matrix as above.
In the list above, I have used lower case letters in place of the conventional Fraktur letters.

Every Lie group G has a Lie algebra g, defined on the tangent space to the identity: g = T eG. Naturally, for each of the Lie groups in the first list, its Lie algebra is the corresponding entry in the second list. So far, we only have enough differential geometry to check that T eG is a subspace of g, as we showed in the example where G = O(1,n), the Lorentz group, and g = o(1,n), the Lorentz algebra. Next time, we will see how to get equality.