An Elementary Introduction to Groups and Representations by Hall B.C.

By Hall B.C.

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8 from Section 3 shows that A(t) = etX . 5. The Lie Algebra of a Matrix Lie Group The Lie algebra is an indispensable tool in studying matrix Lie groups. On the one hand, Lie algebras are simpler than matrix Lie groups, because (as we will see) the Lie algebra is a linear space. Thus we can understand much about Lie algebras just by doing linear algebra. On the other hand, the Lie algebra of a matrix Lie group contains much information about that group. ) Thus many questions about matrix Lie groups can be answered by considering a similar but easier problem for the Lie algebra.

0 1 Thus if X is nilpotent, trace(X) = 0, and det(eX ) = 1. Case 3: X arbitrary. As pointed out in Section 2, every matrix X can be written as the sum of two commuting matrices S and N , with S diagonalizable (over C) and N nilpotent. Since S and N commute, eX = eS eN . So by the two previous cases det eX = det eS det eN = etrace(S) etrace(N ) = etrace(X) , which is what we want. 11. A function A : R → GL(n; C) is called a one-parameter group if 1. A is continuous, 2. A(0) = I, 3. A(t + s) = A(t)A(s) for all t, s ∈ R.

It follows that the Jacobi identity holds if X is in gC, and Y, Z in g. The same argument then shows that we can extend to Y in gC , and then to Z in gC . Thus the Jacobi identity holds in gC . 36. The Lie algebras gl(n; C), sl(n; C), so(n; C), and sp(n; C) are complex Lie algebras, as is the Lie algebra of the complex Heisenberg group. In addition, we have the following isomorphisms of complex Lie algebras gl (n; R)C ∼ = gl(n; C) ∼ u(n)C = gl(n; C) sl (n; R)C ∼ = sl(n; C) ∼ so(n)C = so(n; C) sp(n; R)C ∼ = sp(n; C) ∼ sp(n)C = sp(n; C).

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