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5. THE LIE ALGEBRA OF A MATRIX LIE GROUP 37 Note that even if G is a subgroup of GL(n; C) we do not require that etX be in G for all complex t, but only for all real t. Also, it is definitely not enough to have just eX in G. That is, it is easy to give an example of an X and a G such that eX ∈ G but etX ∈ / G for some values of t. Such an X is not in the Lie algebra of G. It is customary to use lower case Gothic (Fraktur) characters such as g and h to refer to Lie algebras. 1. Physicists’ Convention.
Clearly, the structure constants determine the bracket operation on g. In some of the literature, the structure constants play an important role, although we will not have occasion to use them in this course. ) The structure constants satisfy the following two conditions, cijk + cjik = 0 (cijm cmkl + cjkm cmil + ckim cmjl ) = 0 m for all i, j, k, l. The first of these conditions comes from the skew-symmetry of the bracket, and the second comes from the Jacobi identity. ) 9. 34. If V is a finite-dimensional real vector space, then the complexification of V , denoted VC , is the space of formal linear combinations v1 + iv2 9.
Let X be an n × n real or complex matrix. 1) ∞ Xm . m! m=0 We will follow the convention of using letters such as X and Y for the variable in the matrix exponential. 1. 1) converges. The matrix exponential eX is a continuous function of X. Before proving this, let us review some elementary analysis. Recall that the norm of a vector x in Cn is defined to be x = 2 |xi | . x, x = This norm satisfies the triangle inequality x+y ≤ x + y . The norm of a matrix A is defined to be A = sup x=0 Ax . x Equivalently, A is the smallest number λ such that Ax ≤ λ x for all x ∈ Cn .