We list here some of the basic facts concerning Lie groups and algebras that are needed in this book. Complete proofs and further details can be found in [85]. Since almost all Lie groups that arise in mathematical physics are groups of matrices, we shall confine our attention to local linear Lie groups.

Let W be an open, connected set containing e = (0,…, 0) in the space Rn of all real wn-tuples g = (g1,…,gn).

Definition. An n-dimensional (real) local linear Lie group G is a set of m×m nonsingular complex matrices A(g) = A(g1,…,gn) defined for each g∈ W such that

1. A (e) = Em (the identity matrix).

2. The matrix elements of A (g) are analytic functions of the parameters g1,…,gn and the map g→A(g) is one to one.

3. The n matrices δA(g)/δgj,j= 1,…,n, are linearly independent for each g∈W.

4. There exists a neighborhood Wt of e in Rn, W′ ⊂ W, with the property that for every pair of n-tuples g,h in W′ there is an n-tuple k in W satisfying A (g)A (h) = A (k) where the operation on the left is matrix multiplication.