A representation of a Lie algebra g is a linear transformation ψ:g->M(V), where M(V) is the set of all linear transformations of a vector space V. In particular, if V = R^n, then M(V) is the set of n×n square matrices. The map ψ is required to be a map of Lie algebras so that ψ([A, B]) = ψ(A) ψ(B) - ψ(B) ψ(A) for all A, B element g. Note that the expression A B only makes sense as a matrix product in a representation. For example, if A and B are antisymmetric matrices, then A B - B A is skew-symmetric, but A B may not be antisymmetric.

group representation | irreducible representation | Lie algebra | Lie group | matrix exponential | simple Lie algebra | vector space