ad | ad

A Lie algebra is a vector space g with a Lie bracket [X, Y], satisfying the Jacobi identity. Hence any element X gives a linear transformation given by ad(X)(Y) = [X, Y], which is called the adjoint representation of g. It is a Lie algebra representation because of the Jacobi identity, [ad(X_1), ad(X_2)](Y) | = | [X_1, [X_2, Y]] - [X_2, [X_1, Y]] | = | [[X_1, X_2], Y] | = | ad([X_1, X_2])(Y).

commutator | group representation | Lie algebra | Lie bracket | Lie group | nilpotent Lie algebra | semisimple Lie algebra