so that the Lie bracket with L 0 defines a derivation of the Lie algebra spanned by A, A * and I. Adjoining L 0 gives the semidirect product. The infinitesimal version of the Stone–von Neumann theorem states that the above representation on C [ z ] is the unique irreducible positive energy representation of this Lie algebra with L 0 = A * A + 1/2. For A lowers energy and A * raises energy. So any lowest energy vector v is annihilated by A and the module is exhausted by the powers of A * applied to v. It is thus a non-zero quotient of C [ z ] and hence can be identified with it by irreducibility.