(real or complex) Lie groups, and and are the Lie algebras of G and H respectively, then the differential on tangent spaces at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space V, a representation of Lie groups determines a Lie algebra homomorphism from to the Lie algebra of the general linear group GL(V), i.e. the endomorphism algebra of V. For example, let . Then the differential of at the identity is an element of . Denoting it by one obtains a representation of G on the vector space . This is the adjoint representation of G. Applying the preceding, one gets the Lie algebra representation . It can be shown that , the adjoint representation of . A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras. In quantum physics In quantum theory, one considers "observables" that are self-adjoint operators on a Hilbert space. The commutation relations among these operators are then an important tool. The angular momentum operators, for example, satisfy the commutation relations . Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the rotation group SO(3). Then if is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, will constitute a representation of the Lie algebra so(3). An understanding of the representation theory of so(3) is of great help in, for example, analyzing Hamiltonians with rotational symmetry, such as the hydrogen atom. Many other interesting Lie algebras (and their representations) arise in other parts of quantum physics. Indeed, the history of representation theory is characterized by rich interactions between mathematics and physics. Basic concepts Invariant subspaces and irreducibility Given a representation of a Lie algebra , we say that a subspace of is invariant if for all and . A nonzero representation is said to be irreducible if the only invariant subspaces are itself and the zero space . The term simple module is also used for an irreducible representation. Homomorphisms Let be a Lie algebra. Let V, W be -modules. Then a linear map is a homomorphism of -modules if it is -equivariant; i.e., for any