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Formally, a representation of a Lie group G on a vector space V (over a field K) is a smooth (i.e. respecting the differential structure) group homomorphism G→Aut(V) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL(n,K). This is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W.

On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to End(V) preserving the Lie bracket [ , ]. See representation of Lie algebras for the Lie algebra theory.

If the homomorphism is in fact an monomorphismIn the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. In the more general (and abstract) setting of category theory, a monomorphism (also called a monic morphism is a morphism f : X → Y such that, the representation is said to be faithful.

A unitary representationIn mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π g is a unitary operator for every g ∈ G''. The general theory is well-developed in case G is a locally compact (Hau is defined in the same way, except that G maps to unitary matricesIn mathematics, a unitary matrix is a n by n complex matrix U satisfying the condition U ''U UU I where I is the identity matrix and U is the conjugate transpose (also called the Hermitian adjoint) of U''. Note this condition says that a matrix U is unita; the Lie algebra will then map to skew-hermitian matrices.

If G is a compact Lie group, every finite-dimensional representation is equivalent to a unitary one.

1 Classification

If G is a semisimpleIn mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way (by direct sum). A semisimple module is one in w group, its finite-dimensional representations can be decomposed as direct sumIn abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. In a sense, the direct sum of vector spaces is the "most general" vector space that contains thes of irreducible representations. The irreducibles are indexed by highest weightGiven a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. In basis-free terms, for any set of mutually commuting semisim; the allowable (dominant) highest weights satisfy a suitable positivity condition. In particular, there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights.

If G is a commutative compact Lie group, then its irreducible representations are simply the continuous characters of G: see Pontryagin duality for this case.

A quotient representation is a quotient module of the group ring.