In mathematics, and also theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras (indeed in the physics literature the distinction is often elided).
Formally a representation consists of a vector space V, over one of the fields or , a group homomorphism from the Lie group G to Aut(V). On the Lie algebra level, there will correspond a linear mapping from the Lie algebra of G to End(V) preserving the bracket [ , ].
A unitary representation is defined in the same way, except that G maps to unitary matrices; the Lie algebra will then map to skew-hermitian matrices.
If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL(n,R) or GL(n,C). This is known as a matrix representation.
If the homomorphism is in fact an monomorphism, the representation is said to be faithful.
If G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight; 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 characterss of G: see Pontrygin duality for this case.
See representation of Lie algebras for the Lie algebra theory, as free-standing.
A quotient representation is a quotient module of the group ring.
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