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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