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In mathematics and 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 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 monomorphism, the representation is said to be faithful.
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 G is a compact Lie group, every finite-dimensional representation is equivalent to
a unitary one.
Classification
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 characters of G: see Pontryagin duality for this case.
A quotient representation is a quotient module of the group ring.
Related topics
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