Lie algebra representation

A representationPlanetmathPlanetmath of a Lie algebraMathworldPlanetmath 𝔤 is a Lie algebra homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


where EndV is the commutator Lie algebra of some vector spaceMathworldPlanetmath V. In other words, ρ is a linear mapping that satisfies


Alternatively, one calls V a 𝔤-module, and calls ρ(a),a𝔤 the action of a on V.

We call the representation faithful if ρ is injective.

A invariant subspace or sub-module WV is a subspacePlanetmathPlanetmath of V satisfying ρ(a)(W)W for all a𝔤. A representation is called irreducible or simple if its only invariant subspaces are {0} and the whole representation.

The dimensionPlanetmathPlanetmathPlanetmathPlanetmath of V is called the dimension of the representation. If V is infinite-dimensional, then one speaks of an infinite-dimensional representation.

Given a pair of representations, we can define a new representation, called the direct sumPlanetmathPlanetmathPlanetmath of the two given representations:

If ρ:𝔤End(V) and σ:𝔤End(W) are representations, then VW has the obvious Lie algebra action, by the embedding End(V)×End(W)End(VW).

Title Lie algebra representation
Canonical name LieAlgebraRepresentation
Date of creation 2013-03-22 12:41:13
Last modified on 2013-03-22 12:41:13
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Definition
Classification msc 17B10
Synonym representation
Related topic Dimension3
Defines irreducible
Defines module
Defines dimension
Defines finite dimensional
Defines finite-dimensional
Defines infinite dimensional
Defines infinite-dimensional
Defines faithful
Defines direct sum of representations