Representation of a Lie algebra


Representation of a Lie algebra - Definition

In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ_* on tangent spaces is a homomorphism of Lie algebras, i.e. satisfies

for all x and y in g. In particular, a representation of Lie groups φ: G→GL(V) determines a homomorphism of Lie algebras from g to the Lie algebra of the general linear group GL(V) over the vector space V. (GL(V) is just the endomorphism ring End(V) = Hom(V,V)). Such a homomorphism is called a representation of the Lie algebra g.

More generally (since we can study Lie algebras independently from their incarnation as the tangent space of a Lie group), such a representation may be described as a bilinear map (x,v)→x.v from g×V to V satisfying the Jacobi identity analogue

Equivalently, it is a representation of the universal enveloping algebra.

If the Lie algebra is semisimple, then all reducible reps are decomposable. Otherwise, that's not true in general.

If we have two reps, with V₁ and V₂ as their underlying vector spaces and .[.]₁ and .[.]₂ as the reps, then the product of both reps would have <math>V_1\otimes V_2<math> as the underlying vector space and

<math>x[v_1\otimes v_2]=x[v_1]\otimes v_2+v_1\otimes x[v_2]<math>

If L is a real Lie algebra and <math>\rho:L\times V\rightarrow V<math> is a complex rep of it, we can construct another rep of L called its dual rep as follows.

Let V^* be the dual vector space of V. In other words, V is the set of all linear maps from V to C with addition defined over it in the usual linear way BUT scalar multiplication defined over it such that <math>(z\omega)[X]=\bar{z}\omega[X]<math> for any z in C, ω in V^* and X in V. This is usually rewritten as a contraction with a sesquilinear form <.,>. i.e. <ω,X> is defined to be ω[X].

We define <math>\bar{\rho}<math> as follows: <A[ω],X>+<ω,A[X]>=0

for any A in L, ω in V^* and X in V. This defines <math>\bar{\rho}<math> uniquely.

See also