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Representation of Lie algebras

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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

$\phi_*[x\,y] = [\phi_*x\,\phi_*y]$

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

$[x_1\,x_2].v = x_1.(x_2.v) - x_2.(x_1.v)$

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 $V_1\otimes V_2$ as the underlying vector space and

$x[v_1\otimes v_2]=x[v_1]\otimes v_2+v_1\otimes x[v_2]$

If L is a real Lie algebra and $\rho:L\times V\rightarrow V$ 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 $(z\omega)[X]=\bar{z}\omega[X]$ 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 $\bar{\rho}$ as follows: <A[ω],X>+<ω,A[X]>=0

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

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Representation of Lie algebras

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