Representation of Lie algebras

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In mathematics, if φ: GH 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

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

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

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

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

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

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

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