representation of lie algebras

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_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 ωX. We define

\bar{\rho}

as follows: <Aω,X>+<ω,AX>=0 for any A in L, ω in V^* and X in V. This defines

\bar{\rho}

uniquely.

Representation Of Lie Algebras

See also