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Weights and roots

  1. Feb 1, 2008 #1
    I'm having trouble understanding the idea of a weight space.

    Suppose [itex]\mathfrak{g}[/itex] is the Lie alebra of G with maximal torus T and Cartan subalgebra [itex]\mathfrak{t}[/itex]. The weights are the (1-dimensional) irreducible represenations of T. If we restrict any representation [itex]\rho : G \to GL(V)[/itex] to T ([itex]\rho|_T : T \to GL(V)[/itex]) then we get a direct sum of weights [itex]\alpha_i[/itex]. If [itex]\rho[/itex] is taken to be the adjoint representation, then the roots are defined to be the nontrivial weights of this rep.

    My question concerns the trivial weights. Why exactly is it that T acts trivially on its own tangent space [itex]\mathfrak{t}[/itex]?
     
  2. jcsd
  3. Feb 2, 2008 #2
    I realized that this is pretty obvious since the adjoint action of T on itself is just identity because T is abelian. Thus for any [itex]t \in T[/itex], the linear transformation [itex]Ad|_T (t) : \mathfrak{g} \to \mathfrak{g}[/itex] acts trivially on the Cartan subalgebra [itex]\mathfrak{h}[/itex]. The T-action on the remainder of the Lie algebra, however may be non-trivial, which is where the nontrivial weights (ie roots) enter.
     
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