By Werner Hildbert Greub

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L:Weyl} 34 P. SOSNA Now assume that dim(V ) > 1. If V = V g , there is nothing to prove. So assume that V = V g . 11, ρ(g) is a semisimple subalgebra of gl(V ). We have / V , where we use the bilinear form β(x, y) = tr(xy) for the Casimir operator cφ : V x, y ∈ gl(V ). Since K is algebraically closed, the vector space V decomposes into a direct sum of generalised eigenspaces Vλ of cφ . 1 that cφ commutes with ρ(g), hence every Vλ is a subrepresentation of V . There are now two cases. • cφ has at least two eigenvalues.

To conclude this section, we answer the question whether sl(2, K) does indeed have an irreducible module of each possible highest weight. 3 do define an irreducible representation on an (m + 1)-dimensional vector space V with basis (v0 , . . , vm ) (see Exercise 3 on Sheet 9). Such a representation is denoted by V (m). To make this more explicit, one could take as V (m) the vector space of degree m polynomials in two variables s, t, that is V (m) = spanK (si tm−i ) ⊂ K[s, t] for i = 0, . . , m.

5. Assume that φ is not necessarily faithful. 11. If we denote by g the sum of the remaining simple ideals, then the restriction of φ to g is a faithful representation of g and one can apply the above construction to it. The following result will be useful in the proof of Weyl’s Theorem. 6. Let φ : g / gl(V ) be a representation of a semisimple Lie algebra g. Then φ(g) ⊂ sl(V ). In particular, g acts trivially on any one dimensional g-module. {r:CasiFai} {l:SemiSiSL} Proof. 11, g = [g, g]. 1) to see the first claim.