By Yuri Dokshitzer

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4. A Lie subalgebra n of a Lie algebra g is called normal if [x, y] ∈ n for all x ∈ g and y ∈ n. A Lie algebra g is called simple if the only normal subalgebras are the two trivial ones 0 and g itself. Show that a reductive Lie algebra g is a unique direct sum g = z ⊕ g1 ⊕ · · · ⊕ gn with z the Abelian center and the gi all simple Lie algebras. 5. Show that R spans a∗ if and only if the center z of g is zero. 6. 9 we can choose xα , yα , zα in such a way that xα = yα , yα = xα , zα = zα . 7. Consider the natural inclusion so2n → so2n+1 by adding zeros in the last column and row.

Here Ii and Jj work in the ﬁrst and the second factor of L(m) ⊗ L(n) respectively. 2 below I 2 |L(m)⊗L(n) = m(m + 2) 2 /4 , J 2 |L(m)⊗L(n) = n(n + 2) 2 /4 we conclude that m = n and {ψ ∈ H; Hψ = Eψ} = L(n) ⊗ L(n) for some n ∈ N. Finally we shall derive a formula for En as function of n. Rewriting the above formula K 2 = 2mH(L2 + 2 ) + k 2 m2 in the form L2 + (−2mE)−1 K 2 + 2 = −k 2 m/2E and because on the space L(n) ⊗ L(n) L2 + (−2mE)−1 K 2 + we arrive at 2 = 2(I 2 + J 2 ) + 2 = n(n + 2) E = En = −k 2 m/2(n + 1)2 2 + 2 = (n + 1)2 2 2 with n running over the set N.

A vector μ ∈ h∗ is called h-integral if μ maps into Zn . Suppose we have given a Lie algebra g with a ﬁxed Abelian subalgebra h, together with a ﬁxed basis h = {h1 , · · · , hn } of h. Suppose we have given a representation g → gl(V ), that is h-admissable and h-integral, in the sense that all weights of V are h-integral. Under these assumptions we can deﬁne the formal charachter by charV (g, h, q) = trV (qh ) = trV (q1h1 · · · qnhn ) which lies in Z[[q±1 ]] = Z[[q1±1 , · · · , qn±1 ]]. Clearly the familiar relations of characters for the dual representation charV ∗ (g, h, q) = charV (g, h, q−1 ) , and for the tensor product of two representations charU ⊗V (g, h, q) = charU (g, h, q)charV (g, h, q) do hold, like in the case of ﬁnite groups.