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Integrable representations of affine Lie-algebras. (English) Zbl 0603.17011
Let \({\mathfrak g}\) be a finite dimensional simple Lie algebra over \({\mathbb C}\) and let \(L({\mathfrak g})=L\otimes_{{\mathbb C}}{\mathfrak g}\) \((L={\mathbb C}[t,t^{- 1}])\) be the loop algebra and \(G=\hat L({\mathfrak g})=L({\mathfrak g})\oplus {\mathbb C}c\oplus {\mathbb C}d\) be the corresponding affine algebra with Cartan subalgebra \(H\). Let \({\mathcal I}_{\text{fin}}\) denote the category of integrable \(G\)-modules which, in addition, are weight modules with finite dimensional weight spaces. Also let \(\tilde {\mathcal O}\) denote the category of all the \(G\)-modules \(M\) satisfying: (a) \(M\) is a weight module, (b) there exist finitely many \(\lambda_ 1,\dots,\lambda_ p\in H^*\) such that the set of all the weights of \(M\) is contained in \(\cup \tilde D(\lambda_ i)\), where \(\tilde D(\lambda_ i)=\{\lambda_ i-\eta +n\delta:\eta\in {\dot \Gamma}_+\) and \(n\in {\mathbb Z}\}\) (\(\dot \Gamma_+\) is the non-negative integral linear span of the simple roots of \({\mathfrak g})\), and (c) the centre \(c\) acts trivially on \(M\).
Now let \({\mathfrak h}\) denote the set of all the graded ring homomorphisms \(\Lambda: {\mathfrak G}\to L\), such that \(\text{Image}\;\Lambda={\mathbb C}[t^ r,t^{- r}]\), for some \(r>0\) or \(\text{Image}\;\Lambda={\mathbb C}\), where \({\mathfrak G}=U(T_ 0)/U(T_ 0)c\) and \(T_ 0={\mathbb C}c\oplus_{k\neq 0}G_{k\delta}.\)
Let \(V\in {\mathcal I}_{\text{fin}}\) be irreducible. Then of course \(c\) acts by an integral scalar (say) \(k\). If \(k>0\) (resp. \(k<0)\), then the author shows that \(V\) is a (integrable) highest weight (resp. lowest weight) module and if \(k=0\) then \(V\in \tilde {\mathcal O}\). Now the author studies the irreducible objects in the category \(\tilde {\mathcal O}\) and shows that they are precisely the \(G\)-modules \(V(\lambda,\Lambda)\) (defined in the paper), where \(\lambda \in H^*\) is arbitrary satisfying \(\lambda(c)=0\) and \(\Lambda\in {\mathfrak H}\). She further gives a criterion to decide precisely which of \(V(\lambda,\Lambda)\) are integrable modules.
Reviewer: S.Kumar

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
Full Text: DOI EuDML
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