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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

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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##### References:
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