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Irreducible representations for toroidal Lie algebras. (English) Zbl 0942.17016
A toroidal Lie algebra is a Lie algebra $${\mathfrak g}$$ which has the structure ${\mathfrak g} = \dot{\mathfrak g} \otimes \mathbb C[t_0^{\pm 1}, \ldots, t_n^{\pm 1}] \oplus {\mathcal K} \oplus {\mathcal D},$ where $$\dot{\mathfrak g}$$ is a finite-dimensional simple complex Lie algebra, $${\mathcal K}$$ is central and $${\mathcal D}$$ is a certain Lie algebra of derivations of $$\mathbb C[t_0^{\pm 1}, \ldots, t_n^{\pm 1}]$$. The main purpose of the paper under review is the construction of irreducible representations of toroidal Lie algebras with finite-dimensional weight spaces with respect to certain natural gradings. The techniques to obtain these results combine the previously used vertex operator techniques with classical Verma module techniques to a new construction process satisfying all requirements.
More precisely, the construction starts with an irreducible module $$W$$ of $${\mathfrak {gl}}(n,\mathbb C)$$. Then $$T(W) := \mathbb C[t_1^{\pm 1}, \ldots, t_n^{\pm 1}] \otimes W$$ is a module of $$\text{Der}(\mathbb C[t_1^{\pm 1}, \ldots, t_n^{\pm 1}])$$ and, in addition, of the zero part $${\mathfrak v}_0$$ of a certain $$\mathbb Z$$-graded Lie algebra $${\mathfrak v}$$ with decomposition $${\mathfrak v} = {\mathfrak v}_+ \oplus {\mathfrak v}_0 \oplus {\mathfrak v}_-$$. Then we pass to the induced module $$M := \text{Ind}_{{\mathfrak v}_0 + {\mathfrak v}_+}^{\mathfrak v}(T(W))$$ and further to an $${\mathfrak v}$$-irreducible quotient module $$L$$ of $$M$$. In the final step an irreducible highest weight module of an affine Lie algebra $$\dot{\mathfrak g} \otimes \mathbb C[t_0, t_0^{-1}] \oplus \mathbb C k_0 \oplus \mathbb C d_0$$, $$k_0 \in {\mathcal K}$$, $$d_0 \in {\mathcal D}$$ is tensored with $$L$$, which eventually leads to a module of $${\mathfrak g}$$. It is remarkable that the results leading to finite-dimensionality of certain weight spaces work in amazing generality for the class of $$\mathbb Z$$-graded Lie algebras with polynomial multiplication introduced in Section I. Later these results are specialized to the concrete setting described above.

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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