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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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