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Kostant’s formula for a certian class of generalized Kac-Moody algebras. (English) Zbl 0767.17020

Let \({\mathfrak g}(A)\) be the generalized Kac-Moody algebra associated to a symmetrizable GGCM \(A=(a_{ij})_{i,j\in I}\) over the complex number field \(\mathbb{C}\), with \({\mathfrak h}\) its Cartan subalgebra, \(\Delta\) the root system of \(({\mathfrak g}(A),{\mathfrak h})\), and \({\mathfrak n}_ - =\sum_{\alpha\in\Delta_ +} {\mathfrak g}_{-\alpha}\) the sum of root spaces corresponding to all negative roots. Let \(L(\Lambda)\) be the irreducible highest weight \({\mathfrak g}(A)\)-module with dominant integral highest weight \(\Lambda\). In this paper, we derive an extension of Kostant’s homology theorem to symmetrizable generalized Kac-Moody algebras under the condition \(a_{ii}=2\), or \(a_{ii}=0\) (\(i\in I\)) on the Cartan matrix \(A=(a_{ij})_{i,j\in I}\), by using the Weyl-Kac- Borcherds character formula. This theorem determines the \({\mathfrak h}\)- module structure of the Lie algebra homology \(H_ i({\mathfrak n}_ - ,L(\Lambda))\) (\(i\geq 0\)) of \({\mathfrak n}_ -\) with coefficients in \(L(\Lambda)\). This paper has cast a new light upon the relations between Kostant’s homology theorem and Weyl-Kac-Morcherds character formula.
Reviewer: S.Naito (Shizuoka)

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
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