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Local projective resolutions and translation functors for Kac-Moody algebras. (English) Zbl 0644.17012

In this paper the translation functors are generalized to the case of Kac-Moody algebras, not necessarily defined by a symmetrizable generalized Cartan matrix. These functors are used to show that the multiplicities of irreducible composition factors in Verma modules are independent of the dominant integral weight.
After the basic definitions about a Kac-Moody algebra \(\mathfrak g\), so-called local projective resolutions are introduced and their natural properties are proved. Define the category \(C(\lambda)\), \(\lambda\in\mathfrak h^*\) (\(\mathfrak h\) is the Cartan subalgebra of \(\mathfrak g)\), as the full subcategory of the category of \(\mathfrak g\)-modules \(M\) such that \(M\) is a weight module and \(\mu\leq \lambda\) for each weight \(\mu\) of \(M\). Studying the structure of filtered modules, whose factors are Verma modules, the author introduces certain functors on \(C(\lambda)\) which make it possible to define translation functors in Section 7. In the same Section the multiplicity property is proved.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B55 Homological methods in Lie (super)algebras
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