Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 026, 14 pages      arXiv:0810.3458      https://doi.org/10.3842/SIGMA.2009.026
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Induced Modules for Affine Lie Algebras

Vyacheslav Futorny and Iryna Kashuba
Institute of Mathematics, University of São Paulo, Caixa Postal 66281 CEP 05314-970, São Paulo, Brazil

Received October 20, 2008, in final form March 01, 2009; Published online March 04, 2009

Abstract
We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra P of an affine Lie algebra G, our main result establishes the equivalence between a certain category of P-induced G-modules and the category of weight P-modules with injective action of the central element of G. In particular, the induction functor preserves irreducible modules. If P is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra Pps, P Ì Pps. The structure of P-induced modules in this case is fully determined by the structure of Pps-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. König, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63].

Key words: affine Kac-Moody algebras; induced modules; parabolic subalgebras; Borel subalgebras.

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