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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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