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

SIGMA 5 (2009), 026, 14 pages      arXiv:0810.3458
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

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 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.

pdf (265 kb)   ps (183 kb)   tex (17 kb)


  1. Bekkert V., Benkart G., Futorny V., Weyl algebra modules, Kac-Moody Lie Algebras and Related Topics, Contemp. Math. 343 (2004), 17-42.
  2. Chari V., Pressley A., New unitary representations of loop groups, Math. Ann. 275 (1986), 87-104.
  3. Chari V., Integrable representations of affine Lie algebras, Invent. Math. 85 (1986), 317-335.
  4. Cox B., Verma modules induced from nonstandard Borel subalgebras, Pacific J. Math. 165 (1994), 269-294.
  5. Dimitrov I., Futorny V., Penkov I., A reduction theorem for highest weight modules over toroidal Lie algebras, Comm. Math. Phys. 250 (2004), 47-63.
  6. Dimitrov I., Grantcharov D., Private communication.
  7. Dimitrov I., Mathieu O., Penkov I., On the structure of weight modules, Trans. Amer. Math. Soc. 352 (2000), 2857-2869.
  8. Dimitrov I., Penkov I., Partially integrable highest weight modules, Transform. Groups 3 (1998), 241-253.
  9. Fernando S., Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc. 322 (1990), 757-781.
  10. Futorny V., The weight representations of semisimple finite-dimensional Lie algebras, in Algebraic Structures and Applications, Kiev University, 1988, 142-155.
  11. Futorny V., Representations of affine Lie algebras, Queen's Papers in Pure and Applied Mathematics, Vol. 106, Queen's University, Kingston, ON, 1997.
  12. Futorny V., Irreducible non-dense A1(1)-modules, Pacific J. Math. 172 (1996), 83-99.
  13. Futorny V., The parabolic subsets of root systems and corresponding representations of affine Lie algebras, in Proceedings of the International Conference on Algebra (Novosibirsk, 1989), Contemp. Math. 131 (1992), part 2, 45-52.
  14. Futorny V., Imaginary Verma modules for affine Lie algebras, Canad. Math. Bull. 37 (1994), 213-218.
  15. Futorny V., König S., Mazorchuk V., Categories of induced modules for Lie algebras with triangular decomposition, Forum Math. 13 (2001), 641-661.
  16. Futorny V., König S., Mazorchuk V., Categories of induced modules and projectively stratified algebras, Algebr. Represent. Theory 5 (2002), 259-276.
  17. Futorny V., Saifi H., Modules of Verma type and new irreducible representations for affine Lie algebras, in Representations of Algebras (Ottawa, ON, 1992), CMS Conf. Proc., Vol. 14, Amer. Math. Soc., Providence, RI, 1993, 185-191.
  18. Futorny V., Tsylke A., Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras, J. Algebra 238 (2001), 426-441.
  19. Jakobsen H.P., Kac V.G., A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras, in Nonlinear Equations in Classical and Quantum Field Theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., Vol. 226, Springer, Berlin, 1985, 1-20.
  20. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  21. Mathieu O., Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537-592.
  22. Spirin S., Z2-graded modules with one-dimensional components over the Lie algebra A1(1), Funktsional. Anal. i Prilozhen. 21 (1987), 84-85.

Previous article   Next article   Contents of Volume 5 (2009)