Induced Modules for Affine Lie Algebras

Vyacheslav Futorny ; Iryna Kashuba

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra $\mathcal P$ of an affine Lie algeba $\mathfrak G$, our main result establishes the equivalence between a certain category of $\mathcal P$-induced $\mathfrak G$-modules and the category of weight $\mathcal P$-modules with injective action of the central element of $\mathfrak G$. In particular, the induction functor preserves irreducible modules. If $\mathcal P$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra $\mathcal P^{ps}$, $\mathcal P\subset\mathcal P^{ps}$. The structure of $\mathcal P$-induced modules in this case is fully determined by the structure of $\mathcal P^{ps}$-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].

Détail

Keywords: affine Kac–Moody algebras; induced modules; parabolic subalgebras; Borel subalgebras.

Bibliographie

[1] Bekkert V., Benkart G., Futorny V., “Weyl algebra modules”, Kac–Moody Lie Algebras and Related Topics, Contemp. Math., 343, 2004, 17–42 | MR | Zbl

[2] Chari V., Pressley A., “New unitary representations of loop groups”, Math. Ann., 275 (1986), 87–104 | DOI | MR | Zbl

[3] Chari V., “Integrable representations of affine Lie algebras”, Invent. Math., 85 (1986), 317–335 | DOI | MR | Zbl

[4] Cox B., “Verma modules induced from nonstandard Borel subalgebras”, Pacific J. Math., 165 (1994), 269–294 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[8] Dimitrov I., Penkov I., “Partially integrable highest weight modules”, Transform. Groups, 3 (1998), 241–253 | DOI | MR | Zbl

[9] Fernando S., “Lie algebra modules with finite-dimensional weight spaces. I”, Trans. Amer. Math. Soc., 322 (1990), 757–781 | DOI | MR | Zbl

[10] Futorny V., “The weight representations of semisimple finite-dimensional Lie algebras”, 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, 106, Queen's University, Kingston, ON, 1997 | MR | Zbl

[12] Futorny V., “Irreducible non-dense $A_1^{(1)}$-modules”, Pacific J. Math., 172 (1996), 83–99 | MR | Zbl

[13] Futorny V., “The parabolic subsets of root systems and corresponding representations of affine Lie algebras”, Proceedings of the International Conference on Algebra (Novosibirsk, 1989), Contemp. Math., 131, no. 2, 1992, 45–52 | MR | Zbl

[14] Futorny V., “Imaginary Verma modules for affine Lie algebras”, Canad. Math. Bull., 37 (1994), 213–218 | MR | Zbl

[15] Futorny V., König S., Mazorchuk V., “Categories of induced modules for Lie algebras with triangular decomposition”, Forum Math., 13 (2001), 641–661 | DOI | MR | Zbl

[16] Futorny V., König S., Mazorchuk V., “Categories of induced modules and projectively stratified algebras”, Algebr. Represent. Theory, 5 (2002), 259–276 | DOI | MR | Zbl

[17] Futorny V., Saifi H., “Modules of Verma type and new irreducible representations for affine Lie algebras”, Representations of Algebras (Ottawa, ON, 1992), CMS Conf. Proc., 14, Amer. Math. Soc., Providence, RI, 1993, 185–191 | MR | Zbl

[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 | DOI | MR | Zbl

[19] Jakobsen H. P., Kac V. G., “A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras”, Nonlinear Equations in Classical and Quantum Field Theory (CMeudon/Paris, 1983/1984), Lecture Notes in Phys., 226, Springer, Berlin, 1985, 1–20 | MR

[20] Kac V. G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990 | MR

[21] Mathieu O., “Classification of irreducible weight modules”, Ann. Inst. Fourier (Grenoble), 50 (2000), 537–592 | MR | Zbl

[22] Spirin S., “$\mathbb Z^2$-graded modules with one-dimensional components over the Lie algebra $A_1^{(1)}$”, Funktsional. Anal. i Prilozhen., 21:1 (1987), 84–85 | MR | Zbl