On the properties of branching coefficients for affine Lie groups

M. Ilyin ; P. Kulish ; V. Lyakhovsky

Algebra i analiz, Tome 21 (2009) no. 2, pp. 52-70.

Résumé

It is demonstrated that the decompositions of integrable highest weight modules of a simple Lie algebra (classical or affine) with respect to its reductive subalgebra obey a set of algebraic relations leading to recursive properties for the corresponding branching coefficients. These properties are encoded in a special element $\Gamma _{\mathfrak{g}\supset\mathfrak{a}}$ of the formal algebra $\mathcal{E}_{\mathfrak{a}}$ that describes the injections $\mathfrak{a}\to \mathfrak{g}$ and is called a fan. In the simplest case where $\mathfrak{a}=\mathfrak{h}\left(\mathfrak{g}\right)$, the recursion procedure generates the weight diagram of a module $L_{\mathfrak{g}}$. When the recursion described by a fan is applied to highest weight modules, it provides a highly efficient tool for explicit calculations of branching coefficients.

Détail

Keywords: integrable highest weight modules, simple Lie algebra, reductive subalgebra, branching coefficients, fan, weight diagram.

Bibliographie

[1] Date E., Jimbo M., Kuniba A., Miwa T., Okado M., One-dimensional configuration sums in vertex models and affine Lie algebra characters, Preprint RIMS-631, 1988; Lett. Math. Phys., 17:1 (1989), 69–77 | DOI | MR | Zbl

[2] Faddeev L. D., “How the algebraic Bethe ansatz works for integrable models”, Quantum Symmetries/Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 149–219 ; arXiv: hep-th/9605187 | MR | Zbl

[3] Kazakov V. A., Zarembo K., “Classical/quantum integrability in non-compact sector of AdS/CFT”, J. High Energy Phys., 2004, no. 10, 060, 23 pp., (electronic) | DOI | MR

[4] Beisert N., “The dilatation operator of $N=4$ super Yang–Mills theory and integrability”, Phys. Rep., 405 (2004), 1–202 ; arXiv: hep-th/0407277 | DOI | MR

[5] Kats V., Beskonechnomernye algebry Li, Mir, M., 1993 | MR | Zbl

[6] Wakimoto M., Infinite-dimensional Lie algebras, Transl. Math. Monogr., 195, Amer. Math. Soc., Providnece, RI, 2001 | MR | Zbl

[7] Bernstein I. N., Gel'fand I. M., Gel'fand S. I., “Differential operators on the basic affine space and a study of $\gamma$-modules”, Lie Groups and their Representations, Proc. Summer School (Bolyai János Math. Soc., Budapest, 1971), Halsted Press, New York, 1975, 21–64 | MR

[8] Fauser B., Jarvis P. D., King R. C., Wybourn B. G., New branching rules induced by plethysm, , 2003 arXiv: math-ph/0505037 | MR | Zbl

[9] Hwang S., Rhedin H., General branching functions of affine Lie algebras, , 1994 arXiv: hep-th/9408087 | MR

[10] Feigin B., Feigin E., Jimbo M., Miwa T., Mukhin E., Principal $\widehat{\frak{sl_3}}$ subspaces and quantum Toda Hamiltonians, , 2007 arXiv: 0707.1635v2

[11] Lyakhovsky V. D., Melnikov S. Yu., “Recursion relations and branching rules for simple Lie algebras”, J. Phys. A, 29 (1996), 1075–1087 | DOI | MR | Zbl

[12] Lyakhovsky V. D., “Recurrent properties of affine Lie algebra representations”, Supersymmetry and Quantum Symmetry (Dubna) (to appear)

[13] Bourbaki N., Éléments de mathématique. Fasc. 38. Groupes et algèbres de Lie, Actualités Sci. Indust., 1364, Hermann, Paris, 1975 | MR | Zbl