String Functions for Affine Lie Algebras Integrable Modules

Petr Kulish ; Vladimir Lyakhovsky

Symmetry, integrability and geometry: methods and applications, Tome 4 (2008)

Résumé

The recursion relations of branching coefficients $k_{\xi}^{(\mu)}$ for a module $L_{\mathfrak g\downarrow\mathfrak h}^\mu$ reduced to a Cartan subalgebra $\mathfrak h$ are transformed in order to place the recursion shifts $\gamma\in\Gamma _{\mathfrak a\subset\mathfrak h}$ into the fundamental Weyl chamber. The new ensembles $F\Psi$ (the “folded fans”) of shifts were constructed and the corresponding recursion properties for the weights belonging to the fundamental Weyl chamber were formulated. Being considered simultaneously for the set of string functions (corresponding to the same congruence class $\Xi_{v}$ of modules) the system of recursion relations constitute an equation $\mathbf M_{(u)}^{\Xi _v}\mathbf{m}_{(u)}^{\mu}={\boldsymbol\delta}_{(u)}^{\mu}$ where the operator $\mathbf M_{(u)}^{\Xi _v}$ is an invertible matrix whose elements are defined by the coordinates and multiplicities of the shift weights in the folded fans $F\Psi$ and the components of the vector $\mathbf m_{(u)}^\mu$ are the string function coefficients for $L^\mu$ enlisted up to an arbitrary fixed grade $u$. The examples are presented where the string functions for modules of $\mathfrak g=A_2^{(1)}$ are explicitly constructed demonstrating that the set of folded fans provides a compact and effective tool to study the integrable highest weight modules.

Détail

Keywords: affine Lie algebras; integrable modules; string functions.

Bibliographie

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

[2] Faddeev L. D., “How the algebraic Bethe ansatz works for integrable models”, Symetries Quantique, Proc. Les Houches Summer School (Les Houches, 1995), eds. A. Connes, K. Gawedski, J. Zinn-Justin, North-Holland, 1998, 149–219 ; 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., 2004:10 (2004), 060, 23 pp., ages ; hep-th/0410105 | DOI | MR

[4] Beisert N., “The dilatation operator of $N=4$ super Yang–Mills theory and integrability”, Phys. Rep., 405 (2005), 1–202 ; hep-th/0407277 | DOI | MR

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

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

[7] Wakimoto M., Infinite-dimensional Lie algebras, Translations of Mathematical Monographs, 195, American Mathematical Society, Providence, RI, 2001 | MR | Zbl

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

[9] Hwang S., Rhedin H., “General branching functions of affine Lie algebras”, Modern Phys. Lett. A, 10 (1995), 823–830 ; hep-th/9408087 | DOI | MR | Zbl

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

[11] Ilyin M., Kulish P., Lyakhovsky V., “On the properties of branching coefficients for affine Lie groups”, Algebra i Analiz, 21:2 (2009), 52–70 ; arXiv:0812.2124 | MR

[12] Wakimoto M., Lectures on infinite-dimensional Lie algebra, World Scientific Publishing Co., Inc., River Edge, NJ, 2001 | MR | Zbl

[13] Fulton W., Harris J., Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991 | MR | Zbl

[14] Lyakhovsky V. D., Recurrent properties of affine Lie algebra representations, Supersymmetry and Quantum Symmetry (August, 2007, Dubna)