SIGMA 4 (2008), 081, 23 pages      arXiv:0810.3131      https://doi.org/10.3842/SIGMA.2008.081
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Generating Series for Nested Bethe Vectors

Sergey Khoroshkin ^a and Stanislav Pakuliak ^{b, a
a)} Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia
^b) Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia

Received September 14, 2008; Published online November 24, 2008

Abstract
We reformulate nested relations between off-shell U_q(^gl_N) Bethe vectors as a certain equation on generating series of strings of the composed U_q(^gl_N) currents. Using inversion of the generating series we find a new type of hierarchical relations between universal off-shell Bethe vectors, useful for a derivation of Bethe equation. As an example of application, we use these relations for a derivation of analytical Bethe ansatz equations [Arnaudon D. et al., Ann. Henri Poincaré 7 (2006), 1217-1268, math-ph/0512037] for the parameters of universal Bethe vectors of the algebra U_q(^gl₂).

Key words: Bethe ansatz; current algebras; quantum integrable models.

pdf (384 kb)   ps (252 kb)   tex (28 kb)

References

1. Arnaudon D., Crampè N., Doikou A., Frappat L., Ragoucy E., Spectrum and Bethe ansatz equations for the U_q (gl(N)) closed and open spin chains in any representation, Ann. Henri Poincaré 7 (2006), 1217-1268, math-ph/0512037.
2. Drinfel'd V.G., New realization of Yangians and quantum affine algebras, Soviet Math. Dokl. 36 (1988), 212-216.
3. Ding J.T., Frenkel I.B., Isomorphism of two realizations of quantum affine algebra U_q(^gl_N), Comm. Math. Phys. 156 (1993), 277-300.
4. Ding J., Khoroshkin S., Weyl group extension of quantized current algebras, Transform. Groups 5 (2000), 35-59, math.QA/9804139.
5. Enriquez B., On correlation functions of Drinfeld currents and shuffle algebras, Transform. Groups 5 (2000), 111-120, math.QA/9809036.
6. Enriquez B., Khoroshkin S., Pakuliak S., Weight functions and Drinfeld currents, Comm. Math. Phys. 276 (2007), 691-725, math.QA/0610398.
7. Enriquez B., Rubtsov V., Quasi-Hopf algebras associated with sl₂ and complex curves, Israel J. Math. 112 (1999), 61-108, q-alg/9608005.
8. Frappat L., Khoroshkin S., Pakuliak S., Ragoucy É., Bethe ansatz for the universal weight function, arXiv:0810.3135.
9. Khoroshkin S., Pakuliak S., The weight function for the quantum affine algebra U_q(^sl₃), Theor. and Math. Phys. 145 (2005), 1373-1399, math.QA/0610433.
10. Khoroshkin S., Pakuliak S., Tarasov V., Off-shell Bethe vectors and Drinfeld currents, J. Geom. Phys. 57 (2007), 1713-1732, math.QA/0610517.
11. Khoroshkin S., Pakuliak S., A computation of an universal weight function for the quantum affine algebra U_q(^gl_N), J. Math. Kyoto Univ. 48 (2008), 277-322, arXiv:0711.2819.
12. Kulish P., Reshetikhin N., Diagonalization of GL(N) invariant transfer matrices and quantum N-wave system (Lee model), J. Phys. A: Math. Gen. 16 (1983), L591-L596.
13. Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. 2006 (2006), no. 8, P08002, 44 pages, math.QA/0605015.
14. Oskin A., Pakuliak S., Silantyev A., On the universal weight function for the quantum affine algebra U_q(^gl_N), arXiv:0711.2821.
15. Reshetikhin N., Semenov-Tian-Shansky M., Central extentions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142.
16. Tarasov V., Varchenko A., Jackson integrals for the solutions to Knizhnik-Zamolodchikov equation, St. Petersburg Math. J. 2 (1995), no. 2, 275-313.
17. Tarasov V., Varchenko A., Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), 1-135, q-alg/9703044.
18. Tarasov V., Varchenko A., Combinatorial formulae for nested Bethe vectors, math.QA/0702277.