zbMATH — the first resource for mathematics

The argument shift method and the Gaudin model. (English. Russian original) Zbl 1112.17018
Funct. Anal. Appl. 40, No. 3, 188-199 (2006); translation from Funkts. Anal. Prilozh. 40, No. 3, 30-43 (2005).
Summary: We construct a family of maximal commutative subalgebras in the tensor product of \(n\) copies of the universal enveloping algebra \(U(\mathfrak{g})\) of a semisimple Lie algebra \(\mathfrak{g}\) . This family is parameterized by finite sequences \(\mu\) , \(z_1,\ldots,z_n\), where \(\mu \in \mathfrak{g}^*\) and \(z_i \in \mathbb C\). The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For \(n = 1\), the corresponding commutative subalgebras in the Poisson algebra \(S(\mathfrak{g})\) were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional \(\mathfrak{g}\)-modules and the Gaudin model.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B63 Poisson algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
PDF BibTeX Cite
Full Text: DOI
[1] A. V. Bolsinov and A. V. Borisov, ”Compatible Poisson brackets on Lie algebras,” Mat. Zametki, 72:1 (2002), 11–34; English transl.: Math. Notes, 72:1–2 (2002), 10–30. · Zbl 1042.37041
[2] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint, http://www.ma.utexas.edu/enzvi/BD. · Zbl 0864.14007
[3] B. Enriquez and V. Rubtsov, ”Hitchin systems, higher Gaudin Hamiltonians and r-matrices,” Math. Res. Lett., 3:3 (1996), 343–357; http://arxiv.org/abs/alg-geom/9503010. · Zbl 0871.58038
[4] B. Feigin and E. Frenkel, ”Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras,” Int. Jour. Mod. Phys., A7, Supplement 1A (1992), 197–215. · Zbl 0925.17022
[5] E. Frenkel, ”Affine Algebras, Langlands Duality and Bethe Ansatz,” in: XIth International Congress of Mathematical Physics (Paris, 1994), Internat. Press, Cambridge, MA, 1995, 606–642; http: //arxiv.org/q-alg/9506003. · Zbl 1052.17504
[6] E. Frenkel, ”Lectures on Wakimoto modules, opers and the center at the critical level,” http://arxiv.org/math.QA/0210029 . · Zbl 1129.17014
[7] B. Feigin, E. Frenkel, and N. Reshetikhin, ”Gaudin model, Bethe Ansatz and critical level,” Comm. Math. Phys., 166 (1994), 27–62; http://arxiv.org/hep-th/9402022 . · Zbl 0812.35103
[8] M. Gaudin, ”Diagonalisation d’une classe d’hamiltoniens de spin,” J. Physique, 37:10 (1976), 1087–1098.
[9] M. Gaudin, La fonction d’onde de Bethe. Collection du Commissariat a’ l’Énergie Atomique: Série Scientifique, Masson, Paris, 1983.
[10] A. S. Mishchenko and A. T. Fomenko, ”Integrability of Euler’s equations on semisimple Lie algebras, ” Trudy Sem. Vektor. Tenzor. Anal., 19 (1979), 3–94. · Zbl 0452.58015
[11] A. I. Molev, ”Yangians and their applications,” in: Handbook of algebra, vol. 3, North-Holland, Amsterdam, 2003, 907–959; http://arxiv.org/math.QA/0211288. · Zbl 1086.17008
[12] E. Mukhin and A. Varchenko, Norm of a Bethe vector and the Hessian of the master function, Preprint, http: //arxiv.org/math.QA/0402349. · Zbl 1072.82012
[13] M. Mustata, ”Jet schemes of locally complete intersection canonical singularities (with an appendix by D. Eisenbud and E. Frenkel),” Invent. Math., 145 (2001), 397–424; http: //arxiv.org/math.AG/0008002. · Zbl 1091.14004
[14] M. Nazarov and G. Olshanski, ”Bethe Subalgebras in Twisted Yangians,” Comm. Math. Phys., 178:2 (1996), 483–506; http://arxiv.org/q-alg/9507003. · Zbl 0876.17015
[15] V. V. Shuvalov, ”On the limits of Mishchenko-Fomenko subalgebras in Poisson algebras of semisimple Lie algebras,” Funkts. Anal. Prilozhen., 36:4 (2002), 55–64; English transl.: Funct. Anal. Appl., 36:4 (2002), 298–305. · Zbl 1022.17016
[16] A. A. Tarasov, ”On some commutative subalgebras in the universal enveloping algebra of the Lie algebra gl(n, C),” Mat. Sb., 191:9 (2000), 115–122; English transl.: Russian Acad. Sci. Sb. Math., 191:9–10 (2000), 1375–1382. · Zbl 0985.17012
[17] A. A. Tarasov, ”The maximality of some commutative subalgebras in Poisson algebras of semisimple Lie algebras,” Uspekhi Mat. Nauk, 57:5 (347) (2002), 165–166; English transl.: Russian Math. Surveys, 57:5 (2002), 1013–1014. · Zbl 1077.17019
[18] A. A. Tarasov, ”On the uniqueness of the lifting of maximal commutative subalgebras of the Poisson-Lie algebra to the enveloping algebra,” Mat. Sb., 194:7 (2003), 155–160; English transl.: Russian Acad. Sci. Sb. Math., 194:7–8 (2003), 1105–1111. · Zbl 1064.17005
[19] E. Vinberg, ”On some commutative subalgebras in universal enveloping algebra,” Izv. AN USSR, Ser. Mat., 54:1 (1990), 3–25.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.