Yangian characters and classical \(\mathcal W\)-algebras. (English) Zbl 1329.17025
Kohnen, Winfried (ed.) et al., Conformal field theory, automorphic forms and related topics. CFT 2011, Heidelberg, Germany, September 19–23, 2011. Berlin: Springer (ISBN 978-3-662-43830-5/hbk; 978-3-662-43831-2/ebook). Contributions in Mathematical and Computational Sciences 8, 287-334 (2014).
Let \(\mathfrak{g}\) be a simple finite dimensional complex Lie algebra and \(\hat{\mathfrak{g}}\) the corresponding affine Lie algebra. An appropriate version of the Harish-Chandra homomorphism for \(\hat{\mathfrak{g}}\) induces an isomorphism between the Feigin-Frenkel center of the vertex algebra associated with a certain vacuum \(\hat{\mathfrak{g}}\)-module and the classical \(\mathcal{W}\)-algebra for the Langlands dual of \(\mathfrak{g}\).
Explicit generators of the Feigin-Frenkel center are known. In the paper under review, the authors describe the Harish-Chandra images of certain generators of these generators in types \(B\), \(C\) and \(D\). These images turn out to be elements of the \(\mathcal{W}\)-algebra written in terms of non-commutative analogues of the complete and elementary symmetric functions.
For the entire collection see [Zbl 1297.00041].
Explicit generators of the Feigin-Frenkel center are known. In the paper under review, the authors describe the Harish-Chandra images of certain generators of these generators in types \(B\), \(C\) and \(D\). These images turn out to be elements of the \(\mathcal{W}\)-algebra written in terms of non-commutative analogues of the complete and elementary symmetric functions.
For the entire collection see [Zbl 1297.00041].
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 17B68 | Virasoro and related algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Keywords:
affine vertex algebra; Kirillov-Reshetikhin module; Feigin-Frenkel center; Harish-Chandra isomorphism; Yangian; character; classical \(\mathcal{W}\)-algebraReferences:
| [1] | Arnaudon, D.; Avan, J.; Crampé, N.; Frappat, L.; Ragoucy, E., R-matrix presentation for super-Yangians Y (osp(m | 2n)), J. Math. Phys., 44, 302-308 (2003) · Zbl 1061.17014 · doi:10.1063/1.1525406 |
| [2] | Arnaudon, D.; Molev, A.; Ragoucy, E., On the R-matrix realization of Yangians and their representations, Annales Henri Poincaré, 7, 1269-1325 (2006) · Zbl 1227.17008 · doi:10.1007/s00023-006-0281-9 |
| [3] | Brundan, J., Kleshchev, A.: Representations of shifted Yangians and finite W-algebras. Mem. Am. Math. Soc. 196(918) (2008) · Zbl 1169.17009 |
| [4] | Chervov, A.V., Molev, A.I.: On higher order Sugawara operators. Int. Math. Res. Not. 2009, no. 9, 1612-1635 · Zbl 1225.17031 |
| [5] | Chervov, A., Talalaev, D.: Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence. arXiv:hep-th/0604128 · Zbl 1179.82052 |
| [6] | Chervov, A.; Falqui, G.; Rubtsov, V., Algebraic properties of Manin matrices 1, Adv. Appl. Math., 43, 239-315 (2009) · Zbl 1230.05043 · doi:10.1016/j.aam.2009.02.003 |
| [7] | Dixmier, J., Algèbres Enveloppantes (1974), Paris: Gauthier-Villars, Paris · Zbl 0308.17007 |
| [8] | Drinfeld, V.G.: Quantum groups. In: International Congress of Mathematicians (Berkeley, 1986), pp. 798-820. American Mathematical Society, Providence (1987) · Zbl 0667.16003 |
| [9] | Drinfeld, V. G.; Sokolov, V. V., Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math., 30, 1975-2036 (1985) · Zbl 0578.58040 · doi:10.1007/BF02105860 |
| [10] | Feigin, B.; Frenkel, E., Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Int. J. Mod. Phys. A, 7, suppl. 1A, 197-215 (1992) · Zbl 0925.17022 · doi:10.1142/S0217751X92003781 |
| [11] | Frenkel, E.: Langlands Correspondence for Loop Groups. Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007) · Zbl 1133.22009 |
| [12] | Frenkel, E.; Mukhin, E., Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Commun. Math. Phys., 216, 23-57 (2001) · Zbl 1051.17013 · doi:10.1007/s002200000323 |
| [13] | Frenkel, E., Mukhin, E.: The Hopf algebra \(\text{Rep}\,U_q\widehat{\mathfrak{g}\mathfrak{l}}_{\infty } \) . Selecta Math. 8, 537-635 (2002) · Zbl 1034.17009 |
| [14] | Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of W-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, 1998). Contemporary Mathematics, vol. 248, pp. 163-205. American Mathematical Society, Providence (1999) · Zbl 0973.17015 |
| [15] | Hernandez, D., The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math., 596, 63-87 (2006) · Zbl 1160.17010 |
| [16] | Iorgov, N.; Molev, A. I.; Ragoucy, E., Casimir elements from the Brauer-Schur-Weyl duality, J. Algebra, 387, 144-159 (2013) · Zbl 1355.17008 · doi:10.1016/j.jalgebra.2013.02.041 |
| [17] | Isaev, A.P., Molev, A.I., Ogievetsky, O.V.: A new fusion procedure for the Brauer algebra and evaluation homomorphisms. Int. Math. Res. Not. 2012, no. 11, 2571-2606 · Zbl 1276.20006 |
| [18] | Knight, H., Spectra of tensor products of finite-dimensional representations of Yangians, J. Algebra, 174, 187-196 (1995) · Zbl 0868.17009 · doi:10.1006/jabr.1995.1123 |
| [19] | Kuniba, A.; Suzuki, J., Analytic Bethe ansatz for fundamental representations of Yangians, Commun. Math. Phys., 173, 225-264 (1995) · Zbl 0834.58045 · doi:10.1007/BF02101234 |
| [20] | Kuniba, A.; Ohta, Y.; Suzuki, J., Quantum Jacobi-Trudi and Giambelli formulae for U_q(B_r^(1)) from the analytic Bethe ansatz, J. Phys. A, 28, 6211-6226 (1995) · Zbl 0871.17015 · doi:10.1088/0305-4470/28/21/024 |
| [21] | Kuniba, A.; Okado, M.; Suzuki, J.; Yamada, Y., Difference L operators related to q-characters, J. Phys. A, 35, 1415-1435 (2002) · Zbl 1012.17009 · doi:10.1088/0305-4470/35/6/307 |
| [22] | Kuniba, A.; Nakanishi, T.; Suzuki, J., T-systems and Y -systems in integrable systems, J. Phys. A, 44, 10, 103001 (2011) · Zbl 1222.82041 · doi:10.1088/1751-8113/44/10/103001 |
| [23] | Molev, A.: Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs, vol. 143. American Mathematical Society, Providence (2007) · Zbl 1141.17001 |
| [24] | Molev, A.I.: Feigin-Frenkel center in types B, C and D. Invent. Math. 191, 1-34 (2013) · Zbl 1266.17016 |
| [25] | Molev, A. I.; Ragoucy, E., The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for \(\widehat{\mathfrak{g}\mathfrak{l}}_{m\vert n} \), Moscow Math. J., 14, 83-119 (2014) · Zbl 1342.17016 |
| [26] | Mukhin, E.; Young, C. A.S., Path description of type Bq-characters., Adv. Math., 231, 1119-1150 (2012) · Zbl 1323.17015 · doi:10.1016/j.aim.2012.06.012 |
| [27] | Mukhin, E., Young, C.A.S.: Extended T-systems. Selecta Math. (N.S.) 18, 591-631 (2012) · Zbl 1326.17014 |
| [28] | Mukhin, E., Tarasov, V., Varchenko, A.: Bethe eigenvectors of higher transfer matrices. J. Stat. Mech. Theory Exp. 2006, no. 8, P08002, 44 pp · Zbl 1456.82301 |
| [29] | Mukhin, E., Tarasov, V., Varchenko, A.: Bethe algebra of Gaudin model, Calogero-Moser space and Cherednik algebra. Int. Math. Res. Not. 2014, no. 5, 1174-1204 · Zbl 1304.82021 |
| [30] | Nakai, W.; Nakanishi, T., Paths, tableaux and q-characters of quantum affine algebras: the C_n case, J. Phys. A, 39, 2083-2115 (2006) · Zbl 1085.17011 · doi:10.1088/0305-4470/39/9/007 |
| [31] | Nakai, W.; Nakanishi, T., Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type D_n, J. Algebraic Combin., 26, 253-290 (2007) · Zbl 1171.17004 · doi:10.1007/s10801-007-0057-4 |
| [32] | Nakajima, H., t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent, Theory, 7, 259-274 (2003) · Zbl 1078.17008 |
| [33] | Nazarov, M. L., Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys., 21, 123-131 (1991) · Zbl 0722.17004 · doi:10.1007/BF00401646 |
| [34] | Okounkov, A., Quantum immanants and higher Capelli identities, Transform. Groups, 1, 99-126 (1996) · Zbl 0864.17014 · doi:10.1007/BF02587738 |
| [35] | Reshetikhin, N. Yu.; Takhtajan, L. A.; Faddeev, L. D., Quantization of Lie groups and Lie algebras, Leningrad Math. J., 1, 193-225 (1990) · Zbl 0715.17015 |
| [36] | Talalaev, D. V., The quantum Gaudin system, Funct. Anal. Appl., 40, 73-77 (2006) · Zbl 1111.82015 · doi:10.1007/s10688-006-0012-5 |
| [37] | Zamolodchikov, A.B., Zamolodchikov, Al.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253-291 (1979) |
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