### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 043, 31 pages      arXiv:1702.08710      https://doi.org/10.3842/SIGMA.2017.043
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

### Highest $\ell$-Weight Representations and Functional Relations

Khazret S. Nirov ab and Alexander V. Razumov c
a) Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Ave. 7a, 117312 Moscow, Russia
b) Mathematics and Natural Sciences, University of Wuppertal, 42097 Wuppertal, Germany
c) Institute for High Energy Physics, NRC ''Kurchatov Institute'', 142281 Protvino, Moscow region, Russia

Received March 01, 2017, in final form June 06, 2017; Published online June 17, 2017; Misprints corrected August 17, 2017

Abstract
We discuss highest $\ell$-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and $q$-oscillator representations of the positive Borel subalgebras of the quantum group $\mathrm{U}_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the $L$-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.

Key words: quantum loop algebras; Verma modules; highest $\ell$-weight representations; $q$-oscillators.

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References

1. Asherova R.M., Smirnov Yu.F., Tolstoy V.N., Description of a class of projection operators for semisimple complex Lie algebras, Math. Notes 26 (1979), 499-504.
2. Bazhanov V.V., Hibberd A.N., Khoroshkin S.M., Integrable structure of ${\mathcal W}_3$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory, Nuclear Phys. B 622 (2002), 475-547, hep-th/0105177.
3. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Comm. Math. Phys. 177 (1996), 381-398, hep-th/9412229.
4. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory. II. ${\rm Q}$-operator and DDV equation, Comm. Math. Phys. 190 (1997), 247-278, hep-th/9604044.
5. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory. III. The Yang-Baxter relation, Comm. Math. Phys. 200 (1999), 297-324, hep-th/9805008.
6. Beck J., Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555-568, hep-th/9404165.
7. Boos H., Göhmann F., Klümper A., Nirov Kh.S., Razumov A.V., Exercises with the universal $R$-matrix, J. Phys. A: Math. Theor. 43 (2010), 415208, 35 pages, arXiv:1004.5342.
8. Boos H., Göhmann F., Klümper A., Nirov Kh.S., Razumov A.V., Universal integrability objects, Theoret. and Math. Phys. 174 (2013), 21-39, arXiv:1205.4399.
9. Boos H., Göhmann F., Klümper A., Nirov Kh.S., Razumov A.V., Quantum groups and functional relations for higher rank, J. Phys. A: Math. Theor. 47 (2014), 275201, 47 pages, arXiv:1312.2484.
10. Boos H., Göhmann F., Klümper A., Nirov Kh.S., Razumov A.V., Universal $R$-matrix and functional relations, Rev. Math. Phys. 26 (2014), 1430005, 66 pages, arXiv:1205.1631.
11. Boos H., Göhmann F., Klümper A., Nirov Kh.S., Razumov A.V., Oscillator versus prefundamental representations, J. Math. Phys. 57 (2016), 111702, 23 pages, arXiv:1512.04446.
12. Boos H., Göhmann F., Klümper A., Nirov Kh.S., Razumov A.V., Oscillator versus prefundamental representations. II. Arbitrary higher ranks, arXiv:1701.0262.
13. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261-283.
14. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
15. Damiani I., Drinfeld realization of affine quantum algebras: the relations, Publ. Res. Inst. Math. Sci. 48 (2012), 661-733, arXiv:1406.6729.
16. Damiani I., From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity, Publ. Res. Inst. Math. Sci. 51 (2015), 131-171, arXiv:1407.0341.
17. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985), 254-258.
18. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
19. Drinfel'd V.G., A new realization of Yangians and of quantum affine algebras, Sov. Math. Dokl. 36 (1988), 212-216.
20. Etingof P.I., Frenkel I.B., Kirillov Jr. A.A., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs, Vol. 58, Amer. Math. Soc., Providence, RI, 1998.
21. Feigin B., Jimbo M., Miwa T., Mukhin E., Finite type modules and Bethe ansatz equations, arXiv:1609.05724.
22. Frenkel E., Hernandez D., Baxter's relations and spectra of quantum integrable models, Duke Math. J. 164 (2015), 2407-2460, arXiv:1308.3444.
23. Frenkel E., Reshetikhin N., The $q$-characters of representations of quantum affine algebras and deformations of $\mathcal W$-algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 163-205, math.QA/9810055.
24. Hernandez D., Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), 163-200, math.QA/0312336.
25. Hernandez D., Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. 95 (2007), 567-608, math.QA/0504269.
26. Hernandez D., Jimbo M., Asymptotic representations and Drinfeld rational fractions, Compos. Math. 148 (2012), 1593-1623, arXiv:1104.1891.
27. Jimbo M., A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
28. Jimbo M., A $q$-analogue of $U({\mathfrak g}{\mathfrak l}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252.
29. Jimbo M., Miwa T., Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, Vol. 85, Amer. Math. Soc., Providence, RI, 1995.
30. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
31. Khoroshkin S.M., Tolstoy V.N., On Drinfel'd's realization of quantum affine algebras, J. Geom. Phys. 11 (1993), 445-452.
32. Khoroshkin S.M., Tolstoy V.N., Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras, hep-th/9404036.
33. Leznov A.N., Savel'ev M.V., A parametrization of compact groups, Funct. Anal. Appl. 8 (1974), 347-348.
34. Meneghelli C., Teschner J., Integrable light-cone lattice discretizations from the universal ${R}$-matrix, arXiv:1504.04572.
35. Mukhin E., Young C.A.S., Affinization of category $\mathcal{O}$ for quantum groups, Trans. Amer. Math. Soc. 366 (2014), 4815-4847, arXiv:1204.2769.
36. Nirov Kh.S., Razumov A.V., Quantum groups and functional relations for lower rank, J. Geom. Phys. 112 (2017), 1-28, arXiv:1412.7342.
37. Nirov Kh.S., Razumov A.V., Quantum groups, Verma modules and $q$-oscillators: General linear case, arXiv:1610.02901.
38. Razumov A.V., Monodromy operators for higher rank, J. Phys. A: Math. Theor. 46 (2013), 385201, 24 pages, arXiv:1211.3590.
39. Tolstoy V.N., Khoroshkin S.M., The universal $R$-matrix for quantum untwisted affine Lie algebras, Funct. Anal. Appl. 26 (1992), 69-71.
40. Yamane H., A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebras of type $A_N$, Publ. Res. Inst. Math. Sci. 25 (1989), 503-520.