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

SIGMA 16 (2020), 043, 49 pages      arXiv:1911.03496      https://doi.org/10.3842/SIGMA.2020.043

Isomorphism between the $R$-Matrix and Drinfeld Presentations of Quantum Affine Algebra: Types $B$ and $D$

Naihuan Jing ^a, Ming Liu ^bc and Alexander Molev ^c
a) Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
^b) School of Mathematics, South China University of Technology, Guangzhou, 510640, China
^c) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Received November 18, 2019, in final form May 10, 2020; Published online May 21, 2020

Abstract
Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type $C$ were given therein, while the present paper deals with types $B$ and $D$. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie algebras.

Key words: $R$-matrix presentation; Drinfeld new presentation; universal $R$-matrix; Gauss decomposition.

pdf (583 kb)   tex (39 kb)  

References

1. Beck J., Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555-568, arXiv:hep-th/9404165.
2. Beck J., Convex bases of PBW type for quantum affine algebras, Comm. Math. Phys. 165 (1994), 193-199, arXiv:hep-th/9407003.
3. Beck J., Chari V., Pressley A., An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455-487, arXiv:math.QA/9808060.
4. Brundan J., Kleshchev A., Parabolic presentations of the Yangian $Y({\mathfrak{gl}}_n)$, Comm. Math. Phys. 254 (2005), 191-220, arXiv:math.QA/0407011.
5. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
6. Damiani I., La $R$-matrice pour les algèbres quantiques de type affine non tordu, Ann. Sci. École Norm. Sup. (4) 31 (1998), 493-523.
7. Damiani I., The $R$-matrix for (twisted) affine quantum algebras, in Representations and Quantizations (Shanghai, 1998), China High. Educ. Press, Beijing, 2000, 89-144.
8. Ding J.T., Frenkel I.B., Isomorphism of two realizations of quantum affine algebra $U_q({\mathfrak{gl}}(n))$, Comm. Math. Phys. 156 (1993), 277-300.
9. Drinfeld V.G., Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254-258.
10. Drinfeld V.G., A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212-216.
11. Frenkel E., Mukhin E., The Hopf algebra ${\rm Rep}\,U_q\widehat{{\mathfrak{gl}}}_\infty$, Selecta Math. (N.S.) 8 (2002), 537-635, arXiv:math.AG/0103126.
12. Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60.
13. Gel'fand I.M., Retakh V.S., Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 (1991), 91-102.
14. Gel'fand I.M., Retakh V.S., A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl. 26 (1992), 231-246.
15. Gow L., Molev A., Representations of twisted $q$-Yangians, Selecta Math. (N.S.) 16 (2010), 439-499, arXiv:0909.4905.
16. Guay N., Regelskis V., Wendlandt C., Equivalences between three presentations of orthogonal and symplectic Yangians, Lett. Math. Phys. 109 (2019), 327-379, arXiv:1706.05176.
17. Jimbo M., A $q$-difference analogue of $U({\mathfrak{g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
18. Jimbo M., Quantum $R$-matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), 537-547.
19. Jing N., Liu M., Molev A., Isomorphism between the $R$-matrix and Drinfeld presentations of Yangian in types $B$, $C$ and $D$, Comm. Math. Phys. 361 (2018), 827-872, arXiv:1705.08155.
20. Jing N., Liu M., Molev A., Isomorphism between the $R$-matrix and Drinfeld presentations of quantum affine algebra: Type $C$, J. Math. Phys. 61 (2020), 031701, 41 pages, arXiv:1903.00204.
21. Khoroshkin S.M., Tolstoy V.N., Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), 599-617.
22. Levendorskiǐ S.Z., On generators and defining relations of Yangians, J. Geom. Phys. 12 (1993), 1-11.
23. Reshetikhin N.Y., Semenov-Tian-Shansky M.A., Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142.
24. Tarasov V.O., Irreducible monodromy matrices for the $R$-matrix of the $XXZ$-model and local lattice quantum Hamiltonians, Theoret. and Math. Phys. 63 (1985), 440-454.
25. Wendlandt C., The $R$-matrix presentation for the Yangian of a simple Lie algebra, Comm. Math. Phys. 363 (2018), 289-332, arXiv:1709.08162.