The $R$-matrix formalism for quantized enveloping algebras

Gautam, Sachin; Rupert, Matthew; Wendlandt, Curtis

doi:10.5802/aif.3688

The

R

-matrix formalism for quantized enveloping algebras
[Le formalisme de la matrice R pour les algèbres enveloppantes quantifiées]

Gautam, Sachin ¹ ; Rupert, Matthew ² ; Wendlandt, Curtis ²

¹ Department of Mathematics The Ohio State University Columbus, OH 43210 (USA)
² Department of Mathematics and Statistics University of Saskatchewan Saskatoon, SK S7N 5E6 (Canada)

Annales de l'Institut Fourier, Online first, 59 p.

Résumé
Abstract

Soit $U_{ℏ} 𝔤$ le groupe quantique de Drinfeld–Jimbo associé à une algèbre de Lie simple $𝔤$ . Nous appliquons une modification du formalisme de Faddeev–Reshetikhin–Takhtajan pour les groupes quantiques à l’évaluation de la matrice $R$ universelle de $U_{ℏ} 𝔤$ sur n’importe quelle de ses représentations de dimension finie. Cela produit une algèbre enveloppante quantifiée $U_{R} (𝔤)$ dont la définition est donnée en termes de deux matrices génératrices satisfaisant des variantes des relations $R L L$ . Nous prouvons que $U_{R} (𝔤)$ est isomorphe au produit tensoriel du double quantique de la sous-algèbre de Borel $U_{ℏ} 𝔟 \subset U_{ℏ} 𝔤$ et d’une algèbre polynomiale quantifiée associée à l’espace des $𝔤$ -invariants de $V$ , où $V$ est la limite semi-classique de la représentation de dimension finie choisi. Grâce à cette description, nous caractérisons $U_{ℏ} 𝔤$ et le double quantique de $U_{ℏ} 𝔟$ comme des quotients de Hopf de $U_{R} (𝔤)$ et comme des sous-algèbres de points fixes par rapport à certains automorphismes naturels. Comme corollaire additionnel, nous déduisons que $U_{R} (𝔤)$ est quasi-triangulaire précisément lorsque $V$ est sans multiplicités.

Let $U_{ℏ} 𝔤$ denote the Drinfeld–Jimbo quantum group associated to a simple Lie algebra $𝔤$ . We apply a modification of the $R$ -matrix construction for quantum groups to the evaluation of the universal $R$ -matrix of $U_{ℏ} 𝔤$ on any of its finite-dimensional representations. This produces a quantized enveloping algebra $U_{R} (𝔤)$ whose definition is given in terms of two generating matrices satisfying variants of the well-known $R L L$ relations. We prove that $U_{R} (𝔤)$ is isomorphic to the tensor product of the quantum double of the Borel subalgebra $U_{ℏ} 𝔟 \subset U_{ℏ} 𝔤$ and a quantized polynomial algebra encoded by the space of $𝔤$ -invariants associated to the semiclassical limit $V$ of the underlying finite-dimensional representation. Using this description, we characterize $U_{ℏ} 𝔤$ and the quantum double of $U_{ℏ} 𝔟$ as Hopf quotients of $U_{R} (𝔤)$ and as fixed-point subalgebras with respect to certain natural automorphisms. As an additional corollary, we deduce that $U_{R} (𝔤)$ is quasitriangular precisely when $V$ is multiplicity free.

Reçu le : 2022-10-19
Révisé le : 2023-10-02
Accepté le : 2023-12-21
Première publication : 2025-02-10

DOI : 10.5802/aif.3688

Classification : 17B37, 17B62
Keywords: Quantum groups, R-matrices, Yang–Baxter equation, Lie bialgebras
Mots-clés : Groupes quantiques, matrices R, équation de Yang–Baxter, bi-algèbres de Lie

Affiliations des auteurs :

Gautam, Sachin ¹ ; Rupert, Matthew ² ; Wendlandt, Curtis ²

¹ Department of Mathematics The Ohio State University Columbus, OH 43210 (USA)
² Department of Mathematics and Statistics University of Saskatchewan Saskatoon, SK S7N 5E6 (Canada)

@unpublished{AIF_0__0_0_A158_0,
     author = {Gautam, Sachin and Rupert, Matthew and Wendlandt, Curtis},
     title = {The $R$-matrix formalism for quantized enveloping algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3688},
     language = {en},
     note = {Online first},
}

TY  - UNPB
AU  - Gautam, Sachin
AU  - Rupert, Matthew
AU  - Wendlandt, Curtis
TI  - The $R$-matrix formalism for quantized enveloping algebras
JO  - Annales de l'Institut Fourier
PY  - 2025
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3688
LA  - en
ID  - AIF_0__0_0_A158_0
ER  -

%0 Unpublished Work
%A Gautam, Sachin
%A Rupert, Matthew
%A Wendlandt, Curtis
%T The $R$-matrix formalism for quantized enveloping algebras
%J Annales de l'Institut Fourier
%D 2025
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3688
%G en
%F AIF_0__0_0_A158_0

Gautam, Sachin; Rupert, Matthew; Wendlandt, Curtis. The $R$-matrix formalism for quantized enveloping algebras. Annales de l'Institut Fourier, Online first, 59 p.

Bibliographie
Cité par

[1] Appel, Andrea; Toledano Laredo, V. A 2-categorical extension of Etingof–Kazhdan quantisation, Sel. Math., New Ser., Volume 24 (2018) no. 4, pp. 3529-3617 | DOI | MR | Zbl

[2] Arnaudon, Daniel; Molev, Alexander; Ragoucy, Eric On the $R$ -matrix realization of Yangians and their representations, Ann. Henri Poincaré, Volume 7 (2006) no. 7-8, pp. 1269-1325 | DOI | MR | Zbl

[3] Chari, Vyjayanthi; Pressley, Andrew A guide to quantum groups, Cambridge University Press, 1994, xvi+651 pages | MR | Zbl

[4] Ding, Jintai; Frenkel, Igor Isomorphism of two realizations of quantum affine algebra $U_{q} (𝔤𝔩 (n))$ , Commun. Math. Phys., Volume 156 (1993) no. 2, pp. 277-300 | DOI | MR | Zbl

[5] Drinfel’d, Vladimir Hopf algebras and the quantum Yang–Baxter equation, Sov. Math., Dokl., Volume 32 (1985) no. 1, pp. 254-258 | Zbl

[6] Drinfel’d, Vladimir Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), American Mathematical Society (1987), pp. 798-820 | MR | Zbl

[7] Drinfel’d, Vladimir Almost cocommutative Hopf algebras, Algebra Anal., Volume 1 (1989) no. 2, pp. 30-46 | MR

[8] Etingof, Pavel; Kazhdan, David Quantization of Lie bialgebras. I, Sel. Math., New Ser., Volume 2 (1996) no. 1, pp. 1-41 | DOI | MR | Zbl

[9] Etingof, Pavel; Schiffmann, Olivier Lectures on quantum groups, Lectures in Mathematical Physics, International Press, 2002, xii+242 pages | MR | Zbl

[10] Faddeev, Ludvig Quantum completely integrable models in field theory, Mathematical physics reviews, Vol. 1 (Soviet Scientific Reviews. Section C), Volume 1, Harwood Academic, 1980, pp. 107-155 | MR | Zbl

[11] Faddeev, Ludvig; Reshetikhin, Nikolai; Takhtajan, Leon Quantization of Lie Groups and Lie Algebras, Leningr. Math. J., Volume 1 (1990) no. 1, pp. 193-225 | MR | Zbl

[12] Faddeev, Ludvig; Sklyanin, Evgeny; Takhtajan, Leon Quantum inverse problem method. I, Teor. Mat. Fiz., Volume 40 (1979) no. 2, pp. 194-220 | MR | Zbl

[13] Faddeev, Ludvig; Takhtajan, Leon The quantum method for the inverse problem and the $X Y Z$ Heisenberg model, Usp. Mat. Nauk, Volume 34 (1979) no. 5(209), p. 13-63, 256 | MR

[14] Farinati, Marco A.; Jancsa, Alejandra Patricia Trivial central extensions of Lie bialgebras, J. Algebra, Volume 390 (2013), pp. 56-76 | DOI | MR | Zbl

[15] Finkelberg, Michael; Tsymbaliuk, Alexander Shifted quantum affine algebras: integral forms in type $A$ , Arnold Math. J., Volume 5 (2019) no. 2-3, pp. 197-283 | DOI | MR | Zbl

[16] Gavarini, Fabio The quantum duality principle, Ann. Inst. Fourier, Volume 52 (2002) no. 3, pp. 809-834 | DOI | Numdam | MR | Zbl

[17] Gerstenhaber, Murray On the deformation of rings and algebras, Ann. Math. (2), Volume 79 (1964), pp. 59-103 | DOI | MR | Zbl

[18] Gow, Lucy; Molev, Alexander Representations of twisted $q$ -Yangians, Sel. Math., New Ser., Volume 16 (2010) no. 3, pp. 439-499 | DOI | MR | Zbl

[19] Guay, Nicolas; Regelskis, Vidas; Wendlandt, Curtis R-matrix presentation of orthogonal and symplectic quantum loop algebras and their representations (2022) (in preparation)

[20] Hayaishi, Norifumi; Miki, Kei $L$ operators and Drinfeld’s generators, J. Math. Phys., Volume 39 (1998) no. 3, pp. 1623-1636 | DOI | MR | Zbl

[21] Jimbo, Michio A $q$ -difference analogue of $U (𝔤)$ and the Yang–Baxter equation, Lett. Math. Phys., Volume 10 (1985) no. 1, pp. 63-69 | DOI | MR | Zbl

[22] Jimbo, Michio A $q$ -analogue of $U (𝔤𝔩 (N + 1))$ , Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys., Volume 11 (1986) no. 3, pp. 247-252 | DOI | MR | Zbl

[23] Jing, Naihuan; Liu, Ming; Molev, Alexander Isomorphism between the $R$ -matrix and Drinfeld presentations of quantum affine algebra: type $C$ , J. Math. Phys., Volume 61 (2020) no. 3, 031701, 41 pages | DOI | MR | Zbl

[24] Jing, Naihuan; Liu, Ming; Molev, Alexander Isomorphism between the $R$ -matrix and Drinfeld presentations of quantum affine algebra: types $B$ and $D$ , SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), 043, 49 pages | DOI | MR | Zbl

[25] Jing, Naihuan; Liu, Ming; Molev, Alexander Representations of quantum affine algebras in their $R$ -matrix realization, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), 145, 25 pages | DOI | MR | Zbl

[26] Kassel, Christian Quantum groups, Graduate Texts in Mathematics, 155, Springer, 1995, xii+531 pages | DOI | MR | Zbl

[27] Kassel, Christian; Turaev, Vladimir Biquantization of Lie bialgebras, Pac. J. Math., Volume 195 (2000) no. 2, pp. 297-369 | DOI | MR | Zbl

[28] Kirillov, Anatol N.; Reshetikhin, Nikolai $q$ -Weyl group and a multiplicative formula for universal $R$ -matrices, Commun. Math. Phys., Volume 134 (1990) no. 2, pp. 421-431 | DOI | MR | Zbl

[29] Klimyk, Anatoli; Schmüdgen, Konrad Quantum groups and their representations, Texts and Monographs in Physics, Springer, 1997, xx+552 pages | DOI | MR | Zbl

[30] Kulish, Petr P.; Sklyanin, Evgeny Quantum spectral transform method. Recent developments, Integrable quantum field theories (Tvärminne, 1981) (Lecture Notes in Physics), Volume 151, Springer (1982), pp. 61-119 | DOI | MR | Zbl

[31] Levendorskiĭ, Sergeĭ Z.; Soĭbelʼman, Yan S. Some applications of the quantum Weyl groups, J. Geom. Phys., Volume 7 (1990) no. 2, pp. 241-254 | DOI | MR | Zbl

[32] Majid, Shahn More examples of bicrossproduct and double cross product Hopf algebras, Isr. J. Math., Volume 72 (1990) no. 1-2, pp. 133-148 | DOI | MR | Zbl

[33] Majid, Shahn Quantum group duality in vertex models and other results in the theory of quasitriangular Hopf algebras, Differential geometric methods in theoretical physics (Davis, CA, 1988) (NATO ASI Series. Series B. Physics), Volume 245, Plenum Press, 1990, pp. 373-385 | DOI | MR | Zbl

[34] Majid, Shahn Quasitriangular Hopf algebras and Yang–Baxter equations, Int. J. Mod. Phys. A, Volume 5 (1990) no. 1, pp. 1-91 | DOI | MR | Zbl

[35] Majid, Shahn Foundations of quantum group theory, Cambridge University Press, 1995, x+607 pages | DOI | MR | Zbl

[36] Molev, Alexander Yangians and classical Lie algebras, Mathematical Surveys and Monographs, 143, American Mathematical Society, 2007, xviii+400 pages | DOI | MR | Zbl

[37] Molev, Alexander; Nazarov, Maxim; Olshanskii, Grigori Yangians and classical Lie algebras, Russ. Math. Surv., Volume 51 (1996) no. 2, pp. 205-282 | DOI | Zbl

[38] Molev, Alexander; Ragoucy, Eric; Sorba, Paul Coideal subalgebras in quantum affine algebras, Rev. Math. Phys., Volume 15 (2003) no. 8, pp. 789-822 | DOI | MR | Zbl

[39] Olshanskii, Grigori Twisted Yangians and infinite-dimensional classical Lie algebras, Quantum groups. Proceedings of workshops, held in the Euler International Mathematical Institute, Leningrad, USSR, Fall 1990 (Lecture Notes in Mathematics), Volume 1510, Springer, 1992, pp. 104-119 | DOI | MR | Zbl

[40] Reshetikhin, Nikolai; Semenov-Tian-Shansky, Michael A. Central extensions of quantum current groups, Lett. Math. Phys., Volume 19 (1990) no. 2, pp. 133-142 | DOI | MR | Zbl

[41] Rosso, Marc Représentations irréductibles de dimension finie du $q$ -analogue de l’algèbre enveloppante d’une algèbre de Lie simple, C. R. Math. Acad. Sci. Paris, Volume 305 (1987) no. 13, pp. 587-590 | MR | Zbl

[42] Rosso, Marc An analogue of P.B.W. theorem and the universal $R$ -matrix for $U_{h} sl (N + 1)$ , Commun. Math. Phys., Volume 124 (1989) no. 2, pp. 307-318 | DOI | MR | Zbl

[43] Sasaki, Norihito Quantization of Lie group and algebra of $G_{2}$ type in the Faddeev–Reshetikhin–Takhtajan approach, J. Math. Phys., Volume 36 (1995) no. 8, pp. 4476-4488 | DOI | MR | Zbl

[44] Tanisaki, Toshiyuki Finite dimensional representations of quantum groups, Osaka J. Math., Volume 28 (1991) no. 1, pp. 37-53 | MR | Zbl

[45] Wendlandt, Curtis The $R$ -matrix presentation for the Yangian of a simple Lie algebra, Commun. Math. Phys., Volume 363 (2018) no. 1, pp. 289-332 | DOI | MR | Zbl

[46] Wendlandt, Curtis The restricted quantum double of the Yangian, Can. J. Math. (2024), 72 pages (first view) | DOI

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT