Formal multiparameter quantum groups, deformations and specializations
[Groupes quantiques formels multiparamétriques, déformations et spécialisations]
García, Gastón Andrés1 ; Gavarini, Fabio2
1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CMaLP-CIC-CONICET, 1900 La Plata (Argentina)
2 Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della ricerca scientifica 1, I-00133 Roma (Italy)
Annales de l'Institut Fourier, Online first, 117 p.

Nous introduisons la de algèbre enveloppante universelle quantifiée (=AEUQ) multiparamétrique formelle – en bref AEUQMpFo – comme généralisation directe du groupe quantique de Drinfeld U ℏ (𝔤). Ensuite, nous prouvons que la classe des AEUQMpFo est fermée par rapport aux déformations par torseurs (“toraux”) et aux déformations 2-cocycles (“toraux”) : par conséquent, toutes les “AEUQ multiparamétrique formelle” considerées jusqu’à ce jour sont retrouvées, comme incluses en cette classe. En particulier, nous prouvons que toute AEUQMpFo est isomorphe à une déformation convenable, par torseur ou par 2-cocycle, de la AEUQ standard de Drinfeld.

Nous introduisons aussi des bigèbres de Lie multiparamétriques (en bref, bGLMp), et nous considérons leur déformations, par torseur et par 2-cocycle. La limite semiclassique de chaque AEUQMpFo est une bGLMp convenable, et à l’envers chaque bGLMp peut être quantifiée à une AEUQMpFo convenable. Finalement, nous montrons que, en gros, les deux procedures de “specialisation” – d’une AEUQMpFo à une bGLMp – et de “déformation (par torseur toral ou 2-cocycle toral)” – au niveau des AEUQMpFO ou des bGLMp – commutent l’une avec l’autre.

We introduce the notion of formal multiparameter QUEA – in short FoMpQUEA – as a straightforward generalization of Drinfeld’s quantum group U ℏ (𝔤). Then we show that the class of FoMpQUEAs is closed under deformations by (“toral”) twists and deformations by (“toral”) 2-cocycles: as a consequence, all “multiparameter formal QUEAs” considered so far are recovered, as falling within this class. In particular, we prove that any FoMpQUEA is isomorphic to a suitable deformation, by twist or by 2-cocycle, of Drinfeld’s standard QUEA.

We introduce also multiparameter Lie bialgebras (in short, MpLbA’s), and we consider their deformations, by twist and by 2-cocycle. The semiclassical limit of every FoMpQUEA is a suitable MpLbA, and conversely each MpLbA can be quantized to a suitable FoMpQUEA. In the end, we prove that, roughly speaking, the two processes of “specialization” – of a FoMpQUEA to a MpLbA – and of “deformation (by toral twist or toral 2-cocycle)” – at the level of FoMpQUEAs or of MpLbA’s – do commute with each other.

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Révisé le :
Accepté le :
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DOI : 10.5802/aif.3675
Classification : 17B37, 17B62
Keywords: Quantum Groups, Quantum Enveloping Algebras
Mots-clés : Groupes Quantiques, algèbres enveloppantes universelles quantifiées

García, Gastón Andrés 1 ; Gavarini, Fabio 2

1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CMaLP-CIC-CONICET, 1900 La Plata (Argentina)
2 Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della ricerca scientifica 1, I-00133 Roma (Italy) 
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García, Gastón Andrés; Gavarini, Fabio. Formal multiparameter quantum groups, deformations and specializations. Annales de l'Institut Fourier, Online first, 117 p.

[1] Andruskiewitsch, Nicolás; Schneider, Hans-Jürgen A characterization of quantum groups, J. Reine Angew. Math., Volume 577 (2004), pp. 81-104 | DOI | MR | Zbl

[2] Andruskiewitsch, Nicolás; Schneider, Hans-Jürgen On the classification of finite-dimensional pointed Hopf algebras, Ann. Math., Volume 171 (2010) no. 1, pp. 375-417 annals.princeton.edu/annals/2010/171-1/p07.xhtml | DOI | Zbl

[3] Angiono, Iván; García Iglesias, Agustín Liftings of Nichols algebras of diagonal type II: All liftings are cocycle deformations, Sel. Math., New Ser., Volume 25 (2019) no. 1, 5, 95 pages | DOI | MR | Zbl

[4] Angiono, Iván; Yamane, Hiroyuki The R-matrix of quantum doubles of Nichols algebras of diagonal type, J. Math. Phys., Volume 56 (2015) no. 2, 021702, 19 pages | DOI | MR | Zbl

[5] Appel, Andrea; Sala, Francesco Quantization of continuum Kac–Moody algebras, Pure Appl. Math. Q., Volume 16 (2020) no. 3, pp. 439-493 | DOI | MR | Zbl

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

[7] Benkart, Georgia; Witherspoon, Sarah Restricted two-parameter quantum groups, Representations of finite dimensional algebras and related topics in Lie theory and geometry. Proceedings from the 10th international conference on algebras and related topics, ICRA X, Toronto, Canada, July 15–August 10, 2002, American Mathematical Society, 2004, pp. 293-318 | DOI | MR | Zbl

[8] Benkart, Georgia; Witherspoon, Sarah Two-parameter quantum groups and Drinfel’d doubles, Algebr. Represent. Theory, Volume 7 (2004) no. 3, pp. 261-286 | DOI | MR | Zbl

[9] Bergeron, Nantel; Gao, Yun; Hu, Naihong Drinfel’d doubles and Lusztig’s symmetries of two-parameter quantum groups, J. Algebra, Volume 301 (2006) no. 1, pp. 378-405 | DOI | MR | Zbl

[10] Borcherds, Richard E. A characterization of generalized Kac–Moody algebras, J. Algebra, Volume 174 (1995) no. 3, pp. 1073-1079 | DOI | MR | Zbl

[11] Chari, Vyjayanthi; Pressley, Andrew A guide to quantum groups, Cambridge University Press, 1995 | MR | Zbl

[12] Chin, William; Musson, Ian M. Multiparameter quantum enveloping algebras, J. Pure Appl. Algebra, Volume 107 (1996) no. 2-3, pp. 171-191 | DOI | MR | Zbl

[13] Ciccoli, Nicola; Guerra, Lucio The variety of Lie bialgebras, J. Lie Theory, Volume 13 (2003) no. 2, pp. 579-590 | MR | Zbl

[14] Costantini, Mauro; Varagnolo, Michela Quantum double and multiparameter quantum groups, Commun. Algebra, Volume 22 (1994) no. 15, pp. 6305-6321 | DOI | MR | Zbl

[15] Costantini, Mauro; Varagnolo, Michela Multiparameter quantum function algebra at roots of 1, Math. Ann., Volume 306 (1996) no. 4, pp. 759-780 https://eudml.org/doc/165479 | DOI | MR | Zbl

[16] Dobrev, Vladimir K.; Parashar, Preeti Duality for multiparametric quantum GL(n), J. Phys. A. Math. Gen., Volume 26 (1993) no. 23, pp. 6991-7002 | DOI | MR | Zbl

[17] Dobrev, Vladimir K.; Tahri, E. H. Duality for Multiparametric Quantum Deformation of the Supergroup GL(m/n), Int. J. Mod. Phys. A, Volume 13 (1998) no. 25, pp. 4339-4366 | DOI | MR | Zbl

[18] Doi, Yukio Braided bialgebras and quadratic bialgebras, Commun. Algebra, Volume 21 (1993) no. 5, pp. 1731-1749 | DOI | MR | Zbl

[19] Du, Jie; Parshall, Brian; Wang, Jianpan Two-parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras, J. Lond. Math. Soc., Volume 44 (1990) no. 3, pp. 420-436 | DOI | MR | Zbl

[20] Etingof, Pavel; Schedler, Travis; Schiffmann, Olivier Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras, J. Am. Math. Soc., Volume 13 (2000) no. 3, pp. 595-609 | DOI | MR | Zbl

[21] Etingof, Pavel; Schiffmann, Olivier Lectures on quantum groups, International Press, 2002 | MR | Zbl

[22] Frenkel, Edward; Hernandez, David Langlands duality for representations of quantum groups, Math. Ann., Volume 349 (2011) no. 3, pp. 705-746 | DOI | MR | Zbl

[23] García, Gastón Andrés Multiparameter quantum groups, bosonizations and cocycle deformations, Rev. Unión Mat. Argent., Volume 57 (2016) no. 2, pp. 1-23 | MR | Zbl

[24] García, Gastón Andrés; Gavarini, Fabio Twisted deformations vs. cocycle deformations for quantum groups, Commun. Contemp. Math., Volume 23 (2021) no. 8, 2050084, 56 pages | DOI | MR | Zbl

[25] García, Gastón Andrés; Gavarini, Fabio Multiparameter quantum groups at roots of unity, J. Noncommut. Geom., Volume 16 (2022) no. 3, pp. 839-926 | DOI | MR | Zbl

[26] García, Gastón Andrés; Gavarini, Fabio Quantum Group Deformations and Quantum R–(co)matrices vs. Quantum Duality Principle (2024) (https://arxiv.org/abs/2403.15096)

[27] Gavarini, Fabio Quantization of Poisson groups, Pac. J. Math., Volume 186 (1998) no. 2, pp. 217-266 | DOI | MR | Zbl

[28] Gavarini, Fabio The quantum duality principle, Ann. Inst. Fourier, Volume 52 (2002) no. 3, pp. 809-834 https://eudml.org/doc/115995 | DOI | Numdam | MR | Zbl

[29] Grunenfelder, L.; Mastnak, M. Pointed Hopf algebras as cocycle deformations (2010) (https://arxiv.org/abs/1010.4976)

[30] Halbout, Gilles Duality construction of symmetrizable Kac–Moody algebras, J. Algebra, Volume 222 (1999) no. 1, pp. 65-81 | DOI | MR | Zbl

[31] Hodges, Timothy J.; Levasseur, Thierry; Toro, Margarita Algebraic structure of multiparameter quantum groups, Adv. Math., Volume 126 (1997) no. 1, pp. 52-92 | DOI | MR | Zbl

[32] Hong, Jin; Kang, Seok-Jin Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, 42, American Mathematical Society, 2002 | DOI | MR | Zbl

[33] Jantzen, Jens Carsten Lectures on quantum groups, Graduate Studies in Mathematics, 6, American Mathematical Society, 1996 | DOI | MR | Zbl

[34] Jing, Nai Huan Quantum groups with two parameters, Deformation theory and quantum groups with applications to mathematical physics. Proceedings of an AMS-IMS-SIAM 1990 joint summer research conference, held June 14-20, 1990, at the University of Massachusetts, Amherst, MA, USA, American Mathematical Society, 1992, pp. 129-138 | DOI | MR | Zbl

[35] Kac, Victor G. Infinite dimensional Lie algebras, Cambridge University Press, 1990 | DOI | MR | Zbl

[36] Kassel, Christian Quantum groups, Graduate Texts in Mathematics, 155, Springer, 1995 | DOI | MR | Zbl

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

[38] Kharchenko, Vladislav K. Quantizations of Kac–Moody algebras, J. Pure Appl. Algebra, Volume 218 (2014) no. 4, pp. 666-683 | DOI | MR | Zbl

[39] Klimyk, Anatoli; Schmüdgen, Konrad Quantum groups and their representations, Texts and Monographs in Physics, Springer, 1997 | DOI | MR | Zbl

[40] Lusztig, George Introduction to quantum groups, Progress in Mathematics, 110, Birkhäuser, 1993 | MR | Zbl

[41] Majid, Shahn Foundations of quantum group theory, Cambridge University Press, 1995 | DOI | MR | Zbl

[42] Manin, Yurii I. Multiparametric quantum deformation of the general linear supergroup, Commun. Math. Phys., Volume 123 (1989) no. 1, pp. 163-175 | DOI | MR | Zbl

[43] Merkulov, Sergei; Willwacher, Thomas Deformation theory of Lie bialgebra properads, Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 1, Oxford University Press, 2018, pp. 219-247 | DOI | Zbl

[44] Montgomery, Susan Hopf algebras and their actions on rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992, Regional Conference Series in Mathematics, 82, American Mathematical Society, 1993 | DOI | MR | Zbl

[45] Okado, Masato; Yamane, Hiroyuki R-matrices with gauge parameters and multi-parameter quantized enveloping algebras, Special functions. Proceedings of the Hayashibara Forum 1990, held in Fujisaki Institute, Okayama, Japan, August 16-20, 1990, Springer, 1991, pp. 289-293 | MR | Zbl

[46] Pei, Yufeng; Hu, Naihong; Rosso, Marc Multi-parameter quantum groups and quantum shuffles. I, Quantum affine algebras, extended affine Lie algebras, and their applications. Proceedings of the workshop, March 2–7, 2008, Banff, Canada, American Mathematical Society, 2010, pp. 145-171 | DOI | MR | Zbl

[47] Pogorelsky, Barbara; Vay, Cristian Verma and simple modules for quantum groups at non-abelian groups, Adv. Math., Volume 301 (2016), pp. 423-457 | DOI | MR | Zbl

[48] Radford, David E. Hopf algebras, Series on Knots and Everything, 49, World Scientific, 2012 | DOI | Zbl

[49] Reshetikhin, Nikolai Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys., Volume 20 (1990) no. 4, pp. 331-335 | DOI | MR | Zbl

[50] Rosso, Marc Quantum groups and quantum shuffles, Invent. Math., Volume 133 (1998) no. 2, pp. 399-416 | DOI | MR | Zbl

[51] Sudbery, Anthony Consistent multiparameter quantisation of GL(n), J. Phys. A. Math. Gen., Volume 23 (1990) no. 15, p. l697-l704 | DOI | MR | Zbl

[52] Sweedler, Moss E. Cohomology of algebras over Hopf algebras, Trans. Am. Math. Soc., Volume 133 (1968), pp. 205-239 | DOI | MR | Zbl

[53] Takeuchi, Mitsuhiro A two-parameter quantization of GL(n), Proc. Japan Acad., Ser. A, Volume 66 (1990) no. 5, pp. 112-114 | DOI | MR | Zbl

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