Journal of the Mathematical Society of Japan

Twisting the $q$-deformations of compact semisimple Lie groups

Sergey NESHVEYEV and Makoto YAMASHITA

Full-text: Open access

Abstract

Given a compact semisimple Lie group $G$ of rank $r$, and a parameter $q$ > 0, we can define new associativity morphisms in ${\rm Rep}(G_q)$ using a $3$-cocycle $\Phi$ on the dual of the center of $G$, thus getting a new tensor category ${\rm Rep}(G_q)^\Phi$. For a class of cocycles $\Phi$ we construct compact quantum groups $G^\tau_q$ with representation categories ${\rm Rep}(G_q)^\Phi$. The construction depends on the choice of an $r$-tuple $\tau$ of elements in the center of $G$. In the simplest case of $G=SU(2)$ and $\tau=-1$, our construction produces Woronowicz's quantum group $SU_{-q}(2)$ out of $SU_q(2)$. More generally, for $G=SU(n)$, we get quantum group realizations of the Kazhdan–Wenzl categories.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 2 (2015), 637-662.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1429624598

Digital Object Identifier
doi:10.2969/jmsj/06720637

Mathematical Reviews number (MathSciNet)
MR3340190

Zentralblatt MATH identifier
1333.46059

Subjects
Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

Keywords
compact quantum group $q$-deformation tensor category

Citation

NESHVEYEV, Sergey; YAMASHITA, Makoto. Twisting the $q$-deformations of compact semisimple Lie groups. J. Math. Soc. Japan 67 (2015), no. 2, 637--662. doi:10.2969/jmsj/06720637. https://projecteuclid.org/euclid.jmsj/1429624598


Export citation

References

  • M. Artin, W. Schelter and J. Tate, Quantum deformations of ${\rm GL}_n$, Comm. Pure Appl. Math., 44 (1991), 879–895.
  • T. Banica, Théorie des représentations du groupe quantique compact libre ${\rm O}(n)$, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 241–244.
  • T. Banica, Representations of compact quantum groups and subfactors, J. Reine Angew. Math., 509 (1999), 167–198.
  • J. Bichon, The representation category of the quantum group of a non-degenerate bilinear form, Comm. Algebra, 31 (2003), 4831–4851.
  • K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994.
  • V. G. Drinfel$'$d, Quantum groups, In: Proceedings of the International Congress of Mathematicians, 1 & 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp.,798–820.
  • V. G. Drinfel$'$d, Quasi-Hopf algebras, Algebra i Analiz, 1 (1989), 114–148. Translation in Leningrad Math. J., 1 (1990), 1419–1457.
  • S. Echterhoff, R. Nest and H. Oyono-Oyono, Fibrations with noncommutative fibers, J. Noncommut. Geom., 3 (2009), 377–417.
  • M. Enock and L. Vaĭnerman, Deformation of a Kac algebra by an abelian subgroup, Comm. Math. Phys., 178 (1996), 571–596.
  • W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991.
  • P. H. Hai, On matrix quantum groups of type $A_n$, Internat. J. Math., 11 (2000), 1115–1146.
  • D. Kazhdan and H Wenzl, Reconstructing monoidal categories, I. M. Gel$'$fand Seminar, Amer. Math. Soc., Providence, RI, 1993, pp.,111–136.
  • S. Levendorskiĭ and Y. Soibelman, Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys., 139 (1991), 141–170.
  • C. Mrozinski, Quantum automorphism groups and $\mathrm{SO}(3)$-deformations, J. Pure Appl. Algebra, 219 (2015), 1–32.
  • S. Neshveyev and L. Tuset, The Dirac operator on compact quantum groups, J. Reine Angew. Math., 641 (2010), 1–20.
  • S. Neshveyev and L. Tuset, $K$-homology class of the Dirac operator on a compact quantum group, Doc. Math., 16 (2011), 767–780.
  • S. Neshveyev and L. Tuset, Quantized algebras of functions on homogeneous spaces with Poisson stabilizers, Comm. Math. Phys., 312 (2012), 223–250.
  • S. Neshveyev and L. Tuset, Compact quantum groups and their representation categories, Cours Spécialsés [Specialized Courses], 20, Société Mathématique de France, Paris, 2013.
  • C. Ohn, Quantum $SL(3,{\bm C})$'s with classical representation theory, J. Algebra, 213 (1999), 721–756.
  • C. Ohn, Quantum $SL(3,\mathbb{C})$'s: the missing case, Hopf algebras in noncommutative geometry and physics, Dekker, New York, 2005, pp.,245–255.
  • C. Pinzari, The representation category of the Woronowicz quantum group $S_\mu U(d)$ as a braided tensor $C^*$-category, Internat. J. Math., 18 (2007), 113–136.
  • C. Pinzari and J. E. Roberts, A rigidity result for extensions of braided tensor $C^*$-categories derived from compact matrix quantum groups, Comm. Math. Phys., 306 (2011), 647–662.
  • N. Yu. Reshetikhin, L. A. Takhtadzhyan and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz, 1 (1989), 178–206, Translation in Leningrad Math. J., 1 (1990), 193–225.
  • A. Sangha, KK-fibrations arising from Rieffel deformations, preprint (2011), arXiv:1109.5968 [math.OA].
  • I. Tuba and H. Wenzl, On braided tensor categories of type $BCD$, J. Reine Angew. Math., 581 (2005), 31–69.
  • D. P. Williams, Crossed products of $C^\ast$-algebras, Mathematical Surveys and Monographs, 134, American Mathematical Society, Providence, RI, 2007.
  • S. L. Woronowicz, Tannaka-Kreĭ n duality for compact matrix pseudogroups, Twisted $SU(N)$ groups, Invent. Math., 93 (1988), 35–76.
  • S. L. Woronowicz and S. Zakrzewski, Quantum deformations of the Lorentz group. The Hopf $^*$-algebra level, Compositio Math., 90 (1994), 211–243.
  • M. Yamashita, Equivariant comparison of quantum homogeneous spaces, Comm. Math. Phys., 317 (2013), 593–614.