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


SIGMA 9 (2013), 081, 20 pages      arXiv:1307.3642      http://dx.doi.org/10.3842/SIGMA.2013.081
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Kenny De Commer
Department of Mathematics, University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France

Received August 18, 2013, in final form December 18, 2013; Published online December 24, 2013

Abstract
Let $\mathfrak{g}$ be a compact simple Lie algebra. We modify the quantized enveloping $^*$-algebra associated to $\mathfrak{g}$ by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping $^*$-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.

Key words: compact quantum homogeneous spaces; quantized universal enveloping algebras; Hopf-Galois theory; Verma modules.

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References

  1. Bichon J., Hopf-Galois systems, J. Algebra 264 (2003), 565-581, math.QA/0204348.
  2. Bichon J., Hopf-Galois objects and cogroupoids, arXiv:1006.3014.
  3. Boca F.P., Ergodic actions of compact matrix pseudogroups on $C^*$-algebras, Astérisque 232 (1995), 93-109.
  4. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995.
  5. De Commer K., Comonoidal ${\rm W}^*$-Morita equivalence for von Neumann bialgebras, J. Noncommut. Geom. 5 (2011), 547-571, arXiv:1004.0824.
  6. De Commer K., On a correspondence between ${\rm SU}_q(2)$, $\widetilde{E}_q(2)$ and $\widetilde{\rm SU}_q(1,1)$, Comm. Math. Phys. 304 (2011), 187-228, arXiv:1004.4307.
  7. De Commer K., On a Morita equivalence between the duals of quantum ${\rm SU}(2)$ and quantum $\widetilde{E}(2)$, Adv. Math. 229 (2012), 1047-1079, arXiv:0912.4350.
  8. De Commer K., On the construction of quantum homogeneous spaces from $^*$-Galois objects, Algebr. Represent. Theory 15 (2012), 795-815, arXiv:1001.2153.
  9. De Concini C., Kac V.G., Representations of quantum groups at roots of 1, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., Vol. 92, Birkhäuser Boston, Boston, MA, 1990, 471-506.
  10. Donin J., Mudrov A., Explicit equivariant quantization on coadjoint orbits of ${\rm GL}(n,{\mathbb C})$, Lett. Math. Phys. 62 (2002), 17-32, math.QA/0206049.
  11. Enock M., Morita equivalence of measured quantum groupoids. Application to deformation of measured quantum groupoids by 2-cocycles, in Operator Algebras and Quantum Groups, Banach Center Publ., Vol. 98, Editors W. Pusz, P.M. Soltan, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 107-198, arXiv:1106.1018.
  12. Gilmore R., Lie groups, physics, and geometry. An introduction for physicists, engineers and chemists, Cambridge University Press, Cambridge, 2008.
  13. Hayashi T., Face algebras. I. A generalization of quantum group theory, J. Math. Soc. Japan 50 (1998), 293-315.
  14. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press Inc., New York, 1978.
  15. Humphreys J.E., Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$, Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008.
  16. Joseph A., Letzter G., Local finiteness of the adjoint action for quantized enveloping algebras, J. Algebra 153 (1992), 289-318.
  17. Joseph A., Letzter G., Separation of variables for quantized enveloping algebras, Amer. J. Math. 116 (1994), 127-177.
  18. Joseph A., Todorić D., On the quantum KPRV determinants for semisimple and affine Lie algebras, Algebr. Represent. Theory 5 (2002), 57-99.
  19. Karolinsky E., Stolin A., Tarasov V., Irreducible highest weight modules and equivariant quantization, Adv. Math. 211 (2007), 266-283, math.QA/0507348.
  20. Kashiwara M., On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516.
  21. Kassel C., Schneider H.J., Homotopy theory of Hopf Galois extensions, Ann. Inst. Fourier (Grenoble) 55 (2005), 2521-2550, math.QA/0402034.
  22. Knapp A.W., Representation theory of semisimple groups. An overview based on examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.
  23. Korogodsky L.I., Representations of quantum algebras arising from non-compact quantum groups: Quantum orbit method and super-tensor products, Ph.D. Thesis, Massachusetts Institute of Technology, 1996, available at http://dspace.mit.edu/handle/1721.1/39076.
  24. Letzter G., Quantum symmetric pairs and their zonal spherical functions, Transform. Groups 8 (2003), 261-292, math.QA/0204103.
  25. Letzter G., Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), 88-147, math.QA/0210447.
  26. Levendorski S., Soibelman Y., Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys. 139 (1991), 141-170.
  27. Masuda T., Nakagami Y., Watanabe J., Noncommutative differential geometry on the quantum two sphere of Podleś. I. An algebraic viewpoint, K-Theory 5 (1991), 151-175.
  28. Mudrov A., Quantum conjugacy classes of simple matrix groups, Comm. Math. Phys. 272 (2007), 635-660, math.QA/0412538.
  29. Reshetikhin N., Yakimov M., Quantum invariant measures, Comm. Math. Phys. 224 (2001), 399-426, math.QA/0101048.
  30. Rosso M., Groupes quantiques, représentations linéaires et applications, Ph.D. Thesis, Universite de Paris VII, 1990.
  31. Vaksman L.L., Quantum bounded symmetric domains, Translations of Mathematical Monographs, Vol. 238, American Mathematical Society, Providence, RI, 2010.
  32. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
  33. Woronowicz S.L., Twisted ${\rm SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181.

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