Quantized semisimple Lie groups

Rita Fioresi; Robert Yuncken

Archivum Mathematicum (2024)

  • Volume: 060, Issue: 5, page 311-349
  • ISSN: 0044-8753

Abstract

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The goal of this expository paper is to give a quick introduction to q -deformations of semisimple Lie groups. We discuss principally the rank one examples of 𝒰 q ( 𝔰𝔩 2 ) , 𝒪 ( SU q ( 2 ) ) , 𝒟 ( SL q ( 2 , ℂ ) ) and related algebras. We treat quantized enveloping algebras, representations of 𝒰 q ( 𝔰𝔩 2 ) , generalities on Hopf algebras and quantum groups, * -structures, quantized algebras of functions on q -deformed compact semisimple groups, the Peter-Weyl theorem, * -Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations of SL q ( 2 , ℂ ) , the Plancherel formula for SL q ( 2 , ℂ ) . This exposition is expanding the material treated in a series of lectures given by the second author at the CaLISTA CA 21100 Training School, “Quantum Groups and Noncommutative Geometry in Prague” in 2023.

How to cite

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Fioresi, Rita, and Yuncken, Robert. "Quantized semisimple Lie groups." Archivum Mathematicum 060.5 (2024): 311-349. <http://eudml.org/doc/299628>.

@article{Fioresi2024,
abstract = {The goal of this expository paper is to give a quick introduction to $q$-deformations of semisimple Lie groups. We discuss principally the rank one examples of $\mathcal \{U\}_q(\mathfrak \{sl\}_2)$, $\mathcal \{O\}(\mathrm \{SU\}_q(2))$, $\mathcal \{D\}(\mathrm \{SL\}_q(2,\mathbb \{C\}))$ and related algebras. We treat quantized enveloping algebras, representations of $\mathcal \{U\}_q(\mathfrak \{sl\}_2)$, generalities on Hopf algebras and quantum groups, $*$-structures, quantized algebras of functions on $q$-deformed compact semisimple groups, the Peter-Weyl theorem, $*$-Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations of $\mathrm \{SL\}_q(2,\mathbb \{C\})$, the Plancherel formula for $\mathrm \{SL\}_q(2,\mathbb \{C\})$. This exposition is expanding the material treated in a series of lectures given by the second author at the CaLISTA CA 21100 Training School, “Quantum Groups and Noncommutative Geometry in Prague” in 2023.},
author = {Fioresi, Rita, Yuncken, Robert},
journal = {Archivum Mathematicum},
keywords = {quantum groups; representation theory; semisimple Lie algebras},
language = {eng},
number = {5},
pages = {311-349},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Quantized semisimple Lie groups},
url = {http://eudml.org/doc/299628},
volume = {060},
year = {2024},
}

TY - JOUR
AU - Fioresi, Rita
AU - Yuncken, Robert
TI - Quantized semisimple Lie groups
JO - Archivum Mathematicum
PY - 2024
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 060
IS - 5
SP - 311
EP - 349
AB - The goal of this expository paper is to give a quick introduction to $q$-deformations of semisimple Lie groups. We discuss principally the rank one examples of $\mathcal {U}_q(\mathfrak {sl}_2)$, $\mathcal {O}(\mathrm {SU}_q(2))$, $\mathcal {D}(\mathrm {SL}_q(2,\mathbb {C}))$ and related algebras. We treat quantized enveloping algebras, representations of $\mathcal {U}_q(\mathfrak {sl}_2)$, generalities on Hopf algebras and quantum groups, $*$-structures, quantized algebras of functions on $q$-deformed compact semisimple groups, the Peter-Weyl theorem, $*$-Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations of $\mathrm {SL}_q(2,\mathbb {C})$, the Plancherel formula for $\mathrm {SL}_q(2,\mathbb {C})$. This exposition is expanding the material treated in a series of lectures given by the second author at the CaLISTA CA 21100 Training School, “Quantum Groups and Noncommutative Geometry in Prague” in 2023.
LA - eng
KW - quantum groups; representation theory; semisimple Lie algebras
UR - http://eudml.org/doc/299628
ER -

References

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  1. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I., Differential operators on the base affine space and a study of 𝔤 -modules, Halsted Press [John Wiley & Sons, Inc.], New York-Toronto, Ont., 1975, 21–64. (1975) MR0578996
  2. Buffenoir, E., Roche, Ph., 10.1007/s002200050736, Comm. Math. Phys. 207 (3) (1999), 499–555. (1999) MR1727241DOI10.1007/s002200050736
  3. Čap, A., Slovák, J., Souček, V., 10.2307/3062111, Ann. of Math. (2) 154 (1) (2001), 97–113. (2001) MR1847589DOI10.2307/3062111
  4. De Commer, K., 10.1007/s00220-011-1208-y, Comm. Math. Phys. 304 (1) (2011), 187–228. (2011) MR2793934DOI10.1007/s00220-011-1208-y
  5. De Commer, K., Dzokou Talla, J.R., Invariant integrals on coideals and their drinfeld doubles, arXiv:2112.07476 [math.QA], 2021. MR4776189
  6. De Commer, K., Dzokou Talla, J.R., Quantum s l ( 2 , ℝ ) and its irreducible representations, arXiv:2107.04258 [math.QA], 2021. MR4750927
  7. Dixmier, J., 10.1090/gsm/011/02, Grad. Stud. Math., vol. 11, Providence, RI: AMS, American Mathematical Society, 1996. (1996) MR1393197DOI10.1090/gsm/011/02
  8. Drinfel’d, V.G., Quantum groups, Proc. Int. Congr. Math., vol. 1, Berkeley/Calif 1986, 1987, pp. 798–820. (1987) MR0934283
  9. Drinfel’d, V.G., 10.1007/BF01247086, Journal of Soviet Mathematics 41 92) (1988), 898–915. (1988) MR0869575DOI10.1007/BF01247086
  10. Faddeev, L.D., Reshetikhin, N.Yu., Takhtadzhan, L.A., Quantization of Lie groups and Lie algebras, Algebraic Analysis 1 (1989), 129–139, Dedicated to Prof. Mikio Sato on the Occas. of his Sixtieth Birthday. (1989) MR1015339
  11. Fioresi, R., Lledó, M.A., The Minkowski and conformal superspaces. The classical and quantum descriptions, Hackensack, NJ: World Scientific, 2015. (2015) MR3328668
  12. Gavarini, F., The global quantum duality principle, J. Reine Angew. Math. 612 (2007), 17–33. (2007) MR2364072
  13. Heckenberger, I., Kolb, S., 10.1007/s00031-007-0059-2, Transform. Groups 12 (4) (2007), 647–655. (2007) MR2365438DOI10.1007/s00031-007-0059-2
  14. Heckenberger, I., Kolb, S., 10.1016/j.geomphys.2007.07.005, J. Geom. Phys. 57 (11) (2007), 2316–2344. (2007) MR2360244DOI10.1016/j.geomphys.2007.07.005
  15. Helgason, S., 10.1090/gsm/034, Grad. Stud. Math., vol. 34, Providence, RI: American Mathematical Society (AMS), 2001, reprint with corrections of the 1978 original edition. (2001) MR1834454DOI10.1090/gsm/034
  16. Jimbo, M., 10.1007/BF00400222, Lett. Math. Phys. 11 (1986), 247–252. (1986) MR0841713DOI10.1007/BF00400222
  17. Joseph, A., Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer Verlag, Berlin, 1995. (1995) MR1315966
  18. Kassel, Ch., Quantum groups, Grad. Texts Math., vol. 155, New York, NY: Springer Verlag, 1995. (1995) MR1321145
  19. Klimyk, A., Schmüdgen, K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. (1997) MR1492989
  20. Knapp, A.W., Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. (2002) MR1920389
  21. Koelink, E., Kustermans, J., 10.1007/s00220-002-0736-x, Comm. Math. Phys. 233 (2) (2003), 231–296. (2003) MR1962042DOI10.1007/s00220-002-0736-x
  22. Kulish, P.P., Reshetikhin, N.Yu., Quantum linear problem for the sine-gordon equation and higher representations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 101 (1981), 101–110. (1981) MR0623928
  23. Montgomery, S., Hopf algebras and their actions on rings, Reg. Conf. Ser. Math., vol. 82, Providence, RI: American Mathematical Society, 1993, 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. (1993) MR1243637
  24. Ó Buachalla, R., Somberg, P., Lusztig’s quantum root vectors and a Dolbeault complex for the A -series full quantum flag manifolds, arXiv:2312.13493 [math.QA], 2023. 
  25. Podleś, P., Woronowicz, S.L., 10.1007/BF02473358, Comm. Math. Phys. 130 (2) (1990), 381–431. (1990) MR1059324DOI10.1007/BF02473358
  26. Sklyanin, E.K., On an algebra generated by quadratic relations, Uspekhi Mat. Nauk 40 (1985), 214. (1985) 
  27. Van Daele, A., 10.1090/S0002-9947-1994-1220906-5, Trans. Amer. Math. Soc. 342 (2) (1994), 917–932. (1994) MR1220906DOI10.1090/S0002-9947-1994-1220906-5
  28. Van Daele, A., 10.1006/aima.1998.1775, Adv. Math. 140 (2) (1998), 323–366. (1998) MR1658585DOI10.1006/aima.1998.1775
  29. Varadarajan, V.S., Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. (1974) MR0376938
  30. Voigt, Ch., Yuncken, R., The plancherel formula for complex semisimple quantum groups, Ann. Sci. Éc. Norm. Sup. (to appear), https://arxiv.org/abs/1906.02672, 2019. MR4637134
  31. Voigt, Ch., Yuncken, R., 10.4171/dm/495, Doc. Math. 20 (2015), 433–490. (2015) MR3398718DOI10.4171/dm/495
  32. Voigt, Ch., Yuncken, R., Complex semisimple quantum groups and representation theory, Lect. Notes in Math., Springer, Cham, 2020. (2020) MR4162277
  33. Woronowicz, S.L., 10.1007/BF01219077, Comm. Math. Phys. 111 (4) (1987), 613–665. (1987) MR0901157DOI10.1007/BF01219077
  34. Woronowicz, S.L., 10.2977/prims/1195176848, Publ. Res. Inst. Math. Sci. 23 (1) (1987), 117–181. (1987) MR0890482DOI10.2977/prims/1195176848

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