Hopf algebras having a dense big cell

Bichon, Julien; Riche, Simon

doi:10.1090/tran/6335

Hopf algebras having a dense big cell
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by Julien Bichon and Simon Riche PDF

Trans. Amer. Math. Soc. 368 (2016), 515-538 Request permission

Abstract:

We discuss some axioms that ensure that a Hopf algebra has its simple comodules classified using an analogue of the Borel–Weil construction. More precisely we show that a Hopf algebra having a dense big cell satisfies the above requirement. This method has its roots in the work of Parshall and Wang in the case of $q$-deformed quantum groups $\textrm {GL}$ and $\textrm {SL}$. Here we examine the example of universal cosovereign Hopf algebras, for which the weight group is the free group on two generators.

References

Nicolás Andruskiewitsch and Hans-Jürgen Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2) 171 (2010), no. 1, 375–417. MR 2630042, DOI 10.4007/annals.2010.171.375
Michael Artin, William Schelter, and John Tate, Quantum deformations of $\textrm {GL}_n$, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879–895. MR 1127037, DOI 10.1002/cpa.3160440804
Teodor Banica, Le groupe quantique compact libre $\textrm {U}(n)$, Comm. Math. Phys. 190 (1997), no. 1, 143–172 (French, with English summary). MR 1484551, DOI 10.1007/s002200050237
Teodor Banica and Julien Bichon, Free product formulae for quantum permutation groups, J. Inst. Math. Jussieu 6 (2007), no. 3, 381–414. MR 2329759, DOI 10.1017/S1474748007000072
Teodor Banica and Benoît Collins, Integration over quantum permutation groups, J. Funct. Anal. 242 (2007), no. 2, 641–657. MR 2274824, DOI 10.1016/j.jfa.2006.09.005
Teodor Banica, Stephen Curran, and Roland Speicher, De Finetti theorems for easy quantum groups, Ann. Probab. 40 (2012), no. 1, 401–435. MR 2917777, DOI 10.1214/10-AOP619
Teodor Banica and Roland Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), no. 4, 1461–1501. MR 2554941, DOI 10.1016/j.aim.2009.06.009
Julien Bichon, Cosovereign Hopf algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 121–133. MR 1812048, DOI 10.1016/S0022-4049(00)00024-4
Julien Bichon, Co-representation theory of universal co-sovereign Hopf algebras, J. Lond. Math. Soc. (2) 75 (2007), no. 1, 83–98. MR 2302731, DOI 10.1112/jlms/jdl007
Julien Bichon, Hopf-Galois objects and cogroupoids, Rev. Un. Mat. Argentina 55 (2014), no. 2, 11–69. MR 3285340
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492, DOI 10.1007/978-3-0348-8205-7
Alexandru Chirvasitu, Grothendieck rings of universal quantum groups, J. Algebra 349 (2012), 80–97. MR 2853627, DOI 10.1016/j.jalgebra.2011.09.020
E. E. Demidov, Yu. I. Manin, E. E. Mukhin, and D. V. Zhdanovich, Nonstandard quantum deformations of $\textrm {GL}(n)$ and constant solutions of the Yang-Baxter equation, Progr. Theoret. Phys. Suppl. 102 (1990), 203–218 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182166, DOI 10.1143/PTPS.102.203
Yukio Doi, Braided bialgebras and quadratic bialgebras, Comm. Algebra 21 (1993), no. 5, 1731–1749. MR 1213985, DOI 10.1080/00927879308824649
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Michel Dubois-Violette and Guy Launer, The quantum group of a nondegenerate bilinear form, Phys. Lett. B 245 (1990), no. 2, 175–177. MR 1068703, DOI 10.1016/0370-2693(90)90129-T
Pavel Etingof and Shlomo Gelaki, On cotriangular Hopf algebras, Amer. J. Math. 123 (2001), no. 4, 699–713. MR 1844575, DOI 10.1353/ajm.2001.0025
Gastón Andrés García, Quantum subgroups of $\textrm {GL}_{\alpha ,\beta }(n)$, J. Algebra 324 (2010), no. 6, 1392–1428. MR 2671812, DOI 10.1016/j.jalgebra.2010.07.020
D. I. Gurevich, Algebraic aspects of the quantum Yang-Baxter equation, Algebra i Analiz 2 (1990), no. 4, 119–148 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 4, 801–828. MR 1080202
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532, DOI 10.1090/gsm/006
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI 10.1007/BF00704588
Anatoli Klimyk and Konrad Schmüdgen, Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1492989, DOI 10.1007/978-3-642-60896-4
Claus Köstler and Roland Speicher, A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys. 291 (2009), no. 2, 473–490. MR 2530168, DOI 10.1007/s00220-009-0802-8
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879, DOI 10.1017/CBO9780511735127
Brian Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439, vi+157. MR 1048073, DOI 10.1090/memo/0439
Peter Schauenburg, Hopf bi-Galois extensions, Comm. Algebra 24 (1996), no. 12, 3797–3825. MR 1408508, DOI 10.1080/00927879608825788
Hans-Jürgen Schneider, Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), no. 2, 289–312. MR 1194305, DOI 10.1016/0021-8693(92)90034-J
Alfons Van Daele and Shuzhou Wang, Universal quantum groups, Internat. J. Math. 7 (1996), no. 2, 255–263. MR 1382726, DOI 10.1142/S0129167X96000153
Shuzhou Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), no. 3, 671–692. MR 1316765, DOI 10.1007/BF02101540
Shuzhou Wang, Problems in the theory of quantum groups, Quantum groups and quantum spaces (Warsaw, 1995) Banach Center Publ., vol. 40, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 67–78. MR 1481735
Shuzhou Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195–211. MR 1637425, DOI 10.1007/s002200050385
S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157, DOI 10.1007/BF01219077
S. Zakrzewski, A Hopf star-algebra of polynomials on the quantum $\textrm {SL}(2,\textbf {R})$ for a “unitary” $R$-matrix, Lett. Math. Phys. 22 (1991), no. 4, 287–289. MR 1131752, DOI 10.1007/BF00405603