Proceedings - Mathematical Sciences

, Volume 127, Issue 5, pp 881–933 | Cite as

Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups

  • Moritz WeberEmail author


This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5–24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz’s quantum version of the Tannaka–Krein theorem. Building on this, we define Banica–Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica–Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica–Speicher quantum groups and we list some open problems.


Compact quantum groups compact matrix quantum groups easy quantum groups Banica–Speicher quantum groups noncrossing partitions categories of partitions tensor categories Tannaka–Krein duality 

Mathematics Subject Classification

20G42 05A18 46LXX 



The author would like to thank V. S. Sunder for his invitation to IMSc, Chennai, for his great hospitality and his vision and support to get these lecture notes produced. He would like to wish him all the best for his retirement, in deepest admiration for all of his mathematical achievements. Furthermore, he thanks Soumya for all his great work regarding the contents and the design of these notes and for providing all the figures. The author thanks Pierre Fima and Issan Patri for their various and valuable critical remarks throughout the lectures at IMSc. He also thanks Felix Leid and Simon Schmidt for improvements regarding the exposition in the chapter on Tannaka–Krein theory. The author was partially supported by the ERC Advanced Grant NCDFP, held by Roland Speicher.


  1. 1.
    Banica T, Belinschi S T, Capitaine M and Collins B, Free Bessel laws, Canad. J. Math. 63(1) (2011) 3–37MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Banica T, Théorie des représentations du groupe quantique compact libre \({{O}}(n)\), C. R. Acad. Sci. Paris Sér. I Math. 322(3) (1996) 241–244MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banica T, Le groupe quantique compact libre \({\mathit{U}}(n)\), Comm. Math. Phys. 190(1) (1997) 143–172MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Banica T, Symmetries of a generic coaction, Math. Ann. 314(4) (1999) 763–780.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Banica T, Quantum groups and Fuss-Catalan algebras, Comm. Math. Phys., 226(1) (2002) 221–232MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Banica T, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224(2) (2005) 243–280MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Banica T, Quantum permutations, Hadamard matrices, and the search for matrix models, in: Operator algebras and quantum groups, volume 98 of Banach Center Publ. (2012) (Polish Acad. Sci. Inst. Math., Warsaw) pp. 11–42Google Scholar
  8. 8.
    Banica T, Liberations and twists of real and complex spheres, J. Geom. Phys. 96 (2015) 1–25MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Banica T, Quantum isometries of noncommutative polygonal spheres, Münster J. Math. 8(1) (2015) 253–284MathSciNetzbMATHGoogle Scholar
  10. 10.
    Banica T, Quantum groups from stationary matrix models, Colloq. Math. 148(2) (2017) 247–267MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Banica T, Super-easy quantum groups: definition and examples, arXiv:1706.00152 (2017)
  12. 12.
    Banica T and Bichon J, Quantum automorphism groups of vertex-transitive graphs of order \(\le \)11, J. Algebraic Combin. 26(1) (2007) 83–105zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Banica T and Bichon J, Random walk questions for linear quantum groups, Int. Math. Res. Not. IMRN 24 (2015) 13406–13436Google Scholar
  14. 14.
    Banica T, Bichon J and Chenevier G, Graphs having no quantum symmetry, Ann. Inst. Fourier (2007) 955–971Google Scholar
  15. 15.
    Banica T, Bichon J and Collins B, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22(4) (2007) 345–384MathSciNetzbMATHGoogle Scholar
  16. 16.
    Banica T, Bichon J and Collins B, Quantum permutation groups: a survey, in: Noncommutative harmonic analysis with applications to probability, volume 78 of Banach Center Publ. (2007) (Polish Acad. Sci. Inst. Math., Warsaw) pp. 13–34Google Scholar
  17. 17.
    Banica T, Bichon J, Collins B and Curran S, A maximality result for orthogonal quantum groups, Comm. Algebra 41(2) (2013) 656–665MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Banica T and Collins B, Integration over compact quantum groups. Publ. Res. Inst. Math. Sci. 43(2) (2007) 277–302MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Banica T and Collins B, Integration over quantum permutation groups, J. Funct. Anal. 242(2) (2007) 641–657MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Banica T and Collins B, Integration over the Pauli quantum group, J. Geom. Phys. 58(8) (2008) 942–961MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Banica T, Collins B and Zinn-Justin P, Spectral analysis of the free orthogonal matrix, Int. Math. Res. Not. IMRN 17 (2009) 3286–3309Google Scholar
  22. 22.
    Banica T, Curran S and Speicher R, Classification results for easy quantum groups, Pacific J. Math. 247(1) (2010) 1–26MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Banica T, Curran S and Speicher R, Stochastic aspects of easy quantum groups, Probab. Theory Related Fields 149(3–4) (2011) 435–462MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Banica T, Curran S and Speicher R, De Finetti theorems for easy quantum groups, Ann. Probab. 40(1) (2012) 401–435MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Banica T and Freslon A, Modelling questions for quantum permutations, arXiv:1704.00290 (2017)
  26. 26.
    Banica T and Goswami D, Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298(2) (2010) 343–356MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Banica T and Nechita I, Flat matrix models for quantum permutation groups, Adv. in Appl. Math. 83 (2017) 24–46MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Banica T and Nicoara R, Quantum groups and Hadamard matrices, Pan. Amer. Math. J. 17(1) (2007) 1–24MathSciNetzbMATHGoogle Scholar
  29. 29.
    Banica T and Skalski A, Two-parameter families of quantum symmetry groups, J. Funct. Anal. 260(11) (2011) 3252–3282MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Banica T, Skalski A and Sołtan P, Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys. 62(6) (2012) 1451–1466MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Banica T and Speicher R, Liberation of orthogonal Lie groupsm, Adv. Math.  222(4) (2009) 1461–1501MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Banica T and Vergnioux R, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3(3) (2009) 327–359MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Banica T and Vergnioux R, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier (Grenoble) 60(6) (2010) 2137–2164MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Bhattacharya A and Wang S, Kirchberg’s factorization property for discrete quantum groups, Bull. Lond. Math. Soc. 48(5) (2016) 866–876MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Bhowmick J and Goswami D, Quantum isometry groups: examples and computations, Comm. Math. Phys. 285(2) (2009) 421–444MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Bichon J, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc.  131(3) (2003) 665–673MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Bichon J, Free wreath product by the quantum permutation group, Algebr. Represent. Theory, 7(4) (2004) 343–362MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Bichon J, Gerstenhaber–Schack and Hochschild cohomologies of Hopf algebras, Doc. Math. 21 (2016) 955–986MathSciNetzbMATHGoogle Scholar
  39. 39.
    Bichon J and Dubois-Violette M, Half-commutative orthogonal Hopf algebras, Pacific J. Math. 263(1) (2013) 13–28MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Bichon J, Franz U and Gerhold M, Homological properties of quantum permutation algebras, arXiv:1704.00589 (2017)
  41. 41.
    Bichon J, Kyed D and Raum S, Higher \(l^2\)-Betti numbers of universal quantum groups, arXiv:1612.07706 (2016)
  42. 42.
    Bisch D and Jones V, Algebras associated to intermediate subfactors, Invent. Math. 128(1) (1997) 89–157MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Brannan M, Approximation properties for free orthogonal and free unitary quantum groups, J. Reine Angew. Math. 672 (2012) 223–251MathSciNetzbMATHGoogle Scholar
  44. 44.
    Brannan M, Quantum symmetries and strong Haagerup inequalities, Comm. Math. Phys. 311(1) (2012) 21–53MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Brannan M, Reduced operator algebras of trace-perserving quantum automorphism groups, Doc. Math. 18 (2013) 1349–1402MathSciNetzbMATHGoogle Scholar
  46. 46.
    Brannan M, Strong asymptotic freeness for free orthogonal quantum groups, Canad. Math. Bull. 57(4) (2014) 708–720MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Brannan M and Collins B, Highly entangled, non-random subspaces of tensor products from quantum groups, arXiv:1612.09598 (2016)
  48. 48.
    Brannan M, Collins B and Vergnioux R, The Connes embedding property for quantum group von Neumann algebras, Trans. Amer. Math. Soc.  369(6) (2017) 3799–3819MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Brannan M and Vergnioux R, Orthogonal free quantum group factors are strongly 1-bounded, arXiv:1703.08134 (2017)
  50. 50.
    Brauer R, On algebras which are connected with the semisimple continuous groups, Ann. Math. (2) , 38(4) (1937) 857–872MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Cébron G and Weber M, Quantum groups based on spatial partitions, arXiv:1609.02321 (2016)
  52. 52.
    Chassaniol A, Quantum automorphism group of the lexicographic product of finite regular graphs, J. Algebra (2016) 23–45Google Scholar
  53. 53.
    Chirvasitu A, Free unitary groups are (almost) simple, J. Math. Phys. 53(12) (2012) 123509, 7MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Chirvasitu A, Residually finite quantum group algebras, J. Funct. Anal. 268(11) (2015) 3508–3533MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Cipriani F, Franz U and Kula A, Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory, J. Funct. Anal. 266(5) (2014) 2789–2844MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Collins B and Sho M, Weingarten calculus via orthogonality relations: new applications, arXiv:1701.04493 (2017)
  57. 57.
    Curran S, Quantum exchangeable sequences of algebras, Indiana Univ. Math. J. 58(3) (2009) 1097–1125MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Curran S. Quantum rotatability, Trans. Amer. Math. Soc. 362(9) (2010) 4831–4851MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Curran S, A characterization of freeness by invariance under quantum spreading, J. Reine Angew. Math. 659 (2011) 43–65MathSciNetzbMATHGoogle Scholar
  60. 60.
    Curran S and Speicher R, Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices, Comm. Math. Phys. 301(3) (2011) 627–659MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Curran S and Speicher R, Quantum invariant families of matrices in free probability, J. Funct. Anal., 261(4) (2011) 897–933Google Scholar
  62. 62.
    De Commer K, Freslon A and Yamashita M, CCAP for universal discrete quantum groups, Comm. Math. Phys. 331(2) (2014) 677–701. With an appendix by Stefaan VaesGoogle Scholar
  63. 63.
    Enock M and Schwartz J-M, Kac algebras and duality of locally compact groups (1992) (Berlin: Springer-Verlag),  With a preface by Alain Connes, with a postface by Adrian OcneanuGoogle Scholar
  64. 64.
    Fima P and Pittau L, The free wreath product of a compact quantum group by a quantum automorphism group, J. Funct. Anal. 271(7) (2016) 1996–2043MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Franz U, Hong G, Lemeux F, Ulrich M and Zhang H, Hypercontractivity of heat semigroups on free quantum groups, J. Operator Theory 77(1) (2017) 61–76MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Franz U, Kula A and Skalski A, Lévy processes on quantum permutation groups, in: Noncommutative analysis, operator theory and applications, volume 252 of Oper. Theory Adv. Appl. (2016) (Birkhäuser/Springer) [Cham] pp. 193–259Google Scholar
  67. 67.
    Freslon A, Examples of weakly amenable discrete quantum groups, J. Funct. Anal. 265(9) (2013) 2164–2187MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Freslon A, Fusion (semi)rings arising from quantum groups, J. Algebra, 417 (2014) 161–197MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Freslon A, On the partition approach to Schur–Weyl duality and free quantum groups, Transformation groups (2016) 1–45Google Scholar
  70. 70.
    Freslon A and Skalski A, Wreath products of finite groups by quantum groups, arXiv:1510.05238 (2015)
  71. 71.
    Freslon A and Vergnioux R, The radial MASA in free orthogonal quantum groups, J. Funct. Anal. 271(10) (2016) 2776–2807MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Freslon A and Weber M, On bi-free de Finetti theorems, Ann. Math. Blaise Pascal  23(1) (2016) 21–51MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Freslon A and Weber M, On the representation theory of partition (easy) quantum groups, J. Reine Angew. Math. 720 (2016) 155–197MathSciNetzbMATHGoogle Scholar
  74. 74.
    Fulton M, The quantum automorphism group and undirected trees. Ph.D. Thesis, Virginia (2006)Google Scholar
  75. 75.
    Gabriel O and Weber M, Fixed point algebras for easy quantum groups. SIGMA Symmetry Integrability Geom. Methods Appl. 12 Paper No. 097, 21 (2016)Google Scholar
  76. 76.
    Goswami D and Bhowmick J, Quantum isometry groups, Infosys Science Foundation Series, Springer, New Delhi (2016), Infosys Science Foundation Series in Mathematical Sciences.Google Scholar
  77. 77.
    José M, Gracia-Bondí A, Várilly J C and Figueroa H, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA (2001)Google Scholar
  78. 78.
    Hayase T, De Finetti theorems for a Boolean analogue of easy quantum groups, arXiv:1507.05563 (2015)
  79. 79.
    Isono Y, Examples of factors which have no Cartan subalgebras, Trans. Amer. Math. Soc. 367(11) (2015) 7917–7937MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Isono Y, Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors, arXiv:1607.02847 (2016)
  81. 81.
    Józiak P, Remark on Hopf images in quantum permutation groups \(S_n^+,\) arXiv:1611.09211 (2016)
  82. 82.
    Kassel C, Quantum groups, volume 155 of Graduate Texts in Mathematics (1995) (Springer-Verlag, New York)Google Scholar
  83. 83.
    Kauffman L H, State models and the Jones polynomial, Topology 26(3) (1987) 395–407MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Klimyk A and Schmüdgen K, Quantum groups and their representations, Texts and Monographs in Physics (1997) (Berlin: Springer-Verlag)Google Scholar
  85. 85.
    Köstler C and Speicher R, A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys. 291(2) (2009) 473–490MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Kustermans J and Tuset L, A survey of \(C^*\)-algebraic quantum groups, I, Irish Math. Soc. Bull. 43 (1999) 8–63Google Scholar
  87. 87.
    Kustermans J and Tuset L, A survey of \(C^*\)-algebraic quantum groups, II, Irish Math. Soc. Bull. 44 (2000) 6–54Google Scholar
  88. 88.
    Kustermans J and Vaes S, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92(1) (2003) 68–92MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Kyed D and Raum S, On the \(l^2\)-Betti numbers of universal quantum groups, arXiv:1610.05474 (2016)
  90. 90.
    Lemeux F, The fusion rules of some free wreath product quantum groups and applications, J. Funct. Anal. 267(7) (2014) 2507–2550MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Lemeux F, Haagerup approximation property for quantum reflection groups, Proc. Amer. Math. Soc. 143(5) (2015) 2017–2031MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Lemeux F and Tarrago P, Free wreath product quantum groups: the monoidal category, approximation properties and free probability, J. Funct. Anal., 270(10) (2016) 3828–3883MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Liu W. Extended de Finetti theorems for boolean independence and monotone independence, arXiv:1505.02215 (2015)
  94. 94.
    Lusztig G, Introduction to quantum groups, volume 110 of Progress in Mathematics (1993) (Boston, MA: Birkhäuser Boston Inc.)Google Scholar
  95. 95.
    Maes A and Van Daele A, Notes on compact quantum groups, Nieuw Arch. Wisk. (4)  16(1–2) (1998) 73–112Google Scholar
  96. 96.
    Malacarne S, Woronowicz’s Tannaka–Krein duality and free orthogonal quantum groups, arXiv:1602.04807 (2016)
  97. 97.
    Malacarne S and Neshveyev S, Probabilistic boundaries of finite extensions of quantum groups, arXiv:1704.04717 (2017)
  98. 98.
    Neshveyev S and Tuset L, Compact quantum groups and their representation categories, volume 20 of Cours Spécialisés [Specialized Courses] (2013) (Paris: Société Mathématique de France)Google Scholar
  99. 99.
    Nica A and Speicher R, Lectures on the combinatorics of free probability, volume 335 of London Mathematical Society Lecture Note Series (2006) (Cambridge: Cambridge University Press)Google Scholar
  100. 100.
    Raum S, Isomorphisms and fusion rules of orthogonal free quantum groups and their free complexifications, Proc. Amer. Math. Soc. 140(9) (2012) 3207–3218MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Raum S and Weber M, The combinatorics of an algebraic class of easy quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17(3) (2014) 1450016, 17Google Scholar
  102. 102.
    Raum S and Weber M, Easy quantum groups and quantum subgroups of a semi-direct product quantum group, J. Noncommut. Geom. 9(4) (2015) 1261–1293MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Raum S and Weber M, The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341(3) (2016) 751–779MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Schmidt S and Weber M, Quantum symmetries of graph \(C\)*-algebras, arXiv:1706.08833 (2017)
  105. 105.
    Speicher R, A new example of ‘independence’ and ‘white noise’, Probab. Theory Related Fields 84(2) (1990) 141–159MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Speicher R and Weber M, Quantum groups with partial commutation relations, arXiv:1603.09192 (2016)
  107. 107.
    Tarrago P and Wahl J, Free wreath product quantum groups and standard invariants of subfactors, arXiv:1609.01931 (2016)
  108. 108.
    Tarrago P and Weber M, Appendix, in: Unitary easy quantum groups: the free case and the group case, arXiv:1512.00195 (2015)
  109. 109.
    Tarrago P and Weber M, The classification of tensor categories of two-colored noncrossing partitions, arXiv:1509.00988 (2015)
  110. 110.
    Tarrago P and Weber M. Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Not. (2016)Google Scholar
  111. 111.
    Timmermann T, An invitation to quantum groups and duality, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, (2008) From Hopf algebras to multiplicative unitaries and beyondGoogle Scholar
  112. 112.
    Vaes S and Vergnioux R, The boundary of universal discrete quantum groups, exactness, and factoriality, Duke Math. J. 140(1) (2007) 35–84MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Van Daele A, Dual pairs of Hopf \(*\)-algebras, Bull. London Math. Soc., 25(3) (1993) 209–230MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Van Daele A, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123(10) (1995) 3125–3128MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Van Daele A and Wang S, Universal quantum groups, Internat. J. Math. 7(2) (1996) 255–263MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Vergnioux R, Paths in quantum Cayley trees and \(L^2\)-cohomology, Adv. Math. 229(5) (2012) 2686–2711MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Vergnioux R and Voigt C, The \(K\)-theory of free quantum groups, Math. Ann. 357(1) (2013) 355–400MathSciNetzbMATHCrossRefGoogle Scholar
  118. 118.
    Voiculescu D-V, Stammeier N and Weber M, Free probability and operator algebras, Münster Lecture Notes in Mathematics (2016) (Zürich: Eur. Math. Soc. (EMS))Google Scholar
  119. 119.
    Voigt C, The Baum-Connes conjecture for free orthogonal quantum groups, Adv. Math., 227(5) (2011) 1873–1913MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Voigt C, On the structure of quantum automorphism groups, J. Reine Angew. Math. (2015)Google Scholar
  121. 121.
    Wahl J, A note on reduced and von Neumann algebraic free wreath products, Illinois J. Math. 59(3) (2015) 801–817MathSciNetzbMATHGoogle Scholar
  122. 122.
    Wang S, Free products of compact quantum groups, Comm. Math. Phys. 167(3) (1995) 671–692MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Wang S, Tensor products and crossed products of compact quantum groups, Proc. London Math. Soc. (3), 71(3) (1995) 695–720Google Scholar
  124. 124.
    Wang S, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195(1) (1998) 195–211MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    Wang S, Simple compact quantum groups, I, J. Funct. Anal. 256(10) (2009) 3313–3341MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    Wang S, On the problem of classifying simple compact quantum groups, in: Operator algebras and quantum groups, volume 98 of Banach Center Publ. (2012) (Warsaw: Polish Acad. Sci. Inst. Math.) pp. 433–453Google Scholar
  127. 127.
    Wang S, Equivalent notions of normal quantum subgroups, compact quantum groups with properties \(F\), and other applications, J. Algebra 397 (2014) 515–534MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    Weber M, On the classification of easy quantum groups, Adv. Math., 245 (2013) 500–533MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    Weber M, Basics in free probability, in: Free probability and operator algebras, Münster Lectures in Mathematics (2016) (Zürich: European Mathematical Society (EMS))Google Scholar
  130. 130.
    Weber M, Easy quantum groups, in: Free probability and operator algebras, Münster Lectures in Mathematics (2016) (Zürich: European Mathematical Society (EMS))Google Scholar
  131. 131.
    Weingarten D. Asymptotic behavior of group integrals in the limit of infinite rank, J. Mathematical Phys. 19(5) (1978) 999–1001MathSciNetzbMATHCrossRefGoogle Scholar
  132. 132.
    Woronowicz S L, Compact matrix pseudogroups, Comm. Math. Phys. 111(4) (1987) 613–665MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Woronowicz S L, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23(1) (1987) 117–181MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    Woronowicz S L, Tannaka–Kreĭn duality for compact matrix pseudogroups, Twisted \(\text{ SU }(N)\) groups, Invent. Math. 93(1) (1988) 35–76MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    Woronowicz S L, A remark on compact matrix quantum groups, Lett. Math. Phys. 21(1) (1991) 35–39MathSciNetzbMATHCrossRefGoogle Scholar
  136. 136.
    Woronowicz S L, Compact quantum groups, in: Symétries quantiques (Les Houches, 1995) (1998) (Amsterdam: North-Holland) pp. 845–884Google Scholar

Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  1. 1.Saarland UniversitySaarbrückenGermany

Personalised recommendations