Advances in Operator Theory

Quantum groups, from a functional analysis perspective

Teodor Banica

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‎It is well-known that any compact Lie group appears as closed subgroup of a unitary group‎, ‎$G\subset U_N$‎. ‎The unitary group $U_N$ has a free analogue $U_N^+$‎, ‎and the study of the closed quantum subgroups $G\subset U_N^+$ is a problem of general interest‎. ‎We review here the basic tools for dealing with such quantum groups‎, ‎with all the needed preliminaries included‎, ‎and we discuss as well a number of more advanced topics‎.

Article information

Adv. Oper. Theory, Volume 4, Number 1 (2019), 164-196.

Received: 11 April 2018
Accepted: 8 May 2018
First available in Project Euclid: 8 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L65: Quantizations, deformations
Secondary: 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22]

quantum group ‎free unitary group ‎operator algebra


Banica, Teodor. Quantum groups, from a functional analysis perspective. Adv. Oper. Theory 4 (2019), no. 1, 164--196. doi:10.15352/aot.1804-1342.

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  • T. Banica, Liberations and twists of real and complex spheres, J. Geom. Phys. 96 (2015), 1–25.
  • T. Banica, Unitary easy quantum groups: geometric aspects, J. Geom. Phys. 126 (2018), 127–147.
  • T. Banica, S. T. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), 3–37.
  • T. Banica and J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. 52 (2010), 677–703.
  • T. Banica and J. Bichon, Matrix models for noncommutative algebraic manifolds, J. Lond. Math. Soc. 95 (2017), 519–540.
  • T. Banica, J. Bichon, and B. Collins, Quantum permutation groups: a survey, Banach Center Publ. 78 (2007), 13–34.
  • T. Banica and A. Chirvasitu, Thoma type results for discrete quantum groups, Internat. J. Math. 28 (2017), 1–23.
  • T. Banica and B. Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. 43 (2007), 277–302.
  • T. Banica and S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. 51 (2010), 1–14.
  • T. Banica, S. Curran, and R. Speicher, Classification results for easy quantum groups, Pacific J. Math. 247 (2010), 1–26.
  • T. Banica, U. Franz, and A. Skalski, Idempotent states and the inner linearity property, Bull. Pol. Acad. Sci. Math. 60 (2012), 123–132.
  • T. Banica and I. Nechita, Block-modified Wishart matrices and free Poisson laws, Houston J. Math. 41 (2015), 113–134.
  • T. Banica and I. Patri, Maximal torus theory for compact quantum groups, Illinois J. Math. 61 (2017), 151–170.
  • T. Banica, A. Skalski, and P. M. Sołtan, Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys. 62 (2012), 1451–1466.
  • T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461–1501.
  • T. Banica and R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60 (2010), 2137–2164.
  • H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. 149 (1999), 1023–1060.
  • J. Bhowmick, F. D'Andrea, and L. Dabrowski, Quantum isometries of the finite noncommutative geometry of the standard model, Comm. Math. Phys. 307 (2011), 101–131.
  • J. Bichon and M. Dubois-Violette, Half-commutative orthogonal Hopf algebras, Pacific J. Math. 263 (2013), 13–28.
  • M. Brannan, B. Collins, and R. Vergnioux, The Connes embedding property for quantum group von Neumann algebras, Trans. Amer. Math. Soc. 369 (2017), 3799–3819.
  • R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857–872.
  • A. Chirvasitu, Residually finite quantum group algebras, J. Funct. Anal. 268 (2015), 3508–3533.
  • F. Cipriani, U. Franz, and A. Kula, Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory, J. Funct. Anal. 266 (2014), 2789–2844.
  • A. Connes, Noncommutative Geometry, Academic Press (1994).
  • A. D'Andrea, C. Pinzari, and S. Rossi, Polynomial growth for compact quantum groups, topological dimension and $*$-regularity of the Fourier algebra, preprint 2016.
  • B. Das and D. Goswami, Quantum Brownian motion on noncommutative manifolds: construction, deformation and exit times, Comm. Math. Phys. 309 (2012), 193–228.
  • L. Faddeev, Instructive history of the quantum inverse scattering method, Acta Appl. Math. 39 (1995), 69–84.
  • U. Franz and A. Skalski, On idempotent states on quantum groups, J. Algebra 322 (2009), 1774–1802.
  • A. Freslon, On the partition approach to Schur-Weyl duality and free quantum groups, Transform. Groups 22 (2017), 707–751.
  • M. Fukuda and P. Śniady, Partial transpose of random quantum states: exact formulas and meanders, J. Math. Phys. 54 (2013), 1–31.
  • D. Goswami, Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys. 285 (2009), 141–160.
  • C. Köstler and R. Speicher, A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys. 291 (2009), 473–490.
  • J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup. 33 (2000), 837–934.
  • F. Lemeux and P. Tarrago, Free wreath product quantum groups: the monoidal category, approximation properties and free probability, J. Funct. Anal. 270 (2016), 3828–3883.
  • A. Maes and A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wisk. 16 (1998), 73–112.
  • S. Malacarne, Woronowicz's Tannaka-Krein duality and free orthogonal quantum groups, Math. Scand. 122 (2018), 151–160.
  • J.A. Mingo and M. Popa, Freeness and the partial transposes of Wishart random matrices, preprint 2017.
  • S. Neshveyev and L. Tuset, Compact Quantum Groups and their Representation Categories, SMF (2013).
  • G.K. Pedersen, C$^*$-Algebras and their Automorphism Groups, Academic Press (1979).
  • P. Podleś and S.L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990), 381–431.
  • S. Raum and M. Weber, The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341 (2016), 751–779.
  • P. Tarrago and M. Weber, Unitary easy quantum groups: the free case and the group case, preprint 2015.
  • S. Vaes and R. Vergnioux, The boundary of universal discrete quantum groups, exactness and factoriality, Duke Math. J. 140 (2007), 35–84.
  • R. Vergnioux and C. Voigt, The K-theory of free quantum groups, Math. Ann. 357 (2013), 355–400.
  • D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, AMS (1992).
  • S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671–692.
  • S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211.
  • S. Wang, $L_p$-improving convolution operators on finite quantum groups, Indiana Univ. Math. J. 65 (2016), 1609–1637.
  • H. Wenzl, C$^*$-tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), 261–282.
  • S. L. Woronowicz, Twisted $SU(2)$ group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117–181.
  • S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.
  • S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35–76.