Easy quantum groups and quantum subgroups of a semi-direct product quantum group

We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy quantum groups between the free orthogonal quantum group and the permutation group. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for easy quantum groups in the scope of this article.

[1]  S. Woronowicz.  A remark on compact matrix quantum groups , 1991 .

[2]  S. Popa,et al.  On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra , 2007, 0706.3623.

[3]  Debashish Goswami.  Quantum Group of Isometries in Classical and Noncommutative Geometry , 2007, 0704.0041.

[4]  J. Cooper,et al.  Les Algebres d'Operateurs dans l'Espace Hilbertien , 1958, The Mathematical Gazette.

[5]  Quanhua Xu,et al.  Khintchine type inequalities for reduced free products and applications , 2005, math/0505302.

[6]  T. Timmermann.  An Invitation to Quantum Groups and Duality , 2008 .

[7]  U. Haagerup.  An example of a non nuclearC*-algebra, which has the metric approximation property , 1978 .

[8]  Moritz Weber,et al.  On the classification of easy quantum groups , 2012, 1201.4723.

[9]  A. Sudbery.  QUANTUM GROUPS AND THEIR REPRESENTATIONS (Texts and Monographs in Physics) , 2000 .

[10]  Gizem Karaali.  Book Review: An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond , 2009 .

[11]  Moritz Weber,et al.  The combinatorics of an algebraic class of easy quantum groups , 2013, 1312.1497.

[12]  R. Richardson.  The International Congress of Mathematicians , 1932, Science.

[13]  Teodor Banica,et al.  Liberation of orthogonal Lie groups , 2008, 0808.2628.

[14]  S. Woronowicz.  Compact quantum groups , 2000 .

[15]  R. Speicher,et al.  Classification results for easy quantum groups , 2009, 0906.3890.

[16]  B. Collins,et al.  THE HYPEROCTAHEDRAL QUANTUM GROUP , 2007, math/0701859.

[17]  J. Cooper,et al.  Les Algebres d'Operateurs dans l'Espace Hilbertien , 1958 .

[18]  S. Woronowicz,et al.  Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups , 1988 .

[19]  Stuart White,et al.  The Haagerup property for locally compact quantum groups , 2013, 1303.3261.

[20]  M. Rosso.  Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif , 1990 .

[21]  Shuzhou Wang,et al.  Quantum Symmetry Groups of Finite Spaces , 1998, math/9807091.

[22]  Shahn Majid,et al.  Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction , 1990 .

[23]  A. Ol'šanskiĭ.  ON THE PROBLEM OF A FINITE BASIS OF IDENTITIES IN GROUPS , 1970 .

[24]  A. Connes.  On the cohomology of operator algebras , 1978 .

[25]  Cyril Houdayer.  Strongly solid group factors which are not interpolated free group factors , 2009, 0901.3866.

[26]  B. Collins,et al.  A Maximality Result for Orthogonal Quantum Groups , 2011, 1106.5467.

[27]  H. Neumann.  Varieties of Groups , 1967 .

[28]  Shuzhou Wang,et al.  Free products of compact quantum groups , 1995 .

[29]  Z. Ruan.  Amenability of Hopf von Neumann Algebras and Kac Algebras , 1996 .

[30]  Thomas Sinclair.  ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS SUR LA STRUCTURE DES FACTEURS DE TYPE II1 ASSOCIÉ AVEC LES GROUPES DE COURBURE NÉGATIVE , 2013 .

[31]  S. Woronowicz,et al.  Compact matrix pseudogroups , 1987 .

[32]  A. Valette,et al.  Groups with the Haagerup Property , 2001 .

[33]  Theworkof Alain Connes.  CLASSIFICATION OF INJECTIVE FACTORS , 1981 .

[34]  T. Banica.  Symmetries of a generic coaction , 1998, math/9811060.

[35]  R. Speicher,et al.  De Finetti theorems for easy quantum groups , 2009, 0907.3314.

[36]  Moritz Weber,et al.  The Full Classification of Orthogonal Easy Quantum Groups , 2013, 1312.3857.

[37]  R. Speicher,et al.  A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation , 2008, 0807.0677.

[38]  Sven Raum,et al.  A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups , 2012, 1212.4742.

[39]  K. Schmüdgen,et al.  Quantum Groups and Their Representations , 1998 .

[40]  J. Neumann.  Zur allgemeinen Theorie des Masses , 1929 .

[41]  Extensions of locally compact quantum groups and the bicrossed product construction , 2001, math/0101133.

[42]  B. H. Neumann.  Identical relations in groups. I , 1937 .

[43]  A. Connes,et al.  Classification of Injective Factors Cases II 1 , II ∞ , III λ , λ 1 , 1976 .

[44]  P. Jolissaint.  HAAGERUP APPROXIMATION PROPERTY FOR FINITE VON NEUMANN ALGEBRAS , 2002 .

[45]  Teodor Banica.  Theorie des representations du groupe quantique compact libre O(n) , 1998 .

[46]  U. Haagerup,et al.  Approximation properties for group *-algebras and group von Neumann algebras , 1994 .

[47]  S. Majid.  Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts, and the Classical Yang-Baxter Equations , 1991 .

[48]  Nathanial P. Brown Narutaka Ozawa.  C*-Algebras and Finite-Dimensional Approximations , 2008 .

[49]  M. Choda.  Group factors of the Haagerup type , 1983 .

[50]  U. Haagerup,et al.  Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups , 1985 .