Finite Quantum Groupoids and Their Applications

We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.

[1]  P. Etingof,et al.  Quantum Groups and Lie Theory: Lectures on the dynamical Yang-Baxter Equations , 1999, math/9908064.

[2]  J. Vallin.  Groupoı̈des Quantiques Finis , 2001 .

[3]  Michel Enock.  Inclusions of Von Neumann Algebras and Quantum Groupoı̈ds II , 2000, math/0405480.

[4]  P. Etingof,et al.  Dynamical quantum groups at roots of 1 , 2000, math/0003221.

[5]  D. Nikshych,et al.  A GALOIS CORRESPONDENCE FOR II 1 FACTORS AND QUANTUM GROUPOIDS , 2000, math/0001020.

[6]  G. Maltsiniotis.  Groupoïdes quantiques , 2000 .

[7]  D. Nikshych.  A Duality Theorem for Quantum Groupoids , 1999, math/9912226.

[8]  P. Etingof,et al.  Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras , 1999, math/9912009.

[9]  T. Hayashi.  Galois quantum groups of ${\rm II}_1$-subfactors , 1999 .

[10]  P. Etingof,et al.  Lectures on the dynamical Yang-Baxter equations , 1999, math/9908064.

[11]  G. Bòhm,et al.  Weak Hopf Algebras: II. Representation Theory, Dimensions, and the Markov Trace , 1999, math/9906045.

[12]  G. Böhm.  Doi-hopf modules over weak hopf algebras , 1999 .

[13]  G. Bòhm.  Doi-Hopf Modules over Weak Hopf Algebras , 1999, math/9905027.

[14]  A. Varchenko,et al.  Exchange Dynamical Quantum Groups , 1998, math/9801135.

[15]  M. Enock.  SOUS-FACTEURS INTERMEDIAIRES ET GROUPES QUANTIQUES MESURES , 1999 .

[16]  G. Bòhm,et al.  Weak C^*-Hopf Algebras and Multiplicative Isometries , 1998, math/9810070.

[17]  D. Nikshych.  Duality for actions of weak Kac algebras and crossed product inclusions of II_1 factors , 1998, math/9810049.

[18]  D. Nikshych,et al.  A Characterization of Depth 2 Subfactors of II1 Factors , 1998, math/9810028.

[19]  K. Szlachányi,et al.  Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed Products to Jones Towers , 1998, math/9806130.

[20]  G. Bohm,et al.  Weak Hopf Algebras: I. Integral Theory and C-Structure , 1998, math/9805116.

[21]  F. Nill.  Axioms for Weak Bialgebras , 1998, math/9805104.

[22]  T. Hayashi.  Face algebras I-A generalization of quantum group theory , 1998 .

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

[24]  E. Ragoucy,et al.  Universal Solutions of Quantum Dynamical Yang–Baxter Equations , 1997, q-alg/9712037.

[25]  P. Etingof,et al.  Some properties of finite-dimensional semisimple Hopf algebras , 1997, q-alg/9712033.

[26]  S. Popa,et al.  A Galois Correspondence for Compact Groups of Automorphisms of von Neumann Algebras with a Generalization to Kac Algebras , 1996, funct-an/9604004.

[27]  M. David.  Paragroupe d'Adrian Ocneanu et algèbre de Kac. , 1996 .

[28]  D. Bernard,et al.  A quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations , 1995, q-alg/9511019.

[29]  H. H. Andersen,et al.  Quantum groups at roots of ±1 , 1996 .

[30]  S. Majid.  Foundations of Quantum Group Theory , 1995 .

[31]  G. Bòhm,et al.  A coassociativeC*-quantum group with nonintegral dimensions , 1995, q-alg/9509008.

[32]  Jiang-Hua Lu.  Hopf algebroids and quantum groupoids , 1995, q-alg/9505024.

[33]  G. Felder.  Elliptic quantum groups , 1994, hep-th/9412207.

[34]  S. Popa.  Classification of amenable subfactors of type II , 1994 .

[35]  V. Turaev.  Quantum Invariants of Knots and 3-Manifolds , 1994, hep-th/9409028.

[36]  W. Szymański.  Finite index subfactors and Hopf algebra crossed products , 1994 .

[37]  W. Haboush,et al.  Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods , 1994 .

[38]  T. Yamanouchi.  Duality for Generalized Kac Algebras and a Characterization of Finite Groupoid Algebras , 1994 .

[39]  Susan Montgomery,et al.  Hopf algebras and their actions on rings , 1993 .

[40]  H. Kosaki,et al.  IRREDUCIBLE BIMODULES ASSOCIATED WITH CROSSED PRODUCT ALGEBRAS , 1992 .

[41]  L. Vainerman.  A note on quantum groupoids , 1992 .

[42]  Vladimir Turaev,et al.  Invariants of 3-manifolds via link polynomials and quantum groups , 1991 .

[43]  O. Babelon.  Universal exchange algebra for Bloch waves and Liouville theory , 1991 .

[44]  G. Lusztig.  Quantum groups at roots of 1 , 1990 .

[45]  Y. Watatani.  Index for C*-subalgebras , 1990 .

[46]  Frederick M. Goodman,et al.  Coxeter graphs and towers of algebras , 1989 .

[47]  A. Ocneanu.  Operator Algebras and Applications: Quantized groups, string algebras, and Galois theory for algebras , 1989 .

[48]  S. Popa,et al.  Iterating the basic construction , 1988 .

[49]  S. Popa,et al.  Entropy and index for subfactors , 1986 .

[50]  Susan Montgomery,et al.  A duality theorem for Hopf module algebras , 1985 .

[51]  I. Reiner,et al.  ASSOCIATIVE ALGEBRAS(Graduate Texts in Mathematics, 88) , 1983 .

[52]  V. Jones.  Index for subfactors , 1983 .

[53]  J. Renault.  A Groupoid Approach to C*-Algebras , 1980 .