Module categories, weak Hopf algebras and modular invariants

AbstractWe develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.

[1]  Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases , 1994, hep-th/9405183.

[2]  Isocategorical groups , 2000, math/0007196.

[3]  C. Itzykson,et al.  The A-D-E classification of minimal andA1(1) conformal invariant theories , 1987 .

[4]  J.-B. Zuber,et al.  The many faces of Ocneanu cells , 2001 .

[5]  Alain Bruguières.  Catégories prémodulaires, modularisations et invariants des variétés de dimension 3 , 2000 .

[6]  France,et al.  Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries , 2002 .

[7]  B. Pareigis.  Non-additive Ring and Module Theory , 1977 .

[8]  N. Reshetikhin,et al.  Quantum Groups , 1993, hep-th/9311069.

[9]  A. Kirillov,et al.  Lectures on tensor categories and modular functors , 2000 .

[10]  V. Ostrik,et al.  On a q-Analogue of the McKay Correspondence and the ADE Classification of sl̂2 Conformal Field Theories , 2002 .

[11]  J. Zuber,et al.  SU($N$) Lattice Integrable Models Associated With Graphs , 1990 .

[12]  Boundary conformal field theory and fusion ring representations , 2001, hep-th/0106105.

[13]  On fusion categories , 2002, math/0203060.

[14]  Finite Quantum Groupoids and Their Applications , 2000, math/0006057.

[15]  Representations of tensor categories and Dynkin diagrams , 1994, hep-th/9408078.

[16]  M. Movshev.  Twisting in group algebras of finite groups , 1993 .

[17]  V. Kac.  Infinite dimensional Lie algebras: Frontmatter , 1990 .

[18]  Pierre Mathieu,et al.  Conformal Field Theory , 1999 .

[19]  G. Mack,et al.  Quasi Hopf quantum symmetry in quantum theory , 1992 .

[20]  David E. Evans,et al.  Chiral Structure of Modular Invariants for Subfactors , 1999, math/9907149.

[21]  D. Tambara,et al.  Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups , 1998 .

[22]  Group categories and their field theories , 1998, math/9811047.

[23]  G. Lusztig.  On Quiver Varieties , 1998 .

[24]  Akishi Kato.  Classification of Modular Invariant Partition Functions in Two Dimensions , 1987 .

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

[26]  M. Finkelberg.  An equivalence of fusion categories , 1996 .

[27]  D. Kazhdan,et al.  Reconstructing monoidal categories , 1993 .

[28]  T. Kerler.  Quantum groups, quantum categories, and quantum field theory , 1993 .

[29]  Operator algebras and conformal field theory III. Fusion of positive energy representations of LSU(N) using bounded operators , 1998, math/9806031.

[30]  C. Schweigert,et al.  Category theory for conformal boundary conditions , 2001, math/0106050.

[31]  Boundary conditions in rational conformal field theories , 2000 .

[32]  New Braided Endomorphisms from Conformal Inclusions , 1997, q-alg/9708011.

[33]  V. Ostrik,et al.  On tensor categories attached to cells in affine Weyl groups, III , 2000, math/0010089.