9 research outputs found
Action of a finite quantum group on the algebra of complex NxN matrices
Using the fact that the algebra M := M_N(C) of NxN complex matrices can be
considered as a reduced quantum plane, and that it is a module algebra for a
finite dimensional Hopf algebra quotient H of U_q(sl(2)) when q is a root of
unity, we reduce this algebra M of matrices (assuming N odd) into
indecomposable modules for H. We also show how the same finite dimensional
quantum group acts on the space of generalized differential forms defined as
the reduced Wess Zumino complex associated with the algebra M.Comment: 11 pages, LaTeX, uses diagrams.sty, to be published in "Particles,
Fields and Gravitation" (Lodz conference), AIP proceeding
Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus
After giving a short description, in terms of action of categories, of some
of the structures associated with sl(2) and sl(3) boundary conformal field
theories on a torus, we provide tables of dimensions describing the semisimple
and co-semisimple blocks of the corresponding weak bialgebras (quantum
groupoids), tables of quantum dimensions and orders, and tables describing
induction - restriction. For reasons of size, the sl(3) tables of induction are
only given for theories with self-fusion (existence of a monoidal structure).Comment: 25 pages, 5 tables, 9 figures. Version 2: updated references. Typos
corrected. Several proofs added. Examples of ADE and generalized ADE
trigonometric identities have been removed to shorten the pape
Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings
Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we determine the algebras of quantum symmetries, obtain their generators, and, as a by-product, recover the known graphs E4, E6 and E8 describing exceptional quantum subgroups of type SU(4). We also obtain characteristic numbers (quantum cardinalities, dimensions) for each of them and for their associated quantum groupoïds
From modular invariants to graphs: the modular splitting method
We start with a given modular invariant M of a two dimensional su(n)_k
conformal field theory (CFT) and present a general method for solving the
Ocneanu modular splitting equation and then determine, in a step-by-step
explicit construction, 1) the generalized partition functions corresponding to
the introduction of boundary conditions and defect lines; 2) the quantum
symmetries of the higher ADE graph G associated to the initial modular
invariant M. Notice that one does not suppose here that the graph G is already
known, since it appears as a by-product of the calculations. We analyze several
su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected.
su(2) example has been removed to shorten the paper. Dual annular matrices
for the rejected exceptional su(3) diagram are determine
Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries
For every ADE Dynkin diagram, we give a realization, in terms of usual fusion
algebras (graph algebras), of the algebra of quantum symmetries described by
the associated Ocneanu graph. We give explicitly, in each case, the list of the
corresponding twisted partition functionsComment: 47 pages, 25 figures and 17 tables. Several typos are corrected.
References adde