Triangular dissections, aperiodic tilings and Jones algebras

The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type $A_n$ determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of $\pi/ (n+1).$ There are usually several possible infinite dissections compatible with a given $n$ but a given one makes use of $n/2$ triangle types if $n$ is even. Jones algebra with index $[ 4 \ \cos^2{\pi \over n+1}]^{-1}$ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case $n=4$, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The ``tilings'' associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints) using $n/2$ digits (if $n$ is even) and generalizing the Fibonacci numbers.Comment: 14 pages. Revised version. 18 Postcript figures, a 500 kb uuencoded file called images.uu available by mosaic or gopher from gopher://cpt.univ-mrs.fr/11/preprints/94/fundamental-interactions/94-P.302

Theta functions for lattices of SU(3) hyper-roots

We recall the definition of the hyper-roots that can be associated to modules-categories over the fusion categories defined by the choice of a simple Lie group G together with a positive integer k. This definition was proposed in 2000, using another language, by Adrian Ocneanu. If G=SU(2), the obtained hyper-roots coincide with the usual roots for ADE Dynkin diagrams. We consider the associated lattices when G=SU(3) and determine their theta functions in a number of cases; these functions can be expressed as modular forms twisted by appropriate Dirichlet characters.Comment: 33 pages, 9 figure

Character tables (modular data) for Drinfeld doubles of finite groups

In view of applications to conformal field theory or to other branches of theoretical physics and mathematics, new examples of character tables for Drinfeld doubles of finite groups (modular data) are made available on a website.Comment: 7 pages, 1 figure, 7th International Conference on Mathematical Methods in Physics, Rio de Janeiro, Brazil, April 2012. Version 2: a misleading sentence was removed from section 2. http://pos.sissa.it/archive/conferences/175/024/ICMP%202012_024.pd

Currents on Grassmann algebras

We define currents on a Grassmann algebra $Gr(N)$ with $N$ generators as distributions on its exterior algebra (using the symmetric wedge product). We interpret the currents in terms of ${\Z}_2$-graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on $Gr(N)$). An explicit construction of the vector space of closed currents of degree $p$ on $Gr(N)$ is given by using Berezin integration.Comment: 20 pages, CPT-93/P.2935 and ENSLAPP-440/9

Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups

Global dimensions for fusion categories defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum dimensions of simple objects, for the category of integrable modules over an affine Lie algebra at some level. The same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory WZW models. Similar results are also presented for those associated module-categories that can be obtained via conformal embeddings (they are "quantum subgroups" of a particular kind). As a side result, we express the classical (or quantum) Weyl denominator of simple Lie groups in terms of products of classical (or quantum) factorials calculated for the exponents of the group. Some calculations use the correspondence existing between periodic quivers for simply-laced Lie groups and fusion rules for module-categories (alias nimreps) of type SU(2).Comment: 23 pages. Improvements suggested by a referee: those parts that had much overlap with previous work of the author have been removed, the section discussing the relation between SU2 fusion numbers and roots and weights of Lie groups was enlarged, new reference [9], new appendi

Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups

We obtain formulae giving global dimensions for fusion categories defined by Lie groups G at level k and for the associated module-categories obtained via conformal embeddings. The results can be expressed in terms of Lie quantum superfactorials of type G. The later are related, for the type Ar, to the quantum Barnes function.Comment: 20 pages, talk given at: Coloquio de Algebras de Hopf, Grupos Cuanticos y Categorias Tensoriales, Cordoba, Argentina, 200

Clifford algebras, spinors and fundamental interactions : Twenty Years After

This is a short review of the algebraic properties of Clifford algebras and spinors. Their use in the description of fundamental physics (elementary particles) is also summarized. Lecture given at the ICCA7 conference, Toulouse (23/05/2005)Comment: 14 page

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