294 research outputs found

### 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

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