70 research outputs found
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
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
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
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
On quantum symmetries of ADE graphs
The double triangle algebra(DTA) associated to an ADE graph is considered. A
description of its bialgebra structure based on a reconstruction approach is
given. This approach takes as initial data the representation theory of the DTA
as given by Ocneanu's cell calculus. It is also proved that the resulting DTA
has the structure of a weak *-Hopf algebra. As an illustrative example, the
case of the graph A3 is described in detail.Comment: 15 page
Large scale geometry and evolution of a universe with radiation pressure and cosmological constant
In view of new experimental results that strongly suggest a non-zero
cosmological constant, it becomes interesting to revisit the Friedman-Lemaitre
model of evolution of a universe with cosmological constant and radiation
pressure. In this paper, we discuss the explicit solutions for that model and
perform numerical explorations for reasonable values of cosmological
parameters. We also analyse the behaviour of redshifts in such models and the
description of ``very large scale geometrical features'' when analysed by
distant observers.Comment: 31 pages, Latex, 9 eps figures. Revised version: Introduction was
extended and we have added an application to cosmology of the duplication
formula for the elliptic Weierstrass P functio
From orbital measures to Littlewood-Richardson coefficients and hive polytopes
The volume of the hive polytope (or polytope of honeycombs) associated with a
Littlewood- Richardson coefficient of SU(n), or with a given admissible triple
of highest weights, is expressed, in the generic case, in terms of the Fourier
transform of a convolution product of orbital measures. Several properties of
this function -- a function of three non-necessarily integral weights or of
three multiplets of real eigenvalues for the associated Horn problem-- are
already known. In the integral case it can be thought of as a semi-classical
approximation of Littlewood-Richardson coefficients. We prove that it may be
expressed as a local average of a finite number of such coefficients. We also
relate this function to the Littlewood-Richardson polynomials (stretching
polynomials) i.e., to the Ehrhart polynomials of the relevant hive polytopes.
Several SU(n) examples, for n=2,3,...,6, are explicitly worked out.Comment: 32 pages, 4 figures. This version (V4): a few corrected typo
Conjugation properties of tensor product multiplicities
It was recently proven that the total multiplicity in the decomposition into
irreducibles of the tensor product lambda x mu of two irreducible
representations of a simple Lie algebra is invariant under conjugation of one
of them; at a given level, this also applies to the fusion multiplicities of
affine algebras. Here, we show that, in the case of SU(3), the lists of
multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are
identical up to permutations. This latter property does not hold in general for
other Lie algebras. We conjecture that the same property should hold for the
fusion product of the affine algebra of su(3) at finite levels, but this is not
investigated in the present paper.Comment: 29 pages, 23 figures. v2: Added references. Corrected typos. Some
more explanations and comments have been added : subsections 1.4, 4.2.4 and a
last paragraph in section 3.3. To appear in J Phys
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