152 research outputs found
Triangular dissections, aperiodic tilings and Jones algebras
The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph
of type 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 There are usually several possible
infinite dissections compatible with a given but a given one makes use of
triangle types if is even. Jones algebra with index (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 ,
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 digits (if 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
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
Currents on Grassmann algebras
We define currents on a Grassmann algebra with generators as
distributions on its exterior algebra (using the symmetric wedge product). We
interpret the currents in terms of -graded Hochschild cohomology and
closed currents in terms of cyclic cocycles (they are particular multilinear
forms on ). An explicit construction of the vector space of closed
currents of degree on is given by using Berezin integration.Comment: 20 pages, CPT-93/P.2935 and ENSLAPP-440/9
Algebraic connections on parallel universes
For any manifold , we introduce a \ZZ -graded differential algebra
, which, in particular, is a bi-module over the associative algebra
. We then introduce the corresponding covariant differentials and
show how this construction can be interpreted in terms of Yang-Mills and Higgs
fields. This is a particular example of noncommutative geometry. It differs
from the prescription of Connes in the following way: The definition of
does not rely on a given Dirac-Yukawa operator acting on a space of spinors.Comment: 10 pages, CPT-93/PE 294
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
COMMENTS ABOUT HIGGS FIELDS, NONCOMMUTATIVE GEOMETRY AND THE STANDARD MODEL
We make a short review of the formalism that describes Higgs and Yang Mills
fields as two particular cases of an appropriate generalization of the notion
of connection. We also comment about the several variants of this formalism,
their interest, the relations with noncommutative geometry, the existence (or
lack of existence) of phenomenological predictions, the relation with Lie
super-algebras etc.Comment: pp 20, LaTeX file, no figures, also available via anonymous ftp at
ftp://cpt.univ-mrs.fr/ or via gopher gopher://cpt.univ-mrs.fr
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