190 research outputs found
Symplectic fermions and a quasi-Hopf algebra structure on
We consider the (finite-dimensional) small quantum group at
. We show that does not allow for an R-matrix, even
though holds for all finite-dimensional
representations of . We then give an explicit
coassociator and an R-matrix such that becomes a
quasi-triangular quasi-Hopf algebra.
Our construction is motivated by the two-dimensional chiral conformal field
theory of symplectic fermions with central charge . There, a braided
monoidal category, , has been computed from the factorisation and
monodromy properties of conformal blocks, and we prove that
is braided monoidally equivalent to
.Comment: 40pp, 11 figures; v2: few very minor corrections for the final
version in Journal of Algebr
Fusion and braiding in finite and affine Temperley-Lieb categories
Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the
group algebra of) the famous Artin's braid group , while the affine TL
algebras arise as diagram algebras from a generalized version of the braid
group. We study asymptotic `' representation theory of these
quotients (parametrized by ) from a perspective of
braided monoidal categories. Using certain idempotent subalgebras in the finite
and affine algebras, we construct infinite `arc' towers of the diagram algebras
and the corresponding direct system of representation categories, with terms
labeled by . The corresponding direct-limit category is our
main object of studies.
For the case of the finite TL algebras, we prove that the direct-limit
category is abelian and highest-weight at any and endowed with braided
monoidal structure. The most interesting result is when is a root of unity
where the representation theory is non-semisimple. The resulting braided
monoidal categories we obtain at different roots of unity are new and
interestingly they are not rigid. We observe then a fundamental relation of
these categories to a certain representation category of the Virasoro algebra
and give a conjecture on the existence of a braided monoidal equivalence
between the categories. This should have powerful applications to the study of
the `continuum' limit of critical statistical mechanics systems based on the TL
algebra.
We also introduce a novel class of embeddings for the affine Temperley-Lieb
algebras and related new concept of fusion or bilinear -graded
tensor product of modules for these algebras. We prove that the fusion rules
are stable with the index of the tower and prove that the corresponding
direct-limit category is endowed with an associative tensor product. We also
study the braiding properties of this affine TL fusion.Comment: 50p
Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials
The concept of cyclic tridiagonal pairs is introduced, and explicit examples
are given. For a fairly general class of cyclic tridiagonal pairs with
cyclicity N, we associate a pair of `divided polynomials'. The properties of
this pair generalize the ones of tridiagonal pairs of Racah type. The algebra
generated by the pair of divided polynomials is identified as a higher-order
generalization of the Onsager algebra. It can be viewed as a subalgebra of the
q-Onsager algebra for a proper specialization at q the primitive 2Nth root of
unity. Orthogonal polynomials beyond the Leonard duality are revisited in light
of this framework. In particular, certain second-order Dunkl shift operators
provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of
irreducibility is added; v3: version for Linear Algebra and its Applications,
one assumption added in Appendix about eq. (A.2
The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
The periodic sl(2|1) alternating spin chain encodes (some of) the properties
of hulls of percolation clusters, and is described in the continuum limit by a
logarithmic conformal field theory (LCFT) at central charge c=0. This theory
corresponds to the strong coupling regime of a sigma model on the complex
projective superspace , and the spectrum of critical exponents can be
obtained exactly. In this paper we push the analysis further, and determine the
main representation theoretic (logarithmic) features of this continuum limit by
extending to the periodic case the approach of [N. Read and H. Saleur, Nucl.
Phys. B 777 316 (2007)]. We first focus on determining the representation
theory of the finite size spin chain with respect to the algebra of local
energy densities provided by a representation of the affine Temperley-Lieb
algebra at fugacity one. We then analyze how these algebraic properties carry
over to the continuum limit to deduce the structure of the space of states as a
representation over the product of left and right Virasoro algebras. Our main
result is the full structure of the vacuum module of the theory, which exhibits
Jordan cells of arbitrary rank for the Hamiltonian.Comment: 69pp, 8 fig
Topological defects in lattice models and affine Temperley-Lieb algebra
This paper is the first in a series where we attempt to define defects in
critical lattice models that give rise to conformal field theory topological
defects in the continuum limit. We focus mostly on models based on the
Temperley-Lieb algebra, with future applications to restricted solid-on-solid
(also called anyonic chains) models, as well as non-unitary models like
percolation or self-avoiding walks. Our approach is essentially algebraic and
focusses on the defects from two points of view: the "crossed channel" where
the defect is seen as an operator acting on the Hilbert space of the models,
and the "direct channel" where it corresponds to a modification of the basic
Hamiltonian with some sort of impurity. Algebraic characterizations and
constructions are proposed in both points of view. In the crossed channel, this
leads us to new results about the center of the affine Temperley-Lieb algebra;
in particular we find there a special subalgebra with non-negative integer
structure constants that are interpreted as fusion rules of defects. In the
direct channel, meanwhile, this leads to the introduction of fusion products
and fusion quotients, with interesting mathematical properties that allow to
describe representations content of the lattice model with a defect, and to
describe its spectrum.Comment: 41
The shift of Energy levels of a quantum dot in single electron transistor
The shift of Energy levels of a quantum dot in single electron transistor model is investigated. The self-energy function which related to shift of Energy levels, describing this interaction is added to a bare energy of a dot state. In the standard way of determining the self-interaction corrections to bare energies of quantum dots, the variations of the self-energy functions with energy are ignored, and these corrections are considered to be equal to the values of the self-energy functions for bare energies of state
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