626 research outputs found
Composed inclusions of and subfactors
In this article, we classify all standard invariants that can arise from a
composed inclusion of an with an subfactor. More precisely, if
is the subfactor and
is the subfactor, then only four standard
invariants can arise from the composed inclusion
. This answers a question posed by Bisch and
Haagerup in 1994. The techniques of this paper also show that there are exactly
four standard invariants for the composed inclusion of two subfactors.Comment: 49 pages, 33 figure
Exchange relation planar algebras of small rank
The main purpose of this paper is to classify exchange relation planar
algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the
positivity of subfactor planar algebras based on the Schur product theorem. We
will discuss the lattice of projections of 2-boxes, specifically the rank of
the projections. From this point, several results about biprojections are
obtained. The key break of the classification is to show the existence of a
biprojection. By this method, we also classify another two families of
subfactor planar algebras, subfactor planar algebras generated by 2-boxes with
4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar
algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic
construction ideal is abelian. They extend the classification of singly
generated planar algebras obtained by Bisch, Jones and the author.Comment: 35 pages, 102 figure
Quon language: surface algebras and Fourier duality
Quon language is a 3D picture language that we can apply to simulate
mathematical concepts. We introduce the surface algebras as an extension of the
notion of planar algebras to higher genus surface. We prove that there is a
unique one-parameter extension. The 2D defects on the surfaces are quons, and
surface tangles are transformations. We use quon language to simulate graphic
states that appear in quantum information, and to simulate interesting
quantities in modular tensor categories. This simulation relates the pictorial
Fourier duality of surface tangles and the algebraic Fourier duality induced by
the S matrix of the modular tensor category. The pictorial Fourier duality also
coincides with the graphic duality on the sphere. For each pair of dual graphs,
we obtain an algebraic identity related to the matrix. These identities
include well-known ones, such as the Verlinde formula; partially known ones,
such as the 6j-symbol self-duality; and completely new ones.Comment: 22 page
The generator conjecture for subfactor planar algebras
We state a conjecture for the formulas of the depth 4 low-weight rotational
eigenvectors and their corresponding eigenvalues for the subfactor planar
algebras. We prove the conjecture in the case when is odd. To do so, we
find an action of on the reduced subfactor planar algebra at ,
which is obtained from shading the planar algebra of the even half. We also
show that this reduced subfactor planar algebra is a Yang-Baxter planar
algebra.Comment: 24 pages, many figure
A Mathematical Picture Language Program
We give an overview of our philosophy of pictures in mathematics. We
emphasize a bi-directional process between picture language and mathematical
concepts: abstraction and simulation. This motivates a program to understand
different subjects, using virtual and real mathematical concepts simulated by
pictures.Comment: 15 page
Uncertainty Principles for Kac Algebras
In this paper, we introduce the notation of bi-shift of biprojections in
subfactor theory to unimodular Kac algebras. We characterize the minimizers of
Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle
for unimodular Kac algebras with biprojections and prove Hardy's uncertainty
principle in terms of minimizers.Comment: 15 page
Planar Para Algebras, Reflection Positivity
We define a planar para algebra, which arises naturally from combining planar
algebras with the idea of para symmetry in physics. A
subfactor planar para algebra is a Hilbert space representation of planar
tangles with parafermionic defects, that are invariant under para isotopy. For
each , we construct a family of subfactor planar para algebras
which play the role of Temperley-Lieb-Jones planar algebras. The first example
in this family is the parafermion planar para algebra (PAPPA). Based on this
example, we introduce parafermion Pauli matrices, quaternion relations, and
braided relations for parafermion algebras which one can use in the study of
quantum information. An important ingredient in planar para algebra theory is
the string Fourier transform (SFT), that we use on the matrix algebra generated
by the Pauli matrices. Two different reflections play an important role in the
theory of planar para algebras. One is the adjoint operator; the other is the
modular conjugation in Tomita-Takesaki theory. We use the latter one to define
the double algebra and to introduce reflection positivity. We give a new and
geometric proof of reflection positivity, by relating the two reflections
through the string Fourier transform.Comment: 41 page
Non-commutative R\'{e}nyi Entropic Uncertainty Principles
In this paper, we calculate the norm of the string Fourier transform on
subfactor planar algebras and characterize the extremizers of the inequalities
for parameters . Furthermore, we establish R\'{e}nyi entropic
uncertainty principles for subfactor planar algebras.Comment: 15 page
Reflection Positivity and Levin-Wen Models
The reflection positivity property has played a central role in both
mathematics and physics, as well as providing a crucial link between the two
subjects. In a previous paper we gave a new geometric approach to understanding
reflection positivity in terms of pictures. Here we give a transparent
algebraic formulation of our pictorial approach. We use insights from this
translation to establish the reflection positivity property for the fashionable
Levin-Wen models with respect both to vacuum and to bulk excitations. We
believe these methods will be useful for understanding a variety of other
problems.Comment: 16 page
Classification of Thurston-relation subfactor planar algebras
Bisch and Jones suggested the skein theoretic classification of planar
algebras and investigated the ones generated by 2-boxes with the second author.
In this paper, we consider 3-box generators and classify subfactor planar
algebras generated by a non-trivial 3-box satisfying a relation proposed by
Thurston. The subfactor planar algebras in the classification are either
or the ones from representations of quantum . We introduce a new method
to determine positivity of planar algebras and new techniques to reduce the
complexity of computations.Comment: 21 page
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