512 research outputs found

    Composed inclusions of A3A_3 and A4A_4 subfactors

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    In this article, we classify all standard invariants that can arise from a composed inclusion of an A3A_3 with an A4A_4 subfactor. More precisely, if NβŠ‚P\mathcal{N}\subset \mathcal{P} is the A3A_3 subfactor and PβŠ‚M\mathcal{P}\subset\mathcal{M} is the A4A_4 subfactor, then only four standard invariants can arise from the composed inclusion NβŠ‚M\mathcal{N}\subset\mathcal{M}. 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 A4A_4 subfactors.Comment: 49 pages, 33 figure

    Exchange relation planar algebras of small rank

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    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

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    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 SS 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 3G3^G subfactor planar algebras

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    We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the 3G3^G subfactor planar algebras. We prove the conjecture in the case when ∣G∣|G| is odd. To do so, we find an action of GG on the reduced subfactor planar algebra at f(2)f^{(2)}, 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

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    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

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    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

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    We define a planar para algebra, which arises naturally from combining planar algebras with the idea of ZN\mathbb{Z}_{N} 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 ZN\mathbb{Z}_{N}, 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

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    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 0<p,qβ‰€βˆž0<p,q\leq \infty. Furthermore, we establish R\'{e}nyi entropic uncertainty principles for subfactor planar algebras.Comment: 15 page

    Reflection Positivity and Levin-Wen Models

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    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

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    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 E6E^6 or the ones from representations of quantum SU(N)SU(N). 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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