scholarly journals Ore's theorem on subfactor planar algebras

2020 ◽  
Vol 11 (3) ◽  
pp. 525-543
Author(s):  
Sébastien Palcoux
Keyword(s):  
Planar Algebras

Singly generated planar algebras of small dimension

2000 ◽  
Vol 101 (1) ◽  
pp. 41-75 ◽  
Author(s):  
Vaughan Jones ◽  
Dietmar Bisch
Keyword(s):  
Small Dimension ◽  
Planar Algebras

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals The embedding theorem for finite depth subfactor planar algebras

10.4171/qt/23 ◽  
2011 ◽  
pp. 301-337 ◽  
Author(s):  
Vaughan Jones ◽  
David Penneys
Keyword(s):  
Finite Depth ◽  
Embedding Theorem ◽  
Planar Algebras

Generators for finite depth subfactor planar algebras

2016 ◽  
Vol 126 (2) ◽  
pp. 235-240 ◽  
Author(s):  
VIJAY KODIYALAM ◽  
SRIKANTH TUPURANI
Keyword(s):  
Finite Depth ◽  
Planar Algebras

scholarly journals Skein theory for the D2n planar algebras

2010 ◽  
Vol 214 (2) ◽  
pp. 117-139 ◽  
Author(s):  
Scott Morrison ◽  
Emily Peters ◽  
Noah Snyder
Keyword(s):  
Planar Algebras ◽  
Skein Theory

scholarly journals Planar algebras: A category theoretic point of view

2011 ◽  
Vol 339 (1) ◽  
pp. 27-54 ◽  
Author(s):  
Shamindra Kumar Ghosh
Keyword(s):  
Point Of View ◽  
Planar Algebras ◽  
Theoretic Point

scholarly journals Planar algebras, quantum information theory and subfactors

2020 ◽  
pp. 2050124
Author(s):  
Vijay Kodiyalam ◽  
Sruthymurali ◽  
V. S. Sunder
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Hadamard Matrices ◽  
Quantum Information Theory ◽  
Latin Squares ◽  
Planar Algebras ◽  
Planar Algebra

We define generalized notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum Latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

scholarly journals Affine Modules and the Drinfeld Center

2016 ◽  
Vol 118 (1) ◽  
pp. 119 ◽  
Author(s):  
Paramita Das ◽  
Shamindra Kumar Ghosh ◽  
Ved Prakash Gupta
Keyword(s):  
Finite Index ◽  
Planar Algebras ◽  
Infinite Depth ◽  
Zero Level ◽  
Fusion Algebra ◽  
Planar Algebra ◽  
The Way

Given a finite index subfactor, we show that the affine morphisms at zero level in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a $*$-algebra. This identification paves the way to analyze the structure of affine $P$-modules with weight zero for any subfactor planar algebra $P$ (possibly having infinite depth). Further, for irreducible depth two subfactor planar algebras, we establish an additive equivalence between the category of affine $P$-modules and the center of the category of $N$-$N$-bimodules generated by $L^2(M)$; this partially verifies a conjecture of Jones and Walker.

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