scholarly journals Free oriented extensions of subfactor planar algebras

2018 ◽  
Vol 29 (13) ◽  
pp. 1850093 ◽  
Author(s):  
Shamindra Kumar Ghosh ◽  
Corey Jones ◽  
B. Madhav Reddy
Keyword(s):  
Left Adjoint ◽  
Planar Algebras ◽  
Planar Algebra ◽  
Restriction Functor

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the standard invariant of a hyperfinite [Formula: see text] subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite [Formula: see text] factor.

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.

scholarly journals Planar algebras and the decategorification of bordered Khovanov homology

2017 ◽  
Vol 26 (04) ◽  
pp. 1750023
Author(s):  
Lawrence P. Roberts
Keyword(s):  
Recent Work ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Planar Surface ◽  
Simple Process ◽  
Planar Algebras ◽  
Type D ◽  
Combinatorial Construction ◽  
Planar Algebra ◽  
Compact Planar

We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring [Formula: see text]. This planar algebra is similar in spirit to the Temperley–Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author’s previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov’s tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov’s conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in [Formula: see text] where [Formula: see text] is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov’s approach to the Jones polynomial.

scholarly journals Perturbations of Planar Algebras

2014 ◽  
Vol 114 (1) ◽  
pp. 38 ◽  
Author(s):  
Paramita Das ◽  
Shamindra Kumar Ghosh ◽  
Ved Prakash Gupta
Keyword(s):  
Perturbation Technique ◽  
Planar Algebras ◽  
One To One ◽  
Planar Algebra ◽  
The One

We analyze the effect of pivotal structures (on a $2$-category) on the planar algebra associated to a $1$-cell as in [8] and come up with the notion of perturbations of planar algebras by weights (a concept that appeared earlier in Michael Burns' thesis [6]); we establish a one-to-one correspondence between weights and pivotal structures. Using the construction of [8], to each bifinite bimodule over $\mathit{II}_1$-factors, we associate a bimodule planar algebra in such a way that extremality of the bimodule corresponds to sphericality of the planar algebra. As a consequence of this, we reproduce an extension of Jones' theorem ([13]) (of associating 'subfactor planar algebras' to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with, using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. The perturbation technique helps us to construct an example of a family of non-spherical planar algebras starting from a particular spherical one; we also show that this family is associated to a known family of subfactors constructed by Jones.

scholarly journals FROM SUBFACTOR PLANAR ALGEBRAS TO SUBFACTORS

2009 ◽  
Vol 20 (10) ◽  
pp. 1207-1231 ◽  
Author(s):  
VIJAY KODIYALAM ◽  
V. S. SUNDER
Keyword(s):  
Algebraic Proof ◽  
Planar Algebras ◽  
Planar Algebra

We present a purely planar algebraic proof of the main result of a paper of Guionnet–Jones–Shlaykhtenko which constructs an extremal subfactor from a subfactor planar algebra whose standard invariant is given by that planar algebra.

scholarly journals Drinfeld center of planar algebra

2014 ◽  
Vol 25 (08) ◽  
pp. 1450076 ◽  
Author(s):  
Paramita Das ◽  
Shamindra Kumar Ghosh ◽  
Ved Prakash Gupta
Keyword(s):  
Finite Depth ◽  
Planar Algebras ◽  
Planar Algebra ◽  
Affine Representations ◽  
Finiteness Criterion

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra P (not necessarily having finite depth). We prove that if N ⊂ M is a subfactor realization of P, then the Drinfeld center of the N–N-bimodule category generated byNL2(M)M, is equivalent to the category of Hilbert affine representations of P satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.

scholarly journals Non-semisimple planar algebras from the representation theory of Ūq(𝔰𝔩2)

2018 ◽  
Vol 30 (09) ◽  
pp. 1850017 ◽  
Author(s):  
Stephen Moore
Keyword(s):  
Representation Theory ◽  
Quantum Group ◽  
Two Dimensional ◽  
Planar Algebras ◽  
Planar Algebra

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group [Formula: see text]. This is a diagrammatic description of [Formula: see text], where [Formula: see text] is a two-dimensional [Formula: see text] module.

scholarly journals Planar algebras

10.53733/172 ◽  
2021 ◽  
Vol 52 ◽  
pp. 1-107
Author(s):  
Vaughan Jones
Keyword(s):  
Vector Space ◽  
Finite Index ◽  
Planar Algebras ◽  
Planar Algebra

We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II1 factor, and vice versa.

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