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.

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, 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 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 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.

Strongly and co-strongly minimal abelian structures

2010 ◽  
Vol 75 (2) ◽  
pp. 442-458 ◽  
Author(s):  
Ehud Hrushovski ◽  
James Loveys
Keyword(s):  
Vector Space ◽  
Finite Index ◽  
Division Ring ◽  
First Case ◽  
Special Cases ◽  
Structure Theorems ◽  
Algebraic Element ◽  
Minimal Module ◽  
Induced Structure

AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

ON JONES' PLANAR ALGEBRAS

2004 ◽  
Vol 13 (02) ◽  
pp. 219-247 ◽  
Author(s):  
VIJAY KODIYALAM ◽  
V. S. SUNDER
Keyword(s):  
Finite Index ◽  
Natural Class ◽  
Planar Algebras ◽  
Finite Dimensional

We show that a certain natural class of tangles 'generate the collection of all tangles with respect to composition'. This result is motivated by, and describes the reasoning behind, the 'uniqueness assertion' in Jones' theorem on the equivalence between extremal subfactors of finite index and what we call 'subfactor planar algebras' here. This result is also used to identify the manner in which the planar algebras corresponding to M⊂M1 and Nop⊂Mop are obtained from that of N⊂M. Our results also show that 'duality' in the category of extremal subfactors of finite index extends naturally to the category of 'general' planar algebras (not necessarily finite-dimensional or spherical or connected or C*, in the terminology of Jones).

scholarly journals Temperley–Lieb planar algebra modules arising from the ADE planar algebras

2005 ◽  
Vol 228 (2) ◽  
pp. 445-468 ◽  
Author(s):  
Sarah A. Reznikoff
Keyword(s):  
Planar Algebras ◽  
Planar Algebra

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 Intermediate planar algebra revisited

2018 ◽  
Vol 29 (12) ◽  
pp. 1850077
Author(s):  
Keshab Chandra Bakshi
Keyword(s):  
Direct Product ◽  
Finite Index ◽  
Planar Algebra

In this paper, we explicitly work out the subfactor planar algebra [Formula: see text] for an intermediate subfactor [Formula: see text] of an irreducible subfactor [Formula: see text] of finite index. We do this in terms of the subfactor planar algebra [Formula: see text] by showing that if [Formula: see text] is any planar tangle, the associated operator [Formula: see text] can be read off from [Formula: see text] by a formula involving the so-called biprojection corresponding to the intermediate subfactor [Formula: see text] and a scalar [Formula: see text] carefully chosen so as to ensure that the formula defining [Formula: see text] is multiplicative with respect to composition of tangles. Also, the planar algebra of [Formula: see text] can be obtained by applying these results to [Formula: see text]. We also apply our result to the example of a semi-direct product subgroup-subfactor.

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.

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