planar algebra

A (shaded) planar tangle is the data of finitely many input disks, one output disk, non-intersecting strings giving an even number, say $2n$, intervals per disk and one $\backslash star$-marked interval per disk.

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Here, the mark is shown as a $\backslash star$-shape. On each input disk it is placed between two adjacent outgoing strings, and on the output disk it is placed between two adjacent incoming strings. A planar tangle is defined up to isotopy.

To compose two planar tangles, put the output disk of one into an input of the other, having as many intervals, same shading of marked intervals and such that the $\backslash star$-marked intervals coincide. Finally we remove the coinciding circles. Note that two planar tangles can have zero, one or several possible compositions.

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The planar operad is the set of all the planar tangles (up to isomorphism) with such compositions.

A planar algebra is a representation of the planar operad; more precisely, it is a family of vector spaces $(\backslash mathcal\{P\}\_\{n,\backslash pm\})\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$, called $n$-box spaces, on which acts the planar operad, i.e. for any tangle $T$ (with one output disk and $r$ input disks with $2n\_0$ and $2n\_1,\; \backslash dots,\; 2n\_r$ intervals respectively) there is a multilinear map

: $Z\_T\; :\; \backslash mathcal\{P\}\_\{n\_1,\backslash epsilon\_1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathcal\{P\}\_\{n\_r,\backslash epsilon\_r\}\; \backslash to\; \backslash mathcal\{P\}\_\{n\_0,\backslash epsilon\_0\}$

with $\backslash epsilon\_i\; \backslash in\; \backslash \{+,-\backslash \}$ according to the shading of the $\backslash star$-marked intervals, and these maps (also called partition functions) respect the composition of tangle in such a way that all the diagrams as below commute.

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The family of vector spaces $(\backslash mathcal\{T\}\_\{n,\backslash pm\})\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ generated by the planar tangles having $2n$ intervals on their output disk and a white (or black) $\backslash star$-marked interval, admits a planar algebra structure.

The Temperley-Lieb planar algebra $\backslash mathcal\{TL\}(\backslash delta)$ is generated by the planar tangles without input disk; its $3$-box space $\backslash mathcal\{TL\}\_\{3,+\}(\backslash delta)$ is generated by

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Moreover, a closed string is replaced by a multiplication by $\backslash delta$.

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Note that the dimension of $\backslash mathcal\{TL\}\_\{n,\backslash pm\}(\backslash delta)$ is the Catalan number $\backslash frac\{1\}\{n+1\}\backslash binom\{2n\}\{n\}$.

This planar algebra encodes the notion of Temperley–Lieb algebra.

A semisimple and cosemisimple Hopf algebra over an algebraically closed field is encoded in a planar algebra defined by generators and relations, and "corresponds" (up to isomorphism) to a connected, irreducible, spherical, non degenerate planar algebra with non zero modulus $\backslash delta$ and of depth two.

{{citation

|author=Vijay Kodiyalam |author2=V.S. Sunder

| title= The planar algebra of a semisimple and cosemisimple Hopf algebra

| journal= Proc. Indian Acad. Sci. Math. Sci.

| volume= 116

| year= 2006

| number= 4

| pages= 1–16

| bibcode= 2005math......6153K

| arxiv= math/0506153

}}

Note that connected means $\backslash dim(\backslash mathcal\{P\}\_\{0,\backslash pm\})\; =\; 1$ (as for evaluable below), irreducible means $\backslash dim(\backslash mathcal\{P\}\_\{1,+\})\; =\; 1$, spherical is defined below, and non-degenerate means that the traces (defined below) are non-degenerate.

A subfactor planar algebra is a planar $\backslash star$-algebra $(\backslash mathcal\{P\}\_\{n,\backslash pm\})\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ which is:

: (1) Finite-dimensional:

: (2) Evaluable: $\backslash mathcal\{P\}\_\{0,\backslash pm\}\; =\; \backslash mathbb\{C\}$

: (3) Spherical: $tr:=tr\_r\; =\; tr\_l$

: (4) Positive: $\backslash langle\; a\; \backslash vert\; b\; \backslash rangle\; =\; tr(b^\{\backslash star\}a)$ defines an inner product.

Note that by (2) and (3), any closed string (shaded or not) counts for the same constant $\backslash delta$.

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The tangle action deals with the adjoint by:

: $Z\_T(a\_1\; \backslash otimes\; a\_2\; \backslash otimes\; \backslash cdots\; \backslash otimes\; a\_r)^\{\backslash star\}\; =\; Z\_\{T^\{\backslash star\}\}(a\_1^\{\backslash star\}\; \backslash otimes\; a\_2^\{\backslash star\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; a\_r^\{\backslash star\})$

with $T^\{\backslash star\}$ the mirror image of $T$ and $a\_i^\{\backslash star\}$ the adjoint of $a\_i$ in $\backslash mathcal\{P\}\_\{n\_i,\backslash epsilon\_i\}$.

No-ghost theorem: The planar algebra $\backslash mathcal\{TL\}(\backslash delta)$ has no ghost (i.e. element $a$ with ) if and only if

: $\backslash delta\; \backslash in\; \backslash \{\; 2\backslash cos(\backslash pi/n)\; |\; n=3,4,5,...\; \backslash \}\; \backslash cup\; [2,\; +\backslash infty]$

For $\backslash delta$ as above, let $\backslash mathcal\{I\}$ be the null ideal (generated by elements $a$ with $\backslash langle\; a\; \backslash vert\; a\; \backslash rangle\; =\; 0$). Then the quotient $\backslash mathcal\{TL\}(\backslash delta)/\backslash mathcal\{I\}$ is a subfactor planar algebra, called the Temperley–Lieb-Jones subfactor planar algebra $\backslash mathcal\{TLJ\}(\backslash delta)$. Any subfactor planar algebra with constant $\backslash delta$ admits $\backslash mathcal\{TLJ\}(\backslash delta)$ as planar subalgebra.

A planar algebra $(\backslash mathcal\{P\}\_\{n,\backslash pm\})$ is a subfactor planar algebra if and only if it is the standard invariant of an extremal subfactor $N\; \backslash subseteq\; M$ of index $[M:N]\; =\; \backslash delta^2$, with $\backslash mathcal\{P\}\_\{n,+\}=\; N\text{'}\; \backslash cap\; M\_\{n-1\}$ and $\backslash mathcal\{P\}\_\{n,-\}=\; M\text{'}\; \backslash cap\; M\_\{n\}$.

{{citation

| author = Sorin Popa

| doi = 10.1007/BF01241137

| number= 3

| journal = Inventiones Mathematicae

| mr = 1334479

| pages = 427–445

| title = An axiomatization of the lattice of higher relative commutants of a subfactor

| volume = 120

| year = 1995| bibcode = 1995InMat.120..427P

| s2cid = 1740471

| author-link = Sorin Popa

}}

{{citation

|author=Alice Guionnet |author2=Vaughan F. R. Jones |author3=Dimitri Shlyakhtenko

| journal= Clay Math. Proc.

| volume={11}

| mr = 2732052

| pages = 201–239

| title = Random matrices, free probability, planar algebras and subfactors

| year = 2010}}

{{citation

|author=Vijay Kodiyalam |author2=V.S. Sunder

| doi = 10.1142/S0129167X0900573X

| number= 10

| journal = Internat. J. Math.

| mr = 2574313

| pages = 1207–1231

| title = From subfactor planar algebras to subfactors

| volume = 20

| year = 2009| arxiv = 0807.3704

| s2cid = 115161031

}}

A finite depth or irreducible subfactor is extremal ($tr\_\{N\text{'}\}\; =\; tr\_\{M\}$ on $N\text{'}\; \backslash cap\; M$).

There is a subfactor planar algebra encoding any finite group (and more generally, any finite dimensional Hopf $\{\backslash rm\; C\}^\{\backslash star\}$-algebra, called Kac algebra), defined by generators and relations. A (finite dimensional) Kac algebra "corresponds" (up to isomorphism) to an irreducible subfactor planar algebra of depth two.

{{citation

|author=Paramita Das |author2=Vijay Kodiyalam

| title= Planar algebras and the Ocneanu-Szymanski theorem

| journal= Proc. Amer. Math. Soc.

| volume= 133

| year= 2005

| number= 9

| pages= 2751–2759

| issn= 0002-9939

| mr= 2146224

| doi= 10.1090/S0002-9939-05-07789-0

| doi-access= free

}}

{{citation

|author=Vijay Kodiyalam |author2=Zeph Landau |author3=V.S. Sunder

| title= The planar algebra associated to a Kac algebra

| journal= Proc. Indian Acad. Sci. Math. Sci.

| volume= 113

| year= 2003

| number= 1

| pages= 15–51

| issn= 0253-4142

| mr = 1971553

| doi= 10.1007/BF02829677

| s2cid= 56571515

}}

The subfactor planar algebra associated to an inclusion of finite groups,

{{citation

| author= Ved Prakash Gupta

| title= Planar algebra of the subgroup-subfactor

| journal= Proceedings Mathematical Sciences

| volume= 118

| year= 2008

| number= 4

| pages= 583–612

| doi= 10.1007/s12044-008-0046-0

| arxiv= 0806.1791

| bibcode= 2008arXiv0806.1791G

| s2cid= 5589336

}}

does not always remember the (core-free) inclusion.

{{citation

|author=Vijay Kodiyalam |author2=V.S. Sunder

| title=The subgroup-subfactor

| journal=Math. Scand.

| volume=86

| year=2000

| number=1

| pages=45–74

| issn=0025-5521

| mr=1738515

| doi=10.7146/math.scand.a-14281

| doi-access=free

}}

{{citation

| author=Masaki Izumi

| title=Characterization of isomorphic group-subgroup subfactors

| journal=Int. Math. Res. Not.

| volume=2002

| year=2002

| number=34

| pages=1791–1803

| issn=1073-7928

| mr=1920326

| doi=10.1155/S107379280220402X

| doi-access=

}}

A Bisch-Jones subfactor planar algebra $\backslash mathcal\{BJ\}(\backslash delta\_1,\; \backslash delta\_2)$ (sometimes called Fuss-Catalan) is defined as for $\backslash mathcal\{TLJ\}(\backslash delta)$ but by allowing two colors of string with their own constant $\backslash delta\_1$ and $\backslash delta\_2$, with $\backslash delta\_i$ as above. It is a planar subalgebra of any subfactor planar algebra with an intermediate such that $[K:N]\; =\; \backslash delta\_1^2$ and $[M:K]\; =\; \backslash delta\_2^2$.{{citation

|author=Dietmar Bisch |author2=Vaughan Jones

| doi = 10.1007/s002220050137

| number= 1

| journal = Inventiones Mathematicae

| pages = 89–157

| title = Algebras associated to intermediate subfactors

| volume = 128

| year = 1997| bibcode = 1997InMat.128...89J

| s2cid = 119372640

}}

{{citation

|author=Pinhas Grossman |author2=Vaughan Jones

| doi = 10.1090/S0894-0347-06-00531-5

| number= 1

| journal = J. Amer. Math. Soc.

| mr = 2257402

| pages = 219–265

| title = Intermediate subfactors with no extra structure

| volume = 20

| year = 2007| bibcode = 2007JAMS...20..219G

| doi-access = free

}}

The first finite depth subfactor planar algebra of index $\backslash delta^2\; >\; 4$ is called the Haagerup subfactor planar algebra.

{{citation

| author = Emily Peters

| doi = 10.1142/S0129167X10006380

| number= 8

| journal = Internat. J. Math.

| mr = 2679382

| pages = 987–1045

| title = A planar algebra construction of the Haagerup subfactor

| volume = 21

| year = 2010| arxiv = 0902.1294

| s2cid = 951475

}}

It has index $(5+\backslash sqrt\{13\})/2\; \backslash sim\; 4.303$.

The subfactor planar algebras are completely classified for index at most $5$

{{citation

|author=Vaughan F. R. Jones |author2=Scott Morrison |author3=Noah Snyder

| doi = 10.1090/S0273-0979-2013-01442-3

| number= 2

| journal = Bull. Amer. Math. Soc. (N.S.)

| mr = 3166042

| pages = 277–327

| title = The classification of subfactors of index at most $5$

| volume = 51

| year = 2014| arxiv = 1304.6141

| s2cid = 29962597

}}

and a bit beyond.

{{citation

|author=Narjess Afzaly |author2=Scott Morrison |author3=David Penneys

| arxiv = 1509.00038

| pages = 70pp

| title = The classification of subfactors with index at most $5\; +\; 1/4$

| year = 2015| bibcode = 2015arXiv150900038A

}}

This classification was initiated by Uffe Haagerup.

{{citation

| author = Uffe Haagerup

| mr = 1317352

| pages = 1–38

| title = Principal graphs of subfactors in the index range

| journal = Subfactors (Kyuzeso, 1993)

| year = 1994

| author-link = Uffe Haagerup

}}

It uses (among other things) a listing of possible principal graphs, together with the embedding theorem

{{citation

|author=Vaughan Jones |author2=David Penneys

| doi = 10.4171/QT/23

| number= 3

| journal = Quantum Topol.

| volume = 2

| mr = 2812459

| pages = 301–337

| title = The embedding theorem for finite depth subfactor planar algebras.

| year = 2011

| arxiv = 1007.3173

| s2cid = 59578009

}}

and the jellyfish algorithm.

{{citation

|author=Stephen Bigelow |author2=David Penneys

| doi = 10.1007/s00208-013-0941-2

| number= 1–2

| journal = Math. Ann.

| volume = 358

| mr = 3157990

| pages = 1–24

| title = Principal graph stability and the jellyfish algorithm.

| year = 2014

| arxiv = 1208.1564

| s2cid = 3549669

}}

A subfactor planar algebra remembers the subfactor (i.e. its standard invariant is complete) if it is amenable.

{{citation

| last = Popa | first = Sorin | author-link = Sorin Popa

| doi = 10.1007/BF02392646

| number= 2

| journal = Acta Mathematica

| mr = 1278111

| pages = 163–255

| title = Classification of amenable subfactors of type II

| volume = 172

| year = 1994| doi-access = free

}}

A finite depth hyperfinite subfactor is amenable.

About the non-amenable case: there are unclassifiably many irreducible hyperfinite subfactors of index 6 that all have the same standard invariant.

{{citation

|author=Arnaud Brothier |author2=Stefaan Vaes

| title= Families of hyperfinite subfactors with the same standard invariant and prescribed fundamental group.

| journal= J. Noncommut. Geom.

| volume= 9

| year= 2015

| number= 3

| pages= 775–796

| mr= 3420531

| doi= 10.4171/JNCG/207

| arxiv= 1309.5354

| s2cid= 117853753

}}

Let $N\; \backslash subset\; M$ be a finite index subfactor, and $\backslash mathcal\{P\}$ the corresponding subfactor planar algebra. Assume that $\backslash mathcal\{P\}$ is irreducible (i.e. $\backslash mathcal\{P\}\_\{1,+\}\; =\; N\text{'}\; \backslash cap\; M\_1\; =\; \backslash mathbb\{C\}$). Let $N\; \backslash subset\; K\; \backslash subset\; M$ be an intermediate subfactor. Let the Jones projection $e^M\_K:\; L^2(M)\; \backslash to\; L^2(K)$. Note that $e^M\_K\; \backslash in\; \backslash mathcal\{P\}\_\{2,+\}$. Let $id:=e^M\_M$ and $e\_1:=e^M\_N$.

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Note that $tr(e\_1)\; =\; \backslash delta^\{-2\}\; =\; [M:N]^\{-1\}$ and $tr(id)\; =\; 1$.

Let the bijective linear map $\backslash mathcal\{F\}:\; \backslash mathcal\{P\}\_\{2,\backslash pm\}\; \backslash to\; \backslash mathcal\{P\}\_\{2,\backslash mp\}$ be the Fourier transform, also called $1$-click (of the outer star) or $90^\{\backslash circ\}$ rotation; and let $a\; *\; b$ be the coproduct of $a$ and $b$.

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Note that the word coproduct is a diminutive of convolution product. It is a binary operation.

The coproduct satisfies the equality $a\; *\; b\; =\; \backslash mathcal\{F\}(\backslash mathcal\{F\}^\{-1\}(a)\; \backslash mathcal\{F\}^\{-1\}(b)).$

For any positive operators $a,b$, the coproduct $a*b$ is also positive; this can be seen diagrammatically:

{{citation

| author= Zhengwei Liu

| title= Exchange relation planar algebras of small rank

| journal= Trans. Amer. Math. Soc.

| volume= 368

| year= 2016

| number= 12

| pages= 8303–8348

| issn= 0002-9947

| mr = 3551573

| doi= 10.1090/tran/6582

| arxiv= 1308.5656

| s2cid= 117030298

}}

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Let $\backslash overline\{a\}\; :=\; \backslash mathcal\{F\}(\backslash mathcal\{F\}(a))$ be the contragredient $a$ (also called $180^\{\backslash circ\}$ rotation). The map $\backslash mathcal\{F\}^\{4\}$ corresponds to four $1$-clicks of the outer star, so it's the identity map, and then $\backslash overline\{\backslash overline\{a\}\}\; =\; a$.

In the Kac algebra case, the contragredient is exactly the antipode, which, for a finite group, correspond to the inverse.

A biprojection is a projection $b\; \backslash in\; \backslash mathcal\{P\}\_\{2,+\}\; \backslash setminus\; \backslash \{\; 0\backslash \}$ with $\backslash mathcal\{F\}(b)$ a multiple of a projection.

Note that $e\_1=e^M\_N$ and $id=e^M\_M$ are biprojections; this can be seen as follows:

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A projection $b$ is a biprojection iff it is the Jones projection $e^M\_K$ of an intermediate subfactor $N\; \backslash subset\; K\; \backslash subset\; M$,

{{citation

| author= Dietmar Bisch

| title= A note on intermediate subfactors

| journal= Pacific J. Math.

| volume= 163

| year= 1994

| number= 2

| pages= 201–216

| issn= 0030-8730

| mr= 1262294

| doi= 10.2140/pjm.1994.163.201

| doi-access= free}}

iff $e\_1\; \backslash le\; b=\backslash overline\{b\}\; =\; \backslash lambda\; b\; *\; b,\; \backslash text\{\; with\; \}\; \backslash lambda^\{-1\}\; =\; \backslash delta\; tr(b)$.

{{citation

| author= Zeph A. Landau

| title= Exchange relation planar algebras

| journal= Geom. Dedicata

| volume= 95

| year= 2002

| pages= 183–214

| issn= 0046-5755

| mr= 1950890

| doi= 10.1023/A:1021296230310

| s2cid= 119036175

}}

Galois correspondence:

{{citation

|author=Masaki Izumi |author2=Roberto Longo |author3=Sorin Popa

| title= A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras

| journal= J. Funct. Anal.

| volume= 155

| date= 1998

| number= 1

| pages= 25–63

| issn= 0022-1236

| mr = 1622812

| doi= 10.1006/jfan.1997.3228

| arxiv= funct-an/9604004

| s2cid= 12990106

}}

in the Kac algebra case, the biprojections are 1-1 with the left coideal subalgebras, which, for a finite group, correspond to the subgroups.

For any irreducible subfactor planar algebra, the set of biprojections is a finite lattice,

{{citation

| author=Yasuo Watatani

| title=Lattices of intermediate subfactors

| journal=J. Funct. Anal.

| volume=140

| year=1996

| number=2

| pages=312–334

| issn=0022-1236

| mr=1409040

| doi=10.1006/jfan.1996.0110

| hdl=2115/68899

| hdl-access=free

}}

of the form $[e\_1,id]$, as for an interval of finite groups $[H,G]$.

Using the biprojections, we can make the intermediate subfactor planar algebras.

{{citation

| author = Zeph A. Landau

| journal = Thesis - University of California at Berkeley

| pages = 132pp

| title = Intermediate subfactors

| year = 1998}}

{{citation

| author = Keshab Chandra Bakshi

| arxiv = 1611.05811

| pages = 31pp

| title = Intermediate planar algebra revisited

| journal = International Journal of Mathematics

| year = 2016| volume = 29

| issue = 12

| doi = 10.1142/S0129167X18500775

| bibcode = 2016arXiv161105811B

| s2cid = 119305436

}}

The uncertainty principle extends to any irreducible subfactor planar algebra $\backslash mathcal\{P\}$:

Let $\backslash mathcal\{S\}(x)\; =\; Tr(R(x))$ with $R(x)$ the range projection of $x$ and $Tr$ the unnormalized trace (i.e. $Tr\; =\; \backslash delta^n\; tr$ on $\backslash mathcal\{P\}\_\{n,\backslash pm\}$).

Noncommutative uncertainty principle:

{{citation

|author=Chunlan Jiang |author2=Zhengwei Liu |author3=Jinsong Wu

| doi = 10.1016/j.jfa.2015.08.007

| number= 1

| journal = J. Funct. Anal.

| pages = 264–311

| title = Noncommutative uncertainty principles

| volume = 270

| year = 2016| arxiv = 1408.1165

| s2cid = 16295570

}}

Let $x\; \backslash in\; \backslash mathcal\{P\}\_\{2,\backslash pm\}$, nonzero. Then

: $\backslash mathcal\{S\}(x)\; \backslash mathcal\{S\}(\backslash mathcal\{F\}(x))\; \backslash ge\; \backslash delta^\{2\}$

Assuming $x$ and $\backslash mathcal\{F\}(x)$ positive, the equality holds if and only if $x$ is a biprojection. More generally, the equality holds if and only if $x$ is the bi-shift of a biprojection.

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Category:Operator algebras

Category:Diagram algebras

Category:Knot theory