Planar algebra

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title: "Planar algebra" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["operator-algebras", "diagram-algebras", "knot-theory"] topic_path: "general/operator-algebras" source: "https://en.wikipedia.org/wiki/Planar_algebra" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

Definition

The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant. |author=Vijay Kodiyalam |author2=V.S. Sunder | doi = 10.1142/S021821650400310X | number= 2 | journal = J. Knot Theory Ramifications | mr = 2047470 | pages = 219–247 | title = On Jones' planar algebras | volume = 13 | year = 2004}} |url=https://www.youtube.com/playlist?list=PLjudt2gd3iAe40juwS3dKocAoooXF91Vi |title=Vijay Kodiyalam - Planar algebras - IMSc 2015 |website=youtube.com |date=2015-11-14}}

Planar tangle

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 \star-marked interval per disk.

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Here, the mark is shown as a \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.

Composition

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 \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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Planar operad

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

Planar algebra

A planar algebra is a representation of the planar operad; more precisely, it is a family of vector spaces (\mathcal{P}{n,\pm}){n \in \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, \dots, 2n_r intervals respectively) there is a multilinear map : Z_T : \mathcal{P}{n_1,\epsilon_1} \otimes \cdots \otimes \mathcal{P}{n_r,\epsilon_r} \to \mathcal{P}_{n_0,\epsilon_0}

with \epsilon_i \in {+,-} according to the shading of the \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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Examples

Planar tangles

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

Temperley–Lieb

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

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

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Note that the dimension of \mathcal{TL}_{n,\pm}(\delta) is the Catalan number \frac{1}{n+1}\binom{2n}{n} . This planar algebra encodes the notion of Temperley–Lieb algebra.

Hopf 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 \delta and of depth two. |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 \dim(\mathcal{P}{0,\pm}) = 1 (as for evaluable below), irreducible means \dim(\mathcal{P}{1,+}) = 1, spherical is defined below, and non-degenerate means that the traces (defined below) are non-degenerate.

Subfactor planar algebra

Definition

A subfactor planar algebra is a planar \star-algebra (\mathcal{P}{n,\pm}){n \in \mathbb{N}} which is: : (1) Finite-dimensional: \dim (\mathcal{P}{n,\pm}) : (2) Evaluable: \mathcal{P}{0,\pm} = \mathbb{C} : (3) Spherical: tr:=tr_r = tr_l : (4) Positive: \langle a \vert b \rangle = tr(b^{\star}a) defines an inner product.

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

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The tangle action deals with the adjoint by: : Z_T(a_1 \otimes a_2 \otimes \cdots \otimes a_r)^{\star} = Z_{T^{\star}}(a_1^{\star} \otimes a_2^{\star} \otimes \cdots \otimes a_r^{\star})

with T^{\star} the mirror image of T and a_i^{\star} the adjoint of a_i in \mathcal{P}_{n_i,\epsilon_i} .

Examples and results

No-ghost theorem: The planar algebra \mathcal{TL}(\delta) has no ghost (i.e. element a with \langle a \vert a \rangle ) if and only if : \delta \in { 2\cos(\pi/n) | n=3,4,5,... } \cup [2, +\infty] For \delta as above, let \mathcal{I} be the null ideal (generated by elements a with \langle a \vert a \rangle = 0 ). Then the quotient \mathcal{TL}(\delta)/\mathcal{I} is a subfactor planar algebra, called the Temperley–Lieb-Jones subfactor planar algebra \mathcal{TLJ}(\delta) . Any subfactor planar algebra with constant \delta admits \mathcal{TLJ}(\delta) as planar subalgebra.

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

There is a subfactor planar algebra encoding any finite group (and more generally, any finite dimensional Hopf {\rm C}^{\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. |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 |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

A Bisch-Jones subfactor planar algebra \mathcal{BJ}(\delta_1, \delta_2) (sometimes called Fuss-Catalan) is defined as for \mathcal{TLJ}(\delta) but by allowing two colors of string with their own constant \delta_1 and \delta_2 , with \delta_i as above. It is a planar subalgebra of any subfactor planar algebra with an intermediate such that [K:N] = \delta_1^2 and [M:K] = \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 |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 \delta^2 4 is called the Haagerup subfactor planar algebra. | 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+\sqrt{13})/2 \sim 4.303 .

Fourier transform and biprojections

Let N \subset M be a finite index subfactor, and \mathcal{P} the corresponding subfactor planar algebra. Assume that \mathcal{P} is irreducible (i.e. \mathcal{P}{1,+} = N' \cap M_1 = \mathbb{C} ). Let N \subset K \subset M be an intermediate subfactor. Let the Jones projection e^M_K: L^2(M) \to L^2(K). Note that e^M_K \in \mathcal{P}{2,+} . Let id:=e^M_M and e_1:=e^M_N .

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Note that tr(e_1) = \delta^{-2} = [M:N]^{-1} and tr(id) = 1.

Let the bijective linear map \mathcal{F}: \mathcal{P}{2,\pm} \to \mathcal{P}{2,\mp} be the Fourier transform, also called 1-click (of the outer star) or 90^{\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 = \mathcal{F}(\mathcal{F}^{-1}(a) \mathcal{F}^{-1}(b)).

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Let \overline{a} := \mathcal{F}(\mathcal{F}(a)) be the contragredient a (also called 180^{\circ} rotation). The map \mathcal{F}^{4} corresponds to four 1-clicks of the outer star, so it's the identity map, and then \overline{\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 \in \mathcal{P}_{2,+} \setminus { 0} with \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 \subset K \subset M , | 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 \le b=\overline{b} = \lambda b * b, \text{ with } \lambda^{-1} = \delta tr(b). | 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

Using the biprojections, we can make the intermediate subfactor planar algebras. | author = Zeph A. Landau | journal = Thesis - University of California at Berkeley | pages = 132pp | title = Intermediate subfactors | year = 1998}} | 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 \mathcal{P} :

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

Noncommutative uncertainty principle: |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 \in \mathcal{P}_{2,\pm} , nonzero. Then : \mathcal{S}(x) \mathcal{S}(\mathcal{F}(x)) \ge \delta^{2}
Assuming x and \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.

References

(2004). "Dror Bar-Natan: Publications: Cobordisms".
(2005). "Khovanov's homology for tangles and cobordisms". Geometry & Topology.

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