Planar algebra

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In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. [1] They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition.[2][3] Any finite group (and quantum generalization) can be encoded as a planar algebra.[1]

Definition[edit]

The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant. [1] [4] [5]

Planar tangle[edit]

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

Tangle.png

Here, the mark is shown as a -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[edit]

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 -marked intervals coincide. Finally we remove the coinciding circles. Note that two planar tangles can have zero, one or several possible compositions.

Composition.png

Planar operad[edit]

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

Planar algebra[edit]

A planar algebra is a representation of the planar operad; more precisely, it is a family of vector spaces , called -box spaces, on which acts the planar operad, i.e. for any tangle (with one output disk and input disks with and intervals respectively) there is a multilinear map

with according to the shading of the -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.

PlanarDiagram.png

Examples[edit]

Planar tangles[edit]

The family of vector spaces generated by the planar tangles having intervals on their output disk and a white (or black) -marked interval, admits a planar algebra structure.

Temperley–Lieb-Jones[edit]

The Temperley-Lieb-Jones planar algebra is generated by the planar tangles without input disk; its -box space is generated by

TLJ3.png

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

DeltaMultiplication.png

Note that the dimension of is the Catalan number . This planar algebra encodes the notion of Temperley–Lieb algebra.

Hopf algebra[edit]

A semisimple and cosemisimple Hopf algebra over an algebraically closed field is encoded in a planar algebra defined by generators and relations. [6]

Subfactor planar algebra[edit]

Definition[edit]

A subfactor planar algebra is a planar -algebra which is:

(1) Finite-dimensional:
(2) Evaluable:
(3) Spherical:
(4) Positive: defines an inner product.

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

PlanarOperations.png

The tangle action deals with the adjoint by:

with the mirror image of and the adjoint of in .

Examples and results[edit]

No-ghost theorem: The planar algebra has no ghost (i.e. element with ) if and only if

For as above, and after a quotient by the null ideal (generated by elements with ), we get a subfactor planar algebra. Any subfactor planar algebra with constant admits as planar subalgebra.

A planar algebra is a subfactor planar algebra if and only if it is the standard invariant of an extremal subfactor of index , with and .[7] [8] [9] A finite depth or irreducible subfactor is extremal ( on ).

There is a subfactor planar algebra encoding any finite group (and any finite dimensional Hopf -algebra, called Kac algebra), defined by generators and relations. [10] [11]

The subfactor planar algebra associated to an inclusion of finite groups ,[12] does not always remember the (core-free) inclusion. [13] [14]

A Bisch-Jones planar algebra , initially called Fuss-Catalan, can be defined as for but by allowing two colors of string with their own constant and . For as above, and after the null quotient, it is a planar subalgebra of any subfactor planar algebra with an intermediate such that and . [15] [16]

The first finite depth subfactor planar algebra of index is called the Haagerup subfactor planar algebra. [17] It has index .

The subfactor planar algebras are completely classified for index at most [18] and a bit beyond. [19]

Fourier transform and biprojections[edit]

Let be a finite index subfactor, and the corresponding subfactor planar algebra. Assume that is irreducible (i.e. ). Let be an intermediate subfactor. Let the Jones projection . Note that . Let and .

E1id.png

Note that and .

Let the bijective linear map be the Fourier transform, also called -click (of the outer star) or rotation; and let be the coproduct of and .

FourierCoproduct.png

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

The coproduct satisfies the equality

For any positive operators , the coproduct is also positive; this can be seen diagrammatically: [20]

CoproductPositive.png

Let be the contragredient (also called rotation). The map corresponds to four -clicks of the outer star, so it's the identity map, and then .

In the Kac algebra case, the contragredient is exactly the antipode,[11] which, for a finite group, correspond to the inverse.

A biprojection is a projection with a multiple of a projection. Note that and are biprojections; this can be seen as follows:

F(e1).png

A projection is a biprojection iff it is the Jones projection of an intermediate subfactor [21] , iff . [22] [20]

Galois correspondence: [23] 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, [24] of the form , as for an interval of finite groups .

Using the biprojections, we can make the intermediate subfactor planar algebras. [25] [26]

The uncertainty principle extends to any irreducible subfactor planar algebra :

Let with the range projection of and the unnormalized trace (i.e. on ).

Noncommutative uncertainty principle: [27] Let , nonzero. Then

Assuming and positive, the equality holds if and only if is a biprojection. More generally, the equality holds if and only if is the bi-shift of a biprojection.

References[edit]

  1. ^ a b c Vaughan F. R. Jones (1999), "Planar algebras, I", to appear in New Zealand Journal of Mathematics: 122pp, arXiv:math/9909027Freely accessible 
  2. ^ "Dror Bar-Natan: Publications: Cobordisms". Math.toronto.edu. doi:10.2140/gt.2005.9.1443. Retrieved 2016-11-20. 
  3. ^ "Front: [math/0410495] Khovanov's homology for tangles and cobordisms". Front.math.ucdavis.edu. doi:10.2140/gt.2005.9.1443. Retrieved 2016-11-20. 
  4. ^ Vijay Kodiyalam, V.S. Sunder (2004), "On Jones' planar algebras", J. Knot Theory Ramifications, 13 (2): 219–247, MR 2047470, doi:10.1142/S021821650400310X 
  5. ^ "Vijay Kodiyalam - Planar algebras - IMSc 2015". youtube.com. 2015-11-14. 
  6. ^ Vijay Kodiyalam, V.S. Sunder (2006), "The planar algebra of a semisimple and cosemisimple Hopf algebra", Proc. Indian Acad. Sci. Math. Sci., 116 (4): 1–16 
  7. ^ Sorin Popa (1995), "An axiomatization of the lattice of higher relative commutants of a subfactor", Inventiones Mathematicae, 120 (3): 427–445, MR 1334479, doi:10.1007/BF01241137 
  8. ^ Alice Guionnet, Vaughan F. R. Jones, Dimitri Shlyakhtenko (2010), "Random matrices, free probability, planar algebras and subfactors", Clay Math. Proc., {11}: 201–239, MR 2732052 
  9. ^ Vijay Kodiyalam, V.S. Sunder (2009), "From subfactor planar algebras to subfactors", Internat. J. Math., 20 (10): 1207–1231, MR 2574313, doi:10.1142/S0129167X0900573X 
  10. ^ Paramita Das, Vijay Kodiyalam (2005), "Planar algebras and the Ocneanu-Szymanski theorem", Proc. Amer. Math. Soc., 133 (9): 2751–2759, ISSN 0002-9939, MR 2146224, doi:10.1090/S0002-9939-05-07789-0 
  11. ^ a b Vijay Kodiyalam, Zeph Landau, V.S. Sunder (2003), "The planar algebra associated to a Kac algebra", Proc. Indian Acad. Sci. Math. Sci., 113 (1): 15–51, ISSN 0253-4142, MR 1971553, doi:10.1007/BF02829677 
  12. ^ Ved Prakash Gupta (2008), "Planar algebra of the subgroup-subfactor", Proceedings Mathematical Sciences, 118 (4): 583–612, doi:10.1007/s12044-008-0046-0 
  13. ^ Vijay Kodiyalam, V.S. Sunder (2000), "The subgroup-subfactor", Math. Scand., 86 (1): 45–74, ISSN 0025-5521, MR 1738515 
  14. ^ Masaki Izumi (2002), "Characterization of isomorphic group-subgroup subfactors", Int. Math. Res. Not. (34): 1791–1803, ISSN 1073-7928, MR 1920326, doi:10.1155/S107379280220402X 
  15. ^ Dietmar Bisch, Vaughan Jones (1997), "Algebras associated to intermediate subfactors", Inventiones Mathematicae, 128 (1): 89–157, doi:10.1007/s002220050137 
  16. ^ Pinhas Grossman, Vaughan Jones (2007), "Intermediate subfactors with no extra structure", J. Amer. Math. Soc., 20 (1): 219–265, MR 2257402, doi:10.1090/S0894-0347-06-00531-5 
  17. ^ Emily Peters (2010), "A planar algebra construction of the Haagerup subfactor", Internat. J. Math., 21 (8): 987–1045, MR 2679382, doi:10.1142/S0129167X10006380 
  18. ^ Vaughan F. R. Jones, Scott Morrison, Noah Snyder (2014), "The classification of subfactors of index at most ", Bull. Amer. Math. Soc. (N.S.), 51 (2): 277–327, MR 3166042, doi:10.1090/S0273-0979-2013-01442-3 
  19. ^ Narjess Afzaly, Scott Morrison, David Penneys (2015), The classification of subfactors with index at most , pp. 70pp, arXiv:1509.00038Freely accessible 
  20. ^ a b Zhengwei Liu (2016), "Exchange relation planar algebras of small rank", Trans. Amer. Math. Soc., 368 (12): 8303–8348, ISSN 0002-9947, MR 3551573, doi:10.1090/tran/6582 
  21. ^ Bisch, Dietmar, Bisch (1994), "A note on intermediate subfactors", Pacific J. Math., 163 (2): 201–216, ISSN 0030-8730, MR 1262294, doi:10.2140/pjm.1994.163.201 
  22. ^ Zeph A. Landau (2002), "Exchange relation planar algebras", Geom. Dedicata, 95: 183–214, ISSN 0046-5755, MR 1950890, doi:10.1023/A:1021296230310 
  23. ^ Masaki Izumi, Roberto Longo, Sorin Popa (1998), "A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras", J. Funct. Anal., 155 (1): 25–63, ISSN 0022-1236, MR 1622812, doi:10.1006/jfan.1997.3228 
  24. ^ Yasuo Watatani (1996), "Lattices of intermediate subfactors", J. Funct. Anal., 140 (2): 312–334, ISSN 0022-1236, MR 1409040, doi:10.1006/jfan.1996.0110 
  25. ^ Zeph A. Landau (1998), "Intermediate subfactors", Thesis - University of California at Berkeley: 132pp 
  26. ^ Keshab Chandra Bakshi (2016), Intermediate planar algebra revisited, pp. 31pp, arXiv:1611.05811Freely accessible 
  27. ^ Chunlan Jiang, Zhengwei Liu, Jinsong Wu, (2016), "Noncommutative uncertainty principles", J. Funct. Anal., 270 (1): 264–311, doi:10.1016/j.jfa.2015.08.007