Planar algebra

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor.
 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. Any subfactor planar algebra provides a family of unitary representations of Thompson groups.
Any finite group (and quantum generalization) can be encoded as a planar algebra.
Definition.
The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant.
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 formula_1, intervals per disk and one formula_2-marked interval per disk. 
Here, the mark is shown as a formula_2-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 formula_2-marked intervals coincide. Finally we remove the coinciding circles. Note that two planar tangles can have zero, one or several possible compositions.
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 formula_5, called formula_6-box spaces, on which "acts" the planar operad, i.e. for any tangle formula_7 (with one output disk and formula_8 input disks with formula_9 and formula_10 intervals respectively) there is a multilinear map
with formula_12 according to the shading of the formula_13-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.
Examples.
Planar tangles.
The family of vector spaces formula_14 generated by the planar tangles having formula_1 intervals on their "output" disk and a white (or black) formula_16-marked interval, admits a planar algebra structure.
Temperley–Lieb.
The Temperley-Lieb planar algebra formula_17 is generated by the planar tangles without input disk; its formula_18-box space formula_19 is generated by
Moreover, a closed string is replaced by a multiplication by formula_20.
Note that the dimension of formula_21 is the Catalan number formula_22.
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 formula_20 and of depth two.
Note that "connected" means formula_24 (as for "evaluable" below), "irreducible" means formula_25, "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 formula_2-algebra formula_5 which is: 
Note that by (2) and (3), any closed string (shaded or not) counts for the same constant formula_20.
The tangle action deals with the adjoint by: 
with formula_34 the mirror image of formula_35 and formula_36 the adjoint of formula_37 in formula_38.
Examples and results.
"No-ghost theorem": The planar algebra formula_39 has no ghost (i.e. element formula_40 with formula_41) if and only if 
For formula_20 as above, let formula_44 be the null ideal (generated by elements formula_40 with formula_46). Then the quotient formula_47 is a subfactor planar algebra, called the Temperley–Lieb-Jones subfactor planar algebra formula_48. Any subfactor planar algebra with constant formula_20 admits formula_50 as planar subalgebra.
A planar algebra formula_51 is a subfactor planar algebra if and only if it is the standard invariant of an extremal subfactor formula_52 of index formula_53, with formula_54 and formula_55.
A finite depth or irreducible subfactor is extremal (formula_56 on formula_57). 
There is a subfactor planar algebra encoding any finite group (and more generally, any finite dimensional Hopf formula_58-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.
The subfactor planar algebra associated to an inclusion of finite groups,
does not always remember the (core-free) inclusion.
A Bisch-Jones subfactor planar algebra formula_59 (sometimes called Fuss-Catalan) is defined as for formula_48 but by allowing two colors of string with their own constant formula_61 and formula_62, with formula_63 as above. It is a planar subalgebra of any subfactor planar algebra with an intermediate such that formula_64 and formula_65. 
The first finite depth subfactor planar algebra of index formula_66 is called the Haagerup subfactor planar algebra.
 It has index formula_67.
The subfactor planar algebras are completely classified for index at most formula_68 
and a bit beyond.
This classification was initiated by Uffe Haagerup.
It uses (among other things) a listing of possible principal graphs, together with the embedding theorem
and the jellyfish algorithm.
A subfactor planar algebra remembers the subfactor (i.e. its standard invariant is complete) if it is amenable.
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.
Fourier transform and biprojections.
Let formula_69 be a finite index subfactor, and formula_70 the corresponding subfactor planar algebra. Assume that formula_70 is irreducible (i.e. formula_72). Let formula_73 be an intermediate subfactor. Let the Jones projection formula_74. Note that formula_75. Let formula_76 and formula_77.
Note that formula_78 and formula_79.
Let the bijective linear map formula_80 be the Fourier transform, also called formula_81-click (of the outer star) or formula_82 rotation; and let formula_83 be the coproduct of formula_40 and formula_85.
Note that the word "coproduct" is a diminutive of "convolution product". It is a binary operation.
The coproduct satisfies the equality formula_86
For any positive operators formula_87, the coproduct formula_88 is also positive; this can be seen diagrammatically:
Let formula_89 be the contragredient formula_90 (also called formula_91 rotation). The map formula_92 corresponds to four formula_81-clicks of the outer star, so it's the identity map, and then formula_94.
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 formula_95 with formula_96 a multiple of a projection. 
Note that formula_97 and formula_98 are biprojections; this can be seen as follows:
A projection formula_99 is a biprojection iff it is the Jones projection formula_100 of an intermediate subfactor formula_101
, iff formula_102.
Galois correspondence:
 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, of the form formula_103, as for an interval of finite groups formula_104.
Using the biprojections, we can make the intermediate subfactor planar algebras. 
The uncertainty principle extends to any irreducible subfactor planar algebra formula_105:
Let formula_106 with formula_107 the range projection of formula_108 and formula_109 the unnormalized trace (i.e. formula_110 on formula_111).
Noncommutative uncertainty principle: 
 Let formula_112, nonzero. Then
Assuming formula_108 and formula_115 positive, the equality holds if and only if formula_108 is a biprojection. More generally, the equality holds if and only if formula_108 is the "bi-shift" of a biprojection.