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 subfactor planar algebra provides a family of unitary representations of Thompson groups.
[4] Any finite group (and quantum generalization) can be encoded as a planar algebra.
[1] The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant.
[1][5][6] A (shaded) planar tangle is the data of finitely many input disks, one output disk, non-intersecting strings giving an even number, say
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.
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
-box spaces, on which acts the planar operad, i.e. for any tangle
-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.
intervals on their output disk and a white (or black)
-marked interval, admits a planar algebra structure.
is generated by the planar tangles without input disk; its
which is: Note that by (2) and (3), any closed string (shaded or not) counts for the same constant
[8][9][10] A finite depth or irreducible subfactor is extremal (
There is a subfactor planar algebra encoding any finite group (and more generally, any finite dimensional Hopf
-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.
[16][17] The first finite depth subfactor planar algebra of index
is called the Haagerup subfactor planar algebra.
The subfactor planar algebras are completely classified for index at most
[21] It uses (among other things) a listing of possible principal graphs, together with the embedding theorem[22] and the jellyfish algorithm.
[24] 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.
Note that the word coproduct is a diminutive of convolution product.
In the Kac algebra case, the contragredient is exactly the antipode,[12] which, for a finite group, correspond to the inverse.
[28][26] Galois correspondence:[29] 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,[30] of the form
Using the biprojections, we can make the intermediate subfactor planar algebras.
[31][32] The uncertainty principle extends to any irreducible subfactor planar algebra