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-marked interval per disk.

-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-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-box spaces, on which acts intervals respectively) there is a multilinear map

-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

intervals on their output-marked interval, admits a planar algebra structure.

Temperley–Lieb

is generated by

.

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

Hopf algebra

A semisimple and cosemisimple Hopf algebra over an algebraically closed field and of depth two.

Note that connected (as for evaluable below), irreducible, spherical is defined below, and non-degenerate means that the traces (defined below) are non-degenerate.

Subfactor planar algebra

Definition

A subfactor planar algebra which is:

1. defines an inner product.

.

The tangle action deals with the adjoint by:

.

Examples and results

No-ghost theorem) if and only if

is a subfactor planar algebra, called the Temperley–Lieb-Jones subfactor planar algebra as planar subalgebra.

is a subfactor planar algebra if and only if it is the standard invariant of an extremal subfactor. A finite depth or irreducible subfactor).

There is a subfactor planar algebra encoding any finite group (and more generally, any finite dimensional {\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.

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

.

is called the Haagerup subfactor planar algebra..

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

.

.

be the Fourier transform be the coproduct.

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

is also positive; this can be seen diagrammatically:

be the contragredient.

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

A biprojection are biprojections; this can be seen as follows:

,.

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,.

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

The uncertainty principle:

).

Noncommutative uncertainty principle:, nonzero. Then

is the bi-shift of a biprojection.

References