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]
Definition
The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant.[1][5][6]
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
, intervals per disk and one
-marked interval per disk.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
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.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
, 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
![{\displaystyle Z_{T}:{\mathcal {P}}_{n_{1},\epsilon _{1}}\otimes \cdots \otimes {\mathcal {P}}_{n_{r},\epsilon _{r}}\to {\mathcal {P}}_{n_{0},\epsilon _{0}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Examples
Planar tangles
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
The Temperley-Lieb planar algebra
is generated by the planar tangles without input disk; its
-box space
is generated by
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Moreover, a closed string is replaced by a multiplication by
.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Note that the dimension of
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 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
and of depth two.[7]
Note that connected means
(as for evaluable below), irreducible means
, 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
-algebra
which is:
- (1) Finite-dimensional:
![{\displaystyle \dim({\mathcal {P}}_{n,\pm })<\infty }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- (2) Evaluable:
![{\displaystyle {\mathcal {P}}_{0,\pm }=\mathbb {C} }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- (3) Spherical:
![{\displaystyle tr:=tr_{r}=tr_{l}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- (4) Positive:
defines an inner product.
Note that by (2) and (3), any closed string (shaded or not) counts for the same constant
.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The tangle action deals with the adjoint by:
![{\displaystyle 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 })}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
the mirror image of
and
the adjoint of
in
.
Examples and results
No-ghost theorem: The planar algebra
has no ghost (i.e. element
with
) if and only if
![{\displaystyle \delta \in \{2\cos(\pi /n)|n=3,4,5,...\}\cup [2,+\infty ]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For
as above, let
be the null ideal (generated by elements
with
). Then the quotient
is a subfactor planar algebra, called the Temperley–Lieb-Jones 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
.[8][9][10]A finite depth or irreducible subfactor is extremal (
on
).
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.[11][12]
The subfactor planar algebra associated to an inclusion of finite groups,[13]
does not always remember the (core-free) inclusion.[14][15]
A Bisch-Jones subfactor planar algebra
(sometimes called Fuss-Catalan) is defined as for
but by allowing two colors of string with their own constant
and
, with
as above. It is a planar subalgebra of any subfactor planar algebra with an intermediate such that
and
.[16][17]
The first finite depth subfactor planar algebra of index
is called the Haagerup subfactor planar algebra.[18] It has index
.
The subfactor planar algebras are completely classified for index at most
[19]and a bit beyond.[20]This classification was initiated by Uffe Haagerup.[21]It uses (among other things) a listing of possible principal graphs, together with the embedding theorem[22]and the jellyfish algorithm.[23]
A subfactor planar algebra remembers the subfactor (i.e. its standard invariant is complete) if it is amenable.[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.[25]
Fourier transform and biprojections
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
.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
.
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Note that the word coproduct is a diminutive of convolution product. It is a binary operation.
The coproduct satisfies the equality ![{\displaystyle a*b={\mathcal {F}}({\mathcal {F}}^{-1}(a){\mathcal {F}}^{-1}(b)).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For any positive operators
, the coproduct
is also positive; this can be seen diagrammatically:[26]
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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,[12] 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:
![](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A projection
is a biprojection iff it is the Jones projection
of an intermediate subfactor
,[27] iff
.[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
, as for an interval of finite groups
.
Using the biprojections, we can make the intermediate subfactor planar algebras.[31][32]
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:[33] Let
, nonzero. Then
![{\displaystyle {\mathcal {S}}(x){\mathcal {S}}({\mathcal {F}}(x))\geq \delta ^{2}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
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- ^ "Dror Bar-Natan: Publications: Cobordisms". Math.toronto.edu. arXiv:math/0410495. doi:10.2140/gt.2005.9.1443. Retrieved 2016-11-20.
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