Planar algebra

The family of vector spaces ${\displaystyle ({\mathcal {T}}_{n,\pm })_{n\in \mathbb {N} }}$ generated by the planar tangles having ${\displaystyle 2n}$ intervals on their output disk and a white (or black) ${\displaystyle \star }$-marked interval, admits a planar algebra structure.

The Temperley-Lieb planar algebra ${\displaystyle {\mathcal {TL}}(\delta )}$ is generated by the planar tangles without input disk; its ${\displaystyle 3}$-box space ${\displaystyle {\mathcal {TL}}_{3,+}(\delta )}$ is generated by

Moreover, a closed string is replaced by a multiplication by ${\displaystyle \delta }$.

Note that the dimension of ${\displaystyle {\mathcal {TL}}_{n,\pm }(\delta )}$ is the Catalan number ${\displaystyle {\frac {1}{n+1}}{\binom {2n}{n}}}$. This planar algebra encodes the notion of Temperley–Lieb 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 ${\displaystyle \delta }$ and of depth two.^[7]

Note that connected means ${\displaystyle \dim({\mathcal {P}}_{0,\pm })=1}$ (as for evaluable below), irreducible means ${\displaystyle \dim({\mathcal {P}}_{1,+})=1}$, spherical is defined below, and non-degenerate means that the traces (defined below) are non-degenerate.

A subfactor planar algebra is a planar ${\displaystyle \star }$-algebra ${\displaystyle ({\mathcal {P}}_{n,\pm })_{n\in \mathbb {N} }}$ which is:

(1) Finite-dimensional: ${\displaystyle \dim({\mathcal {P}}_{n,\pm })<\infty }$

(2) Evaluable: ${\displaystyle {\mathcal {P}}_{0,\pm }=\mathbb {C} }$

(3) Spherical: ${\displaystyle tr:=tr_{r}=tr_{l}}$

(4) Positive: ${\displaystyle \langle a\vert b\rangle =tr(b^{\star }a)}$ defines an inner product.

Note that by (2) and (3), any closed string (shaded or not) counts for the same constant ${\displaystyle \delta }$.

The tangle action deals with the adjoint by:

${\displaystyle Z_{T}(a_{1}\otimes a_{2}\otimes \cdots \otimes a_{r})^{\star }=Z_{T^{\star }}(a_{1}^{\star }\otimes a_{2}^{\star }\otimes \cdots \otimes a_{r}^{\star })}$

with ${\displaystyle T^{\star }}$ the mirror image of ${\displaystyle T}$ and ${\displaystyle a_{i}^{\star }}$ the adjoint of ${\displaystyle a_{i}}$ in ${\displaystyle {\mathcal {P}}_{n_{i},\epsilon _{i}}}$.

No-ghost theorem: The planar algebra ${\displaystyle {\mathcal {TL}}(\delta )}$ has no ghost (i.e. element ${\displaystyle a}$ with ${\displaystyle \langle a\vert a\rangle <0}$) if and only if

${\displaystyle \delta \in \{2\cos(\pi /n)|n=3,4,5,...\}\cup [2,+\infty ]}$

For ${\displaystyle \delta }$ as above, let ${\displaystyle {\mathcal {I}}}$ be the null ideal (generated by elements ${\displaystyle a}$ with ${\displaystyle \langle a\vert a\rangle =0}$). Then the quotient ${\displaystyle {\mathcal {TL}}(\delta )/{\mathcal {I}}}$ is a subfactor planar algebra, called the Temperley–Lieb-Jones subfactor planar algebra ${\displaystyle {\mathcal {TLJ}}(\delta )}$. Any subfactor planar algebra with constant ${\displaystyle \delta }$ admits ${\displaystyle {\mathcal {TLJ}}(\delta )}$ as planar subalgebra.

A planar algebra ${\displaystyle ({\mathcal {P}}_{n,\pm })}$ is a subfactor planar algebra if and only if it is the standard invariant of an extremal subfactor ${\displaystyle N\subseteq M}$ of index ${\displaystyle [M:N]=\delta ^{2}}$, with ${\displaystyle {\mathcal {P}}_{n,+}=N'\cap M_{n-1}}$ and ${\displaystyle {\mathcal {P}}_{n,-}=M'\cap M_{n}}$.^[8][9][10] A finite depth or irreducible subfactor is extremal (${\displaystyle tr_{N'}=tr_{M}}$ on ${\displaystyle N'\cap M}$).

There is a subfactor planar algebra encoding any finite group (and more generally, any finite dimensional Hopf ${\displaystyle {\rm {C}}^{\star }}$-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.^[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 ${\displaystyle {\mathcal {BJ}}(\delta _{1},\delta _{2})}$ (sometimes called Fuss-Catalan) is defined as for ${\displaystyle {\mathcal {TLJ}}(\delta )}$ but by allowing two colors of string with their own constant ${\displaystyle \delta _{1}}$ and ${\displaystyle \delta _{2}}$, with ${\displaystyle \delta _{i}}$ as above. It is a planar subalgebra of any subfactor planar algebra with an intermediate such that ${\displaystyle [K:N]=\delta _{1}^{2}}$ and ${\displaystyle [M:K]=\delta _{2}^{2}}$.^[16][17]

The first finite depth subfactor planar algebra of index ${\displaystyle \delta ^{2}>4}$ is called the Haagerup subfactor planar algebra.^[18] It has index ${\displaystyle (5+{\sqrt {13}})/2\sim 4.303}$.

The subfactor planar algebras are completely classified for index at most ${\displaystyle 5}$^[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]

Let ${\displaystyle N\subset M}$ be a finite index subfactor, and ${\displaystyle {\mathcal {P}}}$ the corresponding subfactor planar algebra. Assume that ${\displaystyle {\mathcal {P}}}$ is irreducible (i.e. ${\displaystyle {\mathcal {P}}_{1,+}=N'\cap M_{1}=\mathbb {C} }$). Let ${\displaystyle N\subset K\subset M}$ be an intermediate subfactor. Let the Jones projection ${\displaystyle e_{K}^{M}:L^{2}(M)\to L^{2}(K)}$. Note that ${\displaystyle e_{K}^{M}\in {\mathcal {P}}_{2,+}}$. Let ${\displaystyle id:=e_{M}^{M}}$ and ${\displaystyle e_{1}:=e_{N}^{M}}$.

Note that ${\displaystyle tr(e_{1})=\delta ^{-2}=[M:N]^{-1}}$ and ${\displaystyle tr(id)=1}$.

Let the bijective linear map ${\displaystyle {\mathcal {F}}:{\mathcal {P}}_{2,\pm }\to {\mathcal {P}}_{2,\mp }}$ be the Fourier transform, also called ${\displaystyle 1}$-click (of the outer star) or ${\displaystyle 90^{\circ }}$ rotation; and let ${\displaystyle a*b}$ be the coproduct of ${\displaystyle a}$ and ${\displaystyle b}$.

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

The coproduct satisfies the equality ${\displaystyle a*b={\mathcal {F}}({\mathcal {F}}^{-1}(a){\mathcal {F}}^{-1}(b)).}$

For any positive operators ${\displaystyle a,b}$, the coproduct ${\displaystyle a*b}$ is also positive; this can be seen diagrammatically:^[26]

Let ${\displaystyle {\overline {a}}:={\mathcal {F}}({\mathcal {F}}(a))}$ be the contragredient ${\displaystyle a}$ (also called ${\displaystyle 180^{\circ }}$ rotation). The map ${\displaystyle {\mathcal {F}}^{4}}$ corresponds to four ${\displaystyle 1}$-clicks of the outer star, so it's the identity map, and then ${\displaystyle {\overline {\overline {a}}}=a}$.

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 ${\displaystyle b\in {\mathcal {P}}_{2,+}\setminus \{0\}}$ with ${\displaystyle {\mathcal {F}}(b)}$ a multiple of a projection. Note that ${\displaystyle e_{1}=e_{N}^{M}}$ and ${\displaystyle id=e_{M}^{M}}$ are biprojections; this can be seen as follows:

A projection ${\displaystyle b}$ is a biprojection iff it is the Jones projection ${\displaystyle e_{K}^{M}}$ of an intermediate subfactor ${\displaystyle N\subset K\subset M}$,^[27] iff ${\displaystyle e_{1}\leq b={\overline {b}}=\lambda b*b,{\text{ with }}\lambda ^{-1}=\delta tr(b)}$.^[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 ${\displaystyle [e_{1},id]}$, as for an interval of finite groups ${\displaystyle [H,G]}$.

Using the biprojections, we can make the intermediate subfactor planar algebras.^[31][32]

The uncertainty principle extends to any irreducible subfactor planar algebra ${\displaystyle {\mathcal {P}}}$:

Let ${\displaystyle {\mathcal {S}}(x)=Tr(R(x))}$ with ${\displaystyle R(x)}$ the range projection of ${\displaystyle x}$ and ${\displaystyle Tr}$ the unnormalized trace (i.e. ${\displaystyle Tr=\delta ^{n}tr}$ on ${\displaystyle {\mathcal {P}}_{n,\pm }}$).

Noncommutative uncertainty principle:^[33] Let ${\displaystyle x\in {\mathcal {P}}_{2,\pm }}$, nonzero. Then

${\displaystyle {\mathcal {S}}(x){\mathcal {S}}({\mathcal {F}}(x))\geq \delta ^{2}}$

Assuming ${\displaystyle x}$ and ${\displaystyle {\mathcal {F}}(x)}$ positive, the equality holds if and only if ${\displaystyle x}$ is a biprojection. More generally, the equality holds if and only if ${\displaystyle x}$ is the bi-shift of a biprojection.