The classification of subfactor planar algebras generated by a non-trivial 2-box was initiated by Bisch and Jones [BJ97][BJ03][DBL17]. The complete classification of planar algebras generated solely by a non-trivial 2-box with the Yang-Baxter relation was completed in [Liu15].
Subfactors
One of the main structures to reveal the nature of subfactors is obtained by the basic construction as follows: Let1be the orthogonal projection from L2(M) to the subspace L2(N). The standard invariant is the relative commutants of the Jones tower with respect to N and M, defined as.
Planar algebras
A planar algebra (shaded) is a family of vector spaces scaled by Z2Pn,±, n∈N, with multilinear maps defined by planar perturbations. The standard invariant of an extremal subfactor of finite index is a planar spherical subfactor algebra.
One-way Yang-Baxter planar algebras
Introduction
Due to the classification of Yang-Baxter planar algebras, we are left with the so-called One-way Yang-Baxter case. Does there exist a subfactor planar algebra that satisfies the One-way Yang-Baxter relation without having Yang-Baxter relation.
Subgroup subfactors and Spin models
- Generators
- An evaluable skein theory
Therefore, the subfactorial planar algebra P• is a subalgebra of fixed points of the planar algebra of the spin model V under the action S5. By construction, the black areas of X touching the output disk correspond to the nodes of the Petersen graph. Next, we will show that the third term on the right-hand side of equation (3.1) belongs to the planar subalgebra Q.
The evaluation of Dis is defined as the number of the Petersen coloring of the planar graph ΓD.
Thurston-relation planar algebras
Introduction
Preliminaries
- HOMFLY-PT planar algebras
- Thurston’s skein relations
In this case P•H(q,r) has a unique involution∗, so that its semisimple quotient is a subfactor planar algebra. As emphasized in the introduction, difference theory provides an important perspective from which to understand a planar algebra for many reasons. Moreover, the standard forms in the sense of Thurston [Do17] give a basis of then-box space when their dimension is the maximum ofn! reach.
Also, one can replace them with the other diagrams with two generators using Thurston relation. The Thurston relation is similar to the 6j symbol in a monoidal category, but it provides a much better evaluation algorithm. Indeed, for the example P•H(q,r) appearing in our classification, it is difficult to calculate the 6j symbols.
The Thurston relation requires only partial 6j symbols for two objects, along with the data for the double planar algebra. This combination, instead of considering only one side, seems powerful and completely defines planar algebra. The main goal of this paper is to classify all subfactor planar algebras generated by a 3-box Ssatisfying Thurston relation.
Generic case
- Generators
- Relations in 3-boxes
- Relations in 4-boxes
- Classification
We can consider the elements of the f2 cut as morphisms in the N−N bimodule category (the even part of the subfactor plane algebra). Note that the f2-cut of the 24 diagrams in the basis of P4,+ in Corollary 3.2.7 is in the linear span of B, so that the f2-cut of P4,+ is spanned over B. Suppose that P• is a planar algebra generated by a nontrivial 3-box satisfying Thurston relations, then P• is determined by (δ,γ,ω,a,a0), whereδ2 is the index,γthe ratio of the trace of the two orthogonal minimal idempotents, ω is the rotational eigenvalue and a,a0 are the signs in the unshaded and shaded (2↔2) motions.
Since all coefficients in the strand theory are determined by (δ,γ,ω), φ extends to a planar algebra isomorphism. IfP• is a planar algebra generated by a non-trivial 3-box satisfying the Thurston relation with parameters(δ,γ,ω,a,a0),δ >2, then the rotation eigenvalueω =1 . S , and applying the motion (2↔2) in the dotted line, we can rewrite with respect to the base in Corollary 3.2.7.
The coefficient of SS in the linear representation of SS S with respect to this basis is bω−1. S is an element in the f2 cutdown of P3,+, we apply the (2↔2) motion of the Thurston relation and thus obtain. With the following lemma and theorem we will identify any planar algebra with Thurston relation as P·H(q,r) for some(q,r).
Reduced case
- The case for at most 22 dimensional 4-box space
- The case for 23 dimensional 4-box space
In this section we classify the subfactorial planar algebrasP• generated by the 3-boxSwith Thurston relation such that dim(P4,±) =23. According to the result of Jones [Jon01], 14 Temperley-Lieb diagrams and 8 diagrams in the ring consequence are linearly independent. There we used the fact that S S is linearly independent of B0, which no longer holds for the reduced case.
Whereσ2=ω and[n] = qq−qn−q−1−n is the quantum number (whereq is defined so thatδ = [2] = q+q−1ogq>1), P0 is the projection labeled in our image, and P0 is double projection. One can show that there is no solution with δ >2 in the same way as the case a=1. Assume that P•is a singly generated Thurston relation plane algebra anddimP4,±= 23, then it is isomorphic to the semisimple quotient of P•H(q,r) for some(q,r).
We notice that in the case of 24-dimensional spaces with 4 boxes, our proof did not require the full force of the assumption that we had a planar algebra with a subfactor. In particular, we did not need the canonical inner product on the box spaces to induce in any essential way a positive definite inner product on the hom spaces, and it would have been sufficient in principle to assume that this inner product was simply non-degenerate. . For the cases in this section, where the 4-box space has a size less than 24, we have appealed to triple point obstructions and subfactor classification, which are theorems for planar algebras of subfactors, and so we have the subfactor assumption on used in an essential way.
Positivity
Introduction
The above definition is independent of the choice p,q in the direct limit A(I) and thus, A(I) forms a group with the identity be[(f,f)]for some f ∈D and the inverse of [( f,g)]equal to [(g,f)]. The choice of the category C and the functor Φ gives an action of the group A(I) on the set A(Φ). The above definition is independent of the choice p,q on the direct limit A(I) and thus, this gives an action of A(I) on A(Φ).
From this point of view, there are many actions of the Thompson group F with various choices of the category C and the functorΦ. With a good choice of the functor one can construct unitary representations of the Thompson group F. The Jones subgroup~F and~T was introduced as stabilizer of the vacuum vector in the above representations in [Jon14].
Therefore, applying the construction of the direct limit, we obtain an action of the Thompson groupF on the direct limitA(Φ). Taking the completion of A(Φ), we obtain a Hilbert spaceHR and a mapπR from the Thompson groupF to the bounded operators of the Hilbert spaceHR. From the theory of subfactors one can show that (π,HR) is a unitary representation of the Thompson groupF.
Singly generated subgroups
The vecF subgroup was shown to be isomorphic to the Thompson F3 group by Golan and Sapir [GS17]. For each Xk, we identify it as the same diagram with infinitely many rows on the right. Therefore, we define the multiplication·i Xmand Xnas stacking the tangle from top to bottom as the multiplication inP•.
Motivated by the representation of the pair of binary trees of the group ThompsonF, we consider the group consisting of pairs of elements of a given type by Alg(X). The binary operation◦ is well defined on GX, and GX is a group with the binary operation◦. We drop ◦when there is no confusion and call such a set GX a subgroup generated only by X .
In the following sections, we denote(T,R)for T,R∈Alg(X)nfor the equivalence class of[(T,R)]for elements in GX.
The Classical presentation
If there is one adjacent to the rightmost string at the end of X0, then apply vertical isotopy to get a diagram starting with X0·XN−1. Repeating this procedure and letting α(T) be the number of steps, we obtain that T =Sα(T)·XT, where XT is the rest of the word. By Lemma 4.3.3, we only need to show that every element g∈PX belongs to the subgroup generated by{xn:n∈N}.
It follows that Φ extends to the homomorphism of the surjective group from htn,n∈N|tk−1tntk=tn+N−1,∀k We arrange the pair of trees in R2 so that the leaves eT± are the points n−1)/2,0) with all edges that are line segments sloping either up from left to right or down from e left to right. T+ is in the plane of the upper half and T− is in the plane of the lower half. The vertices (k,0) and (j,0) are connected by an edge if and only if the corresponding regions are separated by an edge sloping up to T+ or down to T− from left to right. From the remark after Proposition 4.4.2, we have that g∈~F if and only if Γ(T+,T−) is a planar bipartite graph. Note that (T+,T−)∼(T+0,T−0) and thus by repeating this procedure, we obtain a pair of tree representations, thereby proving the lemma. Suppose ∈~F is a non-trivial element with some tree representation (T+,T−) as in Lemma 4.4.4 and T+ has 2n−1 leaves. Since every vertex in Γ(T+) is connected to exactly one vertex on its left, there exists ∈N such that (m,0) is connected to (m−1,0) by induction. Jones defined a family of subgroups of F starting with ~F using the matrix R in spin models [Jon14]. These subgroups were redefined by Golan and Sapir denoted by ~Fn,n≥2 and shown to be isomorphic to Fn+1[GS17]. These results can be obtained using the same idea in the proof of Theorem 4.4.5. Assume that g∈F, then g has a tree pair representation(T+,T−) such that the color of the vertices ofΓ(T+,T−) is acbacb· · ·ac. First note that this lemma must hold for the coloring after applying any permutation of {a,b,c}toacbacb· · ·ac. If there are more vertices on the left edge of v00, then the dashed circle part is a tree too. The same argument applies to cases where there are more vertices on the right edge of v00 or the left (or right) edge of v01. Bis97], Bimodules, higher relative commutants and the fusion algebra associated with a subfactor, Operator algebras and their applications (Waterloo, ON Fields Inst. Ocneanu, A new polynomial invariant of knots and links, Bulletin of the American Mathematical Society 12 (1985) , No . Gupta, Planar algebra of the subgroup subfactor, Indian Academy of Sciences Proceedings-Mathematical Sciences, vol Haagerup, Prime graphs of subfactors in the index range4<[M:N]<3+√ 2, Subfactors (Kyuzeso,1993), World Sci. Jon94] , The Potts model and the symmetric group, Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), World Sci.Examples
The Jones subgroup ~ F
The 3-colorable subgroup
