Skein theory for the D_2pi planar algebras

By S. Morrison, E. Peters, Noah Snyder. 2008

65 citations recorded by the source corpus.

Abstract We give a combinatorial description of the “ D 2 n planar algebra”, by generators and relations. We explain how the generator interacts with the Temperley–Lieb braiding. This shows the previously known braiding on the even part extends to a ‘braiding up to sign’ on the entire planar algebra. We give a direct proof that our relations are consistent (using this ‘braiding up to sign’), give a complete description of the associated tensor category and principal graph, and show that the planar algebra is positive definite. These facts allow us to identify our combinatorial construction with the standard invariant of the subfactor D 2 n .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0dc87090f0f8ba526c23fbfd77292237b66a4087

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In Algebra and Number Theory

1. Noah Snyder
2. S. Morrison
3. David Penneys
4. E. Peters
5. Vaughan F. R. Jones
6. James E. Tener
7. A. Henriques
8. Stephen Bigelow

In Geometry and Topology

1. Noah Snyder
2. S. Morrison
3. Vaughan F. R. Jones
4. David Penneys
5. E. Peters
6. A. Henriques
7. James E. Tener
8. Stephen Bigelow

In Mathematical Physics

1. Noah Snyder
2. Vaughan F. R. Jones
3. S. Morrison
4. David Penneys
5. E. Peters
6. A. Henriques
7. James E. Tener
8. Stephen Bigelow

Citations in the corpus

• Constructing the extended Haagerup planar algebra (2009, methodology): “uation (3.1). Nevertheless, we found it useful to look for relations that could play a similar role. In particular, the evaluation algorithm describedin §4 was inspired by theevaluation algorithms in [43] and [4] for planar algebras of types D2n, E6, and E8, where Equation (3.1) holds. Definition 3.3”
• The classification of subfactors of index at most 5 (2013, background): “ and E 8 comes from SU(2) 28 ˆG 2 [Xu98]. The planar algebras for the ADE subfactors are constructed in [Jon01]. A purely generators-and-relations description for the D 2n planar algebras is given in [MPS10], and Bigelow [Big10] gave generators and relations for E 6 and E 8, and gave explicit bases ”
• Categorified trace for module tensor categories over braided tensor categories (2015, methodology): “the braiding (5), only the T 2n arise from commutative algebras, while if we negate (5), only the T 2n+1 arise from commutative algebras. The skein theory for the D 2n planar algebras was computed in [MPS10], and the skein theory for the E 6 and E 8 planar algebras was computed in [Big10]. To give e”
• The embedding theorem for finite depth subfactor planar algebras (2010, background): “the following theorem, which is the main result of this paper: Theorem. A nite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph. (See [MPS10] for the denition of the principal graph of a planar algebra.) We prove this result with the ”
• A PLANAR ALGEBRA CONSTRUCTION OF THE HAAGERUP SUBFACTOR (2009, background): “Proof. By any of [27, 9, 16 ], the existence of a planar algebra with principal graph H implies the existence a subfactor with principal graph H.”
• Categories generated by a trivalent vertex (2015, background): “the even part of the D 6 subfactor and its Galois conjugate. These categories contain a trivalent subcategory SO(3) 20 and a single extra generating 4-box satisfying a version of the relations from [MPS10]. The relationship between D 6 and the tensor products of golden categories is explained by le”