David Penneys

The embedding theorem for finite depth subfactor planar algebras

We define a canonical relative commutant planar algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.

SUBFACTORS OF INDEX LESS THAN 5, PART 4: VINES

We eliminate 39 infinite families of possible principal graphs as part of the classification of subfactors up to index 5. A number-theoretic result of Calegari–Morrison–Snyder, generalizing Asaeda–Yasuda, reduces each infinite family to a finite number of cases. We provide algorithms for computing the effective constants that are required for this result, and we obtain 28 possible principal graphs. The Ostrik d-number test and an algebraic integer test reduce this list to seven graphs in the index range (4,5) which actually occur as principal graphs.

Rigid C*-tensor categories of bimodules over interpolated free group factors

Given a countably generated rigid C*-tensor category C, we construct a planar algebra P• whose category of projections Pro is equivalent to C. From P•, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C*-tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F∞).

CALCULATING TWO-STRAND JELLYFISH RELATIONS

We construct a 3 Z=4 subfactor using an algorithm which, given generators in a spoke graph planar algebra, computes two-strand jellyfish relations. This subfactor was known to Izumi, but has not previously appeared in the literature. We systematically analyze the space of second annular consequences, adapting Jones’ treatment of the space of first annular consequences in his quadratic tangles article. This article is the natural followup to two recent articles on spoke subfactor planar algebras and the jellyfish algorithm. Work of Bigelow and Penneys explains the connection between spoke subfactor planar algebras and the jellyfish algorithm, and work of Morrison and Penneys automates the construction of subfactors where both principal graphs are spoke graphs using one-strand jellyfish. This is the published version of arXiv:1308.5197.

2-supertransitive subfactors at index 3+5

Abstract We introduce a new method for showing that a planar algebra is evaluable. In fact, this method is universal for finite depth subfactor planar algebras. By making careful choices in the methods application, one can often significantly reduce the complexity of the computations. Using our technique, we prove existence and uniqueness of a subfactor planar algebra with principal graph consisting of a diamond with arms of length 2 at opposite sides, which we call “2D2”. This is expected to be the last remaining construction required for the classification of subfactor planar algebras up to index 3 + 5 . This classification will also require showing the uniqueness of the subfactor planar algebra with principal graph 4442. We include a short proof of this fact, known to Izumi but as yet unpublished. This is the published version of arXiv:1406.3401 .

Subfactors of index exactly 5

Masaki Izumi was supported by JSPS, the Grant-in-Aid for Scientific Research (B) 22340032. Scott Morrison was supported by an Australian Research Council Discovery Early Career Researcher Award, DE120100232 and Discovery Project ‘Subfactors and symmetries’ DP140100732. David Penneys was supported in part by the Natural Sciences and Engineering Research Council of Canada. The last four authors were supported by DOD-DARPA grant HR0011-12-1-0009.

Operator Algebras in Rigid C*-Tensor Categories

In this article, we define operator algebras internal to a rigid C*-tensor category

Constructing spoke subfactors using the jellyfish algorithm

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