Exchange relation planar algebras of small rank

Abstract

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras, subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.

35 pages, 102 figures

References in corpus (3)

• The classification of subfactors of index at most 5
• Random matrices, free probability, planar algebras and subfactors
• Subfactors of index less than 5, part 3: quadruple points

Cited by in corpus (10)

• Anyonic Chains, Topological Defects, and Conformal Field Theory
• Planar Para Algebras, Reflection Positivity
• Quantum Fourier Analysis
• Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis
• Fusion Bialgebras and Fourier Analysis
• Quantum Galois groups of subfactors
• Ore's theorem on cyclic subfactor planar algebras and beyond
• Quantum Ruzsa Divergence to Quantify Magic
• Interpolated family of non group-like simple integral fusion rings of Lie type
• Motzkin Algebras and the $A_n$ Tensor Categories of Bimodules