The Annular Structure of Subfactors

By Vaughan F. R. Jones. 2001

126 citations recorded by the source corpus.

Given a planar algebra we show the equivalence of the notions of a module over this algebra (in the operadic sense), and module over a universal annular algebra. We classify such modules, with invariant inner products, in the generic region and give applications to subfactorss, including a planar construction of the $E_6$ and $E_8$ subfactors.

Subfields: Inorganic Chemistry

View Paper: https://www.semanticscholar.org/paper/c72056b59a74706459f32446f55241da770ffe13

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In Inorganic Chemistry

1. E. Peters
2. Sarah Reznikoff
3. S. Ghosh
4. Corey Jones
5. Noah Snyder
6. S. Morrison
7. Ben Elias

Citations in the corpus

• Quadratic Tangles in Planar Algebras (2010, methodology): “(That the annular images are linearly independent is shown in [14].”
• Skein theory for the D_2pi planar algebras (2008, methodology): “Using the techniques in [25] and [30], find the generator and some of the relations for this planar algebra.”
• A PLANAR ALGEBRA CONSTRUCTION OF THE HAAGERUP SUBFACTOR (2009, methodology): “Examples of subfactor planar algebras for which presentations are known include An, D2n, E6 and E8 (a general form for these presentations is given in [ 19 ]; a more combinatorial description of the D2n case is found in [24]), and Bisch-Haagerup subfactors ([4], [3]).”
• Annular representation theory for rigid $C^{*}$-tensor categories (2015, methodology): “The affine annular representations of this planar algebra have been well studied [17],[16], and we will make use of know results to analyze the universal algebra structure on the centralizer algebras of the tube algebra of this category.”
• Hilbert space representations of the annular temperley-lieb algebra (2006, methodology): “(−1)r [r+2k]/[2n] in JW; see [Jones 2001], for example.”
• The two-color Soergel calculus (2013, background): “Rotation in the Temperley-Lieb algebra has been studied before (see [22, 15]), but typically in the negligible quotient.”