an:01787093
Zbl 1019.46036
Jones, Vaughan F. R.
The annular structure of subfactors
EN
Ghys, ??tienne (ed.) et al., Essays on geometry and related topics. M??moires d??di??s ?? Andr?? Haefliger. Vol. 2. Gen??ve: L'Enseignement Math??matique. Monogr. Enseign. Math. 38, 401-463 (2001).
2001
a
46L37
planar algebra; annular structure; subfactor
In the article under review, the author uses the technique of decomposition of any planar algebra which contains the Temperley-Lieb planar algebra.
The author presents two main applications of his technique. The first is a positivity result for the Poincar?? series of a planar algebra, obtained by summing the generating functions of the TL-modules contained in a planar algebra. There are certain restrictions on the principal graph of a subfactor of index close to 4.
The second application is to give a uniform method of the ADE series of subfactors of index less than 4. The author gave two versions of the proof, the first of which interprets the vanishing of a certain determinant as being the flatness of a certain connection in the Ocneanu language, or the computation of the relative commutants for a certain commuting square. The second proof is a purely planar algebraic proof which proceeds by giving a system of ``skein'' relations on a generator of a planar algebra which allow one to calculate the partition function of any closed tangle.
For the entire collection see [Zbl 0988.00115].
Andreiy Kondrat'yev (Red Level)