Spectral measures of small index principal graphs

The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $\Delta$ is the adjacency matrix of $X$ we consider the equation $\Delta=U+U^{-1}$. When $X$ has square norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map $u\to u^{-1}$, and we get a probability measure $\epsilon$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $\epsilon$ for the principal graphs of subfactors with index $\le 4$, the (extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment generating function of $\epsilon$ is closely related to Jones' $\Theta$-series.Comment: 23 page

A planar calculus for infinite index subfactors

We develop an analog of Jones' planar calculus for II_1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns' results on rotations and extremality for infinite index subfactors. These results are obtained without Jones' basic construction and the resulting Jones projections.Comment: 56 pages, many figure

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric

On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.Comment: 24 pages. Final version. To appear in Journal of Algebraic Combinatorics

Subfactors of index less than 5, part 1: the principal graph odometer

In this series of papers we show that there are exactly ten subfactors, other than $A_\infty$ subfactors, of index between 4 and 5. Previously this classification was known up to index $3+\sqrt{3}$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $3+\sqrt{3}$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete

Classical integrable lattice models through quantum group related formalism

We translate effectively our earlier quantum constructions to the classical language and using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra are able to construct Lax operators and associated $r$-matrices of classical integrable models. Thus new as well as known lattice systems of different classes are generated including new types of collective integrable models and canonical models with nonstandard $r$ matrices.Comment: 7 pages; Talk presented at NEEDS'93 (Gallipoli,Italy

Combinatorial expression for universal Vassiliev link invariant

The most general R-matrix type state sum model for link invariants is constructed. It contains in itself all R-matrix invariants and is a generating function for "universal" Vassiliev link invariants. This expression is more simple than Kontsevich's expression for the same quantity, because it is defined combinatorially and does not contain any integrals, except for an expression for "the universal Drinfeld's associator".Comment: 20 page

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.Comment: 33 pages. Has color figure

Weak Riemannian manifolds from finite index subfactors

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $${\cal O}(p) =\{u p u^*: u\in U_M\}$$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p)$ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\|<r\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\cal O}(p)\subset {\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$.Comment: 19 page

On the idempotents of Hecke algebras

We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu--Markov trace of the idempotents.Comment: 11 page