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Vaughan F. R. Jones

Publications and source records attributed to Vaughan F. R. Jones.

At least 19 recordsLinked to original sources
arXiv2020

Motzkin Algebras and the $A_n$ Tensor Categories of Bimodules

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that $\cup_{k\geq 1}M_k(D)$ admits a factor trace if and only if $D\in \{2\cos(π/n)+1|n\geq 3\}\cup [3,\infty)$ and higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over the factors are constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.

math.RT↗
arXiv2020

Bergman space zero sets, modular forms, von Neumann algebras and ordered groups

$A^2_α$ will denote the weighted $L^2$ Bergman space. Given a subset $S$ of the open unit disc we define $Ω(S)$ to be the infimum of $\{s| \exists f \in A^2_{s-2}, f\neq 0, \mbox{ having $S$ as its zero set} \}$.By classical results on Hardy space there are sets $S$ for which $Ω(S)=1$. Using von Neumann dimension techniques and cusp forms we give examples of $S$ where $1<Ω(S)<\infty$. By using a left order on certain Fuchsian groups we are able to calculate $Ω(S)$ exactly if $Ω(S)$ is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms for \pslz.

math.FA↗
arXiv2019

On spectral measures for certain unitary representations of R. Thompson's group F

The Hilbert space $\mathcal H$ of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson's group $F$ via local scale transformations. Given a vector in the canonical dense subspace of $\mathcal H$ we show how to calculate the corresponding spectral measure for any element of $F$ and illustrate with some examples. Introducing the "essential part" of an element we show that the spectral measure of any vector in $\mathcal H$ is, apart from possibly finitely many eigenvalues, absolutely continuous with respect to Lebesgue measure. The same considerations and results hold for the Brown-Thompson groups $F_n$ (for which $F=F_2$).

math.GR↗
arXiv2018

Pythagorean representations of Thompson's groups

We introduce the Pythagorean C*-algebras and use the category/functor method to construct unitary representations of Thompson's groups from representations of them. We calculate several examples.

math.GR↗
arXiv2018

On the Haagerup and Kazhdan properties of R. Thompson's groups

A machine developed by the second author produces a rich family of unitary representations of the Thompson groups F,T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero [F,F]-invariant vectors reproving, at least for T, Reznikov's result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan's property (T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving Farley's result that T has the Haagerup property.

math.GR↗
arXiv2017

The Homflypt polynomial and the oriented Thompson group

We show how to construct unitary representations of the oriented Thompson group $\vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec{F}$.

math.GR↗
arXiv2017

Scale invariant transfer matrices and Hamiltionians

Given a direct system of Hilbert spaces $s\mapsto \mathcal H_s$ (with isometric inclusion maps $ι_s^t:\mathcal H_s\rightarrow \mathcal H_t$ for $s\leq t$) corresponding to quantum systems on scales $s$, we define notions of scale invariant and weakly scale invariant operators. Is some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so. Scale invariance forces spatial inhomogeneity of the spectral parameter. But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.

math.OA↗
arXiv2016

A no-go theorem for the continuum limit of a periodic quantum spin chain

We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.

math.OA↗
arXiv2014

Some unitary representations of Thompson's groups F and T

In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations" of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new "oriented" subgroups $\overrightarrow F <F$ and $\overrightarrow T< T$ which allow us to produce all \emph{oriented} knots and links.

math.GR↗
arXiv2014

Singly generated planar algebras of small dimension, Part III

The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for $\mathbb{Z}_3$, the Fuss-Catalan planar algebra ; and the group/subgroup planar algebra for $\mathbb{Z}_2\subset \mathbb{Z}_5\rtimes \mathbb{Z}_2$. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all BMW. Precisely it contains a depth 3 one from quantum $SO(3)$, and a one-parameter family from quantum $Sp(4)$.

math.OA↗
arXiv2014

Infinite index subfactors and the GICAR categories

Given a II$_1$-subfactor $A\subset B$ of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as $A-A$ bimodule maps among the bimodules $\bigotimes_A^n L^2(B)$. As a corollary, we get a lower bound on the dimension of the centralizer algebras $A_0'\cap A_{2n}$ for infinite index subfactors, and we also get that $A_0'\cap A_{2n}$ is nonabelian for $n\geq 2$, where $(A_n)_{n\geq 0}$ is the Jones tower for $A_0=A\subset B=A_1$. We also show that the annular GICAR/planar rook category acts as maps amongst the $A$-central vectors in $\bigotimes_A^n L^2(B)$, although this action may be degenerate. We prove these results in more generality using bimodules. The embedding of the GICAR category builds on work of Connes and Evans who originally found GICAR algebras inside Temperley-Lieb algebras with finite modulus.

math.OA↗
arXiv2013

The classification of subfactors of index at most 5

A subfactor is an inclusion $N \subset M$ of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action $M^G \subset M$, and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There's a measure of size of a subfactor, called the index. Remarkably the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the '80s and early '90s. The possible index values above 4 are not quantized, but once you exclude a certain family it turns out that again the possibilities are quantized. Recently the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.

math.OA↗
arXiv2011

Subfactors of index less than 5, part 3: quadruple points

One major obstacle in extending the classification of small index subfactors beyond 3+\sqrt{3} is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the "weeds" Q and Q' from Part 1 (arXiv:1007.1730)). Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. In this paper we prove two quadruple point obstructions. The first uses quadratic tangles techniques and eliminates the weed Q' immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds Q and Q'. Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodman-de la Harpe-Jones subfactor using a combination of planar algebra techniques and connections.

math.OA↗
arXiv2011

On the symmetric enveloping algebra of planar algebra subfactors

We give a diagrammatic description of Popa's symmetric enveloping algebras associated to planar algebra subfactors. As an application we construct a natural family of derivations on these factors, and compute a certain free entropy dimension type quantity.

math.OA↗
arXiv2010

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.

math.OA↗
arXiv2010

Quadratic Tangles in Planar Algebras

In planar algebras, we show how to project certain simple "quadratic" tangles onto the linear space spanned by "linear" and "constant" tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.

math.OA↗