arXiv subjects

Arnaud Brothier

Publications and source records attributed to Arnaud Brothier.

At least 19 recordsLinked to original sources

arXiv2026

The Cuntz-Pimsner algebra of the simplest Motzkin subproduct system is 2-subhomogeneous

The Motzkin subproduct system (SPS) is constructed from the Jones Wenzl idempotents of the Motzkin algebras, which generalizes the Temperley Lieb SPS. The simplest Motzkin SPS, which is not a Temperley Lieb SPS, is constructed from a 3 dimensional Hilbert space. We explicitly describe the spectrum of the corresponding Cuntz Pimsner algebra, which remarkably admits only irreducible representations of dimensions 1 and 2, and can thus be viewed as a mildly quantum space. We moreover analyse representations of the Richard Thompson groups and of the Cuntz algebra that are associated to this spectrum.

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arXiv2026

A new perspective on spectra of quantum spaces

We consider a class of C*-algebras C(X) associated with quantum spaces such as spheres, projective spaces, and lens spaces. We introduce a non-self-adjoint operator algebra A together with an explicit functor from the category of representations of A to that of C(X). We then construct explicitly a family of one-dimensional representations of A that parametrise the entire spectrum of C(X).

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arXiv2026

Classification of representations of higher-rank graph C*-algebras

We develop new techniques for the construction and classification of representations of row-finite and locally convex higher-rank graph C*-algebras O. This class includes Cuntz--Krieger algebras associated to row-finite directed graphs. Our approach relies on the representation theory of a certain non-self-adjoint algebra and a lifting process of representations. We introduce a novel dimension vector for representations of O yielding a countable partition of the spectrum. Given a Cuntz--Krieger algebra and a finite dimension vector, we construct a smooth manifold parametrising the corresponding spectral component. Our techniques are both explicit and functorial.

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arXiv2025

Finitely presented simple groups with no piecewise projective actions

We construct an explicit infinite family of pairwise non-isomorphic infinite simple groups of type $\mathrm{F}_\infty$ (in particular, they are finitely presented) that act faithfully on the circle by orientation-preserving homeomorphisms, but that admit no non-trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest-skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras.

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arXiv2024

Moduli of representations of Leavitt path algebras

We transpose Jones' technology and the authors' C*-algebraic techniques to study representations of the Leavitt path algebra L (over an arbitrary row-finite graph) by using its quiver algebra A. We establish an equivalence of categories between certain full subcategories of Rep(A) and Rep(L) that preserves irreducibility and indecomposability. We define a dimension function on Rep(L), and for each finite dimension we provide a moduli space for the irreducible classes by transporting structures of King and Nakajima on quiver representations. Our techniques are both explicit and functorial.

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arXiv2024

McCleary--Rubin reconstruction for simple forest-skein groups

A simple Ore forest-skein category produces three infinite groups analogous to Richard Thompson's groups F,T,V. We prove that reconstruction theorems of McCleary and Rubin apply to them: each of these groups encodes a canonical action by homeomorphisms. This provides powerful invariants that we use to distinguish infinitely many explicit simple groups that are finitely presented (of type $F_\infty$).

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arXiv2024

A tensor product for representations of the Cuntz algebra and of the R. Thompson groups

The authors continue a series of articles studying certain unitary representations of the Richard Thompson groups $F,T,V$ called Pythagorean. They all extend to the Cuntz algebra $\mathcal{O}$ and conversely all representations of $\mathcal{O}$ are of this form. Via this approach we introduce a tensor product for a large class of representations of $F,T,V,\mathcal{O}$. We prove that a sub-category forms a tensor category and perform a number of explicit computations of fusion rules.

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arXiv2024

Irreducible Pythagorean representations of R. Thompson's groups and of the Cuntz algebra

We introduce the Pythagorean dimension: a natural number (or infinity) for all representations of the Cuntz algebra and certain unitary representations of the Richard Thompson groups called Pythagorean. For each natural number d we completely classify (in a functorial manner) all such representations using finite dimensional linear algebra. Their irreducible classes form a real manifold of dimension $2d^2+1$ playing the role of a moduli space. Apart from a finite disjoint union of circles, each point of the manifold corresponds to an irreducible unitary representation of Thompson's group F (which extends to the other Thompson groups and the Cuntz algebra) that is not monomial. The remaining circles provide monomial representations which we previously fully described and classified. We translate in our language a large number of previous results in the literature. We explain how our techniques extend them.

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arXiv2024

Forest-skein groups III: simplicity

An Ore forest-skein category provides three forest-skein groups equipped with a powerful diagrammatic calculus analogous to Richard Thompson's groups F,T,V. We investigate when forest-skein groups have simple derived subgroups and establish two characterisations: a dynamical one and a categorical one. We then construct two classes of examples. The first associates two finitely presented simple groups to every finite binary tree and the second associates two simple groups to every n-ary Higman-Thompson group.

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arXiv2024

Atomic representations of R. Thompson's groups and Cuntz's algebra

We continue to study Pythagorean unitary representation of Richard Thompson's groups $F,T,V$ and their extension to the Cuntz(-Dixmier) algebra. Any linear isometry from a Hilbert space to its direct sum square produces such. We focus on those arising from a finite-dimensional Hilbert space. We show that they decompose as a direct sum of a so-called diffuse part and an atomic part. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. In this article, we fully describe the atomic part: it is a finite direct sum of irreducible monomial representations arising from a precise family of parabolic subgroups.

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arXiv2024

Forest-skein groups II: construction from homogeneously presented monoids

Inspired by the reconstruction program of conformal field theories of Vaughan Jones we recently introduced a vast class of so called forest-skein groups. They are built from a skein presentation: a set of colours and a set of pairs of coloured trees. Each nice skein presentation produces four groups similar to Richard Thompson's group F,T,V and the braided version BV of Brin and Dehornoy. In this article, we consider forest-skein groups obtained from one-dimensional skein presentations; the data of a homogeneous monoid presentation. We decompose these groups as wreath products. This permits to classify them up to isomorphisms. Moreover, we prove that a number of properties of the fraction group of the monoid pass through the forest-skein groups such as the Haagerup property, homological and topological finiteness properties, and orderability.

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arXiv2023

Decomposition of Pythagorean representations of R. Thompson's groups

We continue to study Pythagorean unitary representation of Richard Thompson's groups $F$, $T$ and $V$ that are built from a single isometry from a Hilbert space to its double. By developing powerful diagrammatically based techniques we show that each such representation splits into a diffuse and an atomic parts. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. We fully decompose the atomic part: the building blocks are monomial representations arising from a precise family of parabolic subgroups of $F$.

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arXiv2023

Haagerup property for wreath products constructed with Thompson's groups

Using recent techniques introduced by Jones we prove that a large family of discrete groups and groupoids have the Haagerup property. In particular, we show that if G is a discrete group with the Haagerup property, then the wreath product $\oplus_{Q_2}G\rtimes V$ obtained from the group G and the usual action of Thompson's group V on the dyadic rational $Q_2$ of the unit interval has the Haagerup property.

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arXiv2022

Jones' representations of R. Thompson's groups not induced by finite-dimensional ones

Given any linear isometry from a Hilbert space to its square one can explicitly construct a so-called Pythagorean unitary representation of Richard Thompson's group F. We introduce a condition on the isometry implying that the associated representation does not contain any induced representations by finite-dimensional ones. This provides the first result of this kind. We illustrate this theorem via a family of representations parametrised by the real 3-sphere for which all of them have this property except two sub-circles.

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arXiv2022

Forest-skein groups I: between Vaughan Jones' subfactors and Richard Thompson's groups

Vaughan Jones discovered unexpected connections between Richard Thompson's group and subfactor theory while attempting to construct conformal field theories (in short CFT). Among other this founded Jones' technology: a powerful new method for constructing actions of fraction groups which had numerous applications in mathematical physics, operator algebras, group theory and more surprisingly in knot theory and noncommutative probability theory. We propose and outline a program in the vein of Jones' work but where the Thompson group is replaced by a family of groups that we name forest-skein groups. These groups are constructed from diagrammatic categories, are tailor-made for using Jones' technology, capture key aspects of the Thompson group, and aim to better connect subfactors with CFT. Our program strengthens Jones' visionary work and moreover produces a plethora of concrete groups which satisfy exceptional properties. In this first article we introduce the general theory of forest-skein groups, provide criteria of existence, give explicit presentations, prove that their first L$^2$-Betti number vanishes, construct a canonical action on a totally ordered set, establish a topological finiteness theorem showing that many of our groups are of type $F_\infty$, and finish by studying a beautiful class of explicit examples.

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arXiv2021

Classification of Thompson related groups arising from Jones technology I

In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson's group $V$ is acting. These semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of groups having the Haagerup property. They naturally appear in certain field theories as being generated by local and global symmetries. Moreover, these groups occur in a construction of Tanushevski and can be realised using Brin-Zappa-Szep's products with the technology of cloning systems of Witzel-Zaremsky. We consider in this article the class of groups obtained in that way where one of the endomorphism is trivial leaving the case of two nontrivial endomorphisms to a second article. We provide an explicit description of all these groups as permutational restricted twisted wreath products where $V$ is the group acting and the twist depends on the endomorphism chosen. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.

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arXiv2021

Classification of Thompson related groups arising from Jones technology II

In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups have remarkable diagrammatic properties that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semidirect product. In our first article dedicated to this construction, we classify up to isomorphism all these semidirect products when one of the endomorphisms is trivial and described their automorphism group. In this article we focus on the case where both endomorphisms are automorphisms. The situation is rather different and we obtain semidirect products where the largest Richard Thompson's group $V$ is acting on some discrete analogues of loop groups. Note that these semidirect products appear naturally in recent constructions of quantum field theories. Moreover, they have been previously studied by Tanushevski and can be constructed via the framework of cloning systems of Witzel-Zaremsky. In particular, they provide examples of groups with various finiteness properties and possible counterexamples of a conjecture of Lehnert on co-context-free groups. We provide a partial classification of these semidirect products and describe explicitly their automorphism group. Moreover, we prove that groups studied in the first and second articles are never isomorphic to each other nor admit nice embeddings between them. We end the article with an appendix comparing Jones technology with Witzel-Zaremsky's cloning systems and with Tanushevski's construction. As in the first article, all the results presented were possible to achieve via a surprising rigidity phenomena on isomorphisms between these groups.

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arXiv2020

On Jones' connections between subfactors, conformal field theory, Thompson's groups and knots

Surprisingly Richard Thompson's groups have recently appeared in Jones' subfactor theory. Vaughan Jones is famous for linking theories that are a priori completely disconnected; for instance, his celebrated polynomial for links emanating from subfactor theory. This note is about a new beautiful story in mathematics which results from a fortunate accident in the land of quantum field theory.

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