1,354,175 research outputs found
Classification of Thompson related groups arising from Jones' technology II
No full textIn 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 admit remarkable diagrammatic descriptions that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semi-direct product. In our first article dedicated to this construction, we classify up to isomorphism all these semi-direct products when one of the endomorphisms is trivial and describe their automorphism group. In this article, we focus on the case where both endomorphisms are automorphisms. The situation is rather different, and we obtain semi-direct products where the largest Richard Thompson’s group V is acting on some discrete analogues of loop groups. Note that these semi-direct products appear naturally in recent constructions of quantum field theories. Moreover, they were 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 semi-direct products and describe their automorphism group explicitly. Moreover, we prove that groups studied in the first and second articles are never isomorphic to each other nor do they 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, it was possible to achieve all the results presented via a surprising rigidity phenomenon on isomorphisms between these groups
The cup subalgebra of a II1 factor given by a subfactor planar algebra is maximal amenable
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Operator-algebraic construction of gauge theories and Jones' actions of Thompson's groups
No full textUsing ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1 + 1 -dimensional gauge theories on a spacetime cylinder. Given a separable compact group G, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of G over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson’s group T as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of G we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian G, we provide a very explicit description of our algebras. For a single measure on the dual of G, we have a state-preserving action of Thompson’s group and semi-finite von Neumann algebras. For G= S the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita–Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson’s group T, for geometrically motivated choices of families of heat-kernel states
A Tensor Product for Representations of the Cuntz Algebra and of the R. Thompson Groups
No full textThe 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 O and conversely all representations of O are of this form. Via this approach we introduce a tensor product for a large class of representations of F,T,V,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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