Related papers: Trees, Amalgams and Calogero-Moser Spaces

Let $A_1=K < X, Y | [Y,X]=1>$ be the (first) Weyl algebra over a field $K$ of characteristic zero. It is known that the set of eigenvalues of the inner derivation $\ad (YX)$ of $A_1$ is $\Z$. Let $ A_1\ra A_1$, $X\mapsto x$, $Y\mapsto y$,…

Let $A_1:=K\langle x, \frac{d}{dx} \rangle$ be the Weyl algebra and $\mI_1:= K\langle x, \frac{d}{dx}, \int \rangle$ be the algebra of polynomial integro-differential operators over a field $K$ of characteristic zero. The Conjecture/Problem…

We introduce $C^*$-algebras associated to directed graphs of groups. In particular, we associate a combinatorial $C^*$-algebra to each row-finite directed graph of groups with no sources, and show that this $C^*$-algebra is Morita…

This paper addresses several structural aspects of the insertion-elimination algebra, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the…

Kanel-Belov and Kontsevich's conjecture in \cite[Conjecture 1]{BeKo} is proved: The automorphism group of the $n$-th Weyl algebra is isomorphic to the Poisson automorphism group of the $n$-th Poisson Weyl algebra.

We obtain a kind of structure theorem for the automorphism group ${\rm Aut}{\cal A}$ of a unital C$^{*}$-algebra ${\cal A}$. According to it, ${\rm Aut}{\cal A}$ can be regarded as a subgroup of the semi-direct product of direct product…

Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local…

Let $K$ be a {\em perfect} field of characteristic $p>0$, $A_1:=K< x, \der | \der x- x\der =1>$ be the first Weyl algebra and $Z:=K[X:=x^p, Y:=\der^p]$ be its centre. It is proved that $(i)$ the restriction map $\res :\Aut_K(A_1)\ra…

In the first part of the article we introduce $C^*$-algebras associated to self-similar groups and study their properties and relations to known algebras. The algebras are constructed as sub-algebras of the Cuntz-Pimsner algebra (and its…

In this paper, we study Bass-Serre theory from the perspectives of $C^*$-algebras and topological dynamics. In particular, we investigate the actions of fundamental groups of graphs of groups on their Bass-Serre trees and the associated…

The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we…

In this paper, we study a family of generalized Weyl algebras $\{\A\}$ and their polynomial extensions. We will show that the algebra $\A$ has a simple localization $\A_{\mathbb{S}}$ when none of $p$ and $q$ is a root of unity. As an…

Let $\mathbb{K}$ be a field of characteristic $p$ and $G$ be a cyclic $p$-group which acts on a finite acyclic quiver $Q$. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew…

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the…

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We…

This is the first of two papers on the superselection sectors of the conformal model in the title, in a time zero formulation. A classification of the sectors of the net of observables as restrictions of solitonic (twisted) and…

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is…

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity…

We generalise the Dixmier-Douady classification of continuous-trace C*-algebras to Fell algebras. To do so, we show that C*-diagonals in Fell algebras are precisely abelian subalgebras with the extension property, and use this to prove that…

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several…