396 research outputs found
Fixed points of endomorphisms and relations between metrics in preGarside monoids
Rodaro and Silva proved that the fixed points submonoid and the periodic
points submonoid of a trace monoid endomorphism are always finitely generated.
We show that for finitely generated left preGarside monoids, that includs
finitely generated preGarside monoids, Garside monoids and Artin monoids, the
fixed and periodic points submonoids of any endomorphism are also finitely
generated left preGarside monoids under some condition, and in the case of
Artin monoids, these submonoids are always Artin monoids too. We also prove
algebraically some inequalities, equivalences and non-equivalences between
three metrics in finitely generated preGarside monoids, and especially in trace
monoids and Garside monoids
On periodic points of free inverse monoid endomorphisms
It is proved that the periodic point submonoid of a free inverse monoid
endomorphism is always finitely generated. Using Chomsky's hierarchy of
languages, we prove that the fixed point submonoid of an endomorphism of a free
inverse monoid can be represented by a context-sensitive language but, in
general, it cannot be represented by a context-free language.Comment: 18 page
AQFT from n-functorial QFT
There are essentially two different approaches to the axiomatization of
quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and
functorial QFT, going back to Atiyah and Segal. More recently, based on ideas
by Baez and Dolan, the latter is being refined to "extended" functorial QFT by
Freed, Hopkins, Lurie and others. The first approach uses local nets of
operator algebras which assign to each patch an algebra "of observables", the
latter uses n-functors which assign to each patch a "propagator of states".
In this note we present an observation about how these two axiom systems are
naturally related: we demonstrate under mild assumptions that every
2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport")
naturally yields a local net. This is obtained by postcomposing the propagation
2-functor with an operation that mimics the passage from the Schroedinger
picture to the Heisenberg picture in quantum mechanics.
The argument has a straightforward generalization to general
pseudo-Riemannian structure and higher dimensions.Comment: 39 pages; further examples added: Hopf spin chains and asymptotic
inclusion of subfactors; references adde
Equilibrium states on right LCM semigroup C*-algebras
We determine the structure of equilibrium states for a natural dynamics on
the boundary quotient diagram of -algebras for a large class of right LCM
semigroups. The approach is based on abstract properties of the semigroup and
covers the previous case studies on ,
dilation matrices, self-similar actions, and Baumslag-Solitar monoids. At the
same time, it provides new results for large classes of right LCM semigroups,
including those associated to algebraic dynamical systems.Comment: 43 pages, to appear in Int. Math. Res. No
Fixed points of endomorphisms of graph groups
It is shown, for a given graph group , that the fixed point subgroup
Fix is finitely generated for every endomorphism of if
and only if is a free product of free abelian groups. The same conditions
hold for the subgroup of periodic points. Similar results are obtained for
automorphisms, if the dependence graph of is a transitive forest.Comment: 9 page
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