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Register a new userThe groupoids of adaptable separated graphs and their type semigroup
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Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorff etale groupoid. We show that this groupoid is always amenable, and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the Realization Problem for von Neumann regular rings, in the finitely generated case.
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Given a higher-rank graph $Lambda$, we investigate the relationship between the cohomology of $Lambda$ and the cohomology of the associated groupoid $G_Lambda$. We define an exact functor between the abelian category of right modules over a higher-rank graph $Lambda$ and the category of $G_Lambda$-sheaves, where $G_Lambda$ is the path groupoid of $Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $Lambda$ with coefficients in a right $Lambda$-module, and the continuous cocycle cohomology of $G_Lambda$ with values in the corresponding $G_Lambda$-sheaf, into the sheaf cohomology of $G_Lambda$.
We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.
Inspired by results for graph $C^*$-algebras, we investigate connections between the ideal structure of an inverse semigroup $S$ and that of its tight $C^*$-algebra by relating ideals in $S$ to certain open invariant sets in the associated tight groupoid. We also develop analogues of Conditions (L) and (K) for inverse semigroups, which are related to certain congruences on $S$. We finish with applications to the inverse semigroups of self-similar graph actions and some relevant comments on the authors earlier uniqueness theorems for inverse semigroups.
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{ipi/ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C^*-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras {mathfrak g} eq E_8 and certain roots of unity. Our main tools here are Drinfelds coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfelds associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case {mathfrak g}={mathfrak sl}_N we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.
The structure of the automorphism group of the sandwich semigroup IS_n is described in terms of standard group constructions.
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