Related papers: Groupoids decomposition, propagation and K-theory
In this master's thesis, we will go into the (projective) blow-up construction for Lie groupoids and Lie algebroids. In the literature, there are different methods to be found on how to do this, especially for Lie groupoids. The main goal…
We construct a sequence of $n-1$ cyclic exact sequences that can be used to compute the $K$-theory of the $C^\star$-algebra crossed product $A \ltimes {\mathbb Z}_n$.
We discuss the interplay between K-theoretical dynamics and the structure theory for certain C*-algebras arising from crossed products. For noncommutative C*-systems we present notions of minimality and topological transitivity in the…
We describe how Lie groupoids are used in singular analysis, index theory and non-commutative geometry and give a brief overview of the theory. We also expose groupoid proofs of the Atiyah-Singer index theorem and discuss the Baum-Connes…
k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop a theory of the fundamental groupoid of a k-graph, and relate it…
In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the…
In this paper we refine a version of bivariant $K$-theory developed by Cuntz to define symmetric spectra representing the $KK$-theory of $C^\ast$-categories and discrete groupoid $C^\ast$-algebras. In both cases, the Kasparov product can be…
For an $r$-discrete Hausdorff groupoid ${\cal G}$ and an inverse semigroup $S$ of slices of ${\cal G}$ there is an isomorphism between ${\cal G}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use it to define…
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher…
In this paper, we develop a groupoid approach to the equivariant coarse Baum--Connes conjecture. For a bounded geometry metric space $X$ equipped with a proper, free, and isometric action of a countable discrete group $\Gamma$, we introduce…
We construct a possibly non-commutative groupoid from the failure of $3$-uniqueness of a strong type. The commutative groupoid constructed by John Goodrick and Alexei Kolesnikov in \cite{GK} lives in the center of the groupoid. A certain…
The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, \textit{Complexity of finite semigroups}, Annals of Mathematics (2) \textbf{88} (1968), 128--160, motivated by the…
We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts…
We prove a decomposition theorem for the equivariant K-theory of actions of affine group schemes G of finite type over a field on regular separated noetherian algebraic spaces, under the hypothesis that the actions have finite geometric…
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a…
This is the first of a series of three papers where we prove the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups and an Ichino--Ikeda type refinement. Our strategy is based on the comparison of relative trace…
We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of…
We present an understandable, efficient, and streamlined proof of the Holonomy Decomposition for finite transformation semigroups and automata. This constructive proof closely follows the existing computational implementation. Its novelty…
Decompositions of the unitary group U(n) are useful tools in quantum information theory as they allow one to decompose unitary evolutions into local evolutions and evolutions causing entanglement. Several recursive decompositions have been…
We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum…