The purpose of this article is to present a "Groupoid proof" the Lefschetz fixed point formula for elliptic complexes. We shall define "relative version" tangent groupoid, describe corresponding pseudodifferential calculi and explain relation with formula.

in this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid g(n) . also we show that g(n) contains commuting regular subsemigroup and give a necessary and sucient condition for the groupoid g(n) to be commuting regular.

Motivated by Harper Hamiltonians on skeletal graphs and their C∗–geometry, we study a certain class of graph Hamiltonians. These Hamiltonians can be thought of as a finite groupoid representation in separable Hilbert spaces. Here the groupoid is the path groupoid of a finite graph. Given such a setup, we consider the possible matrix versions of the Hamiltonian, which are indexed by the choice o...

In the framework of C-algebraic deformation quantization we propose a notion of deformation groupoid which could apply to known examples e.g. Connes’ tangent groupoid of a manifold, its generalisation by Landsman and Ramazan, Rieffel’s noncommutative torus, and even Landi’s noncommutative 4-sphere. We construct such groupoid for a wide class of T-spaces, that generalizes the one given for C by ...

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by reflections and Coxeter relations. This answers a question of Serganova. Our w...

We show that the graded set of filter homotopy classes rel vertices of maps from the n-globe to a filtered space may be given the structure of (strict) globular ω-groupoid. The proofs use an analogous fundamental cubical ω-groupoid due to the author and Philip Higgins in 1981. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation all...

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in th...

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C-algebra may be regarded as a result of a quantization procedure. The C-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra G) is the C-algebra of a continuous field of C-algebras over R with fibers ...

Suppose G is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let G/G denote the orbit space of G and C∗(G) denote the groupoid C∗-algebra. Suppose that the isotropy groups of G are amenable. We show that C∗(G) is CCR if and only if G/G is a T1 topological space and all of the isotropy groups are CCR. We also show that C∗(G) is GCR if and only if G/G is a T...