Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a normal form theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous normal forms exist around such transversa...

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 main aim of this paper is to give a Hopf algebroid approach the Picard-Vessiot theory linear differential matrix equations with coefficients in polynomial complex algebra. To end, we introduce general construction what call here \emph{the finite dual} co-commutative (right) and then apply first Weyl algebra viewed as universal enveloping Lie all vector fields on affine line. In way, for fix...

We introduce a new method to perform reduction of contact manifolds that extends Willett’s and Albert’s results. To carry out our reduction procedure all we need is a complete Jacobi map J : M → Γ0 from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of Γ0 on M , and we show that the quotients of fibers J(x) by suitable Lie subgroups Γx are e...

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...

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...

For a finitely generated discrete group Γ, the Γ-sectors of an orb-ifold Q are a disjoint union of orbifolds corresponding to homomorphisms from Γ into a groupoid presenting Q. Here, we show that the inertia orbifold and k-multi-sectors are special cases of the Γ-sectors, and that the Γ-sectors are orbifold covers of Leida's fixed-point sectors. In the case of a global quotient, we show that th...