The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties ...

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These rep...

Let f : G → A be a surjective homomorphism of transitive groupoid schemes and let L denote the kernel of f . The exact sequence of groupoid schemes 1 → L → G → A → 1 induces a sequence of functors between the categories of finite representations of these groupoid schemes Repf (A) → Repf (G) → Repf (L). We show that the category Repf (L) is a quotient category of Repf (G) by Repf (A) in an appro...

In this paper, we have introduced the concept of fuzzy ordered AG-groupoids which is the generalization of fuzzy ordered semigroups first considered by Kehayopulu and Tsingelis (2002). We have studied some important features of a left regular ordered AG-groupoid in terms of fuzzy left ideals, fuzzy right ideals, fuzzy two-sided ideals, fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interio...

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...