Related papers: Delta-groupoids and ideal triangulations

The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension is related to the rank of a matrix originating from a regular left/right…

We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions…

Symmetry is at the heart of much of mathematics, physics, and art. Traditional geometric symmetry groups are defined in terms of isometries of the ambient space of a shape or pattern. If we slightly generalize this notion to allow the…

This article continues the study of diagrams in the bicategory of \'etale groupoid correspondences. We prove that any such diagram has a groupoid model and that the groupoid model is a locally compact \'etale groupoid if the diagram is…

Loop braid groups characterize the exchange of extended objects, namely loops, in three dimensional space generalizing the notion of braid groups that describe the exchange of point particles in two dimensional space. Their interest in…

Let $\Lambda = \mathbb{Z}^n$ with lexicographic ordering. $\Lambda$ is a totally ordered group. Let $X = \Lambda^+ * \Lambda^+$. Then $X$ is a $\Lambda$-tree. Analogous to the construction of graph $C^*$-algebras, we form a groupoid whose…

In this paper we have discusses {\Gamma}-left, {\Gamma}-right, {\Gamma}-bi-, {\Gamma}-quasi-, {\Gamma}-interior and {\Gamma}-ideals in {\Gamma}-AG^{**}-groupoids and regular {\Gamma}-AG^{**}-groupoids. Moreover we have proved that the set…

Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with…

Let $\Delta$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $\Delta$ is that $g_2$ of $\Delta$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $\Delta$ has…

We define groupoidal and $(n+k)$-truncated $n$-quasi-categories, which are the translation to the world of $n$-quasi-categories of groupoidal and truncated $(\infty, n)$-$\Theta$-spaces defined by Rezk. We show that these objects are the…

We introduce a new group action on set systems, constructed as a semidirect product of a permutation group and a group generated by twist and loop complementation operations on a single element. This action extends the ribbon group…

Let $\Delta$ be a building of type $\widetilde A_2$ and order $q$, with maximal boundary $\Omega$. Let $\Gamma$ be a group of type preserving automorphisms of $\Delta$ which acts regularly on the chambers of $\Delta$. Then the crossed…

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an…

In this note we show that the known relation between double groupoids and matched pairs of groups may be extended, or seems to extend, to the triple case. The references give some other occurrences of double groupoids.

We study 3d theories determined by three-manifolds. Previously, we found that some basic 3d dualities relate to the surgeries of three-manifolds and defined gauge circles and matter circles. In this note, we discuss some operations…

The antiprism triangulation provides a natural way to subdivide a simplicial complex $\Delta$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as…

It is well-known that complex hyperbolic triangle groups $\Delta(3,3,4)$ generated by three complex reflections $I_1,I_2,I_3$ in $\mbox{PU(2,1)}$ has 1-dimensional moduli space. Deforming the representations from the classical…

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector…

A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $\Delta$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a…

Factorization algebras are local-to-global objects living on manifolds, and they arise naturally in mathematics and physics. Their local structure encompasses examples like associative algebras and vertex algebras; in these examples, their…