Related papers: Universal Associative Geometry
Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental…
We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pask algebras are universally defined and have a graded uniqueness theorem. We also…
Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. In the literature,…
This paper constructs the geometrically natural objects which are associated with any projection tensor field on a manifold with any affine connection. The approaches to projection tensor fields which have been used in general relativity…
We provide a brief overview of tensor models and group field theories, focusing on their main common features. Both frameworks arose in the context of quantum gravity research, and can be understood as higher-dimensional generalizations of…
We prove a generalization of a result of Peres and Schlag on the dimensions of certain exceptional sets of projections and then apply it to a geometric problem.
Many homogeneous, four-dimensional space-time geometries can be considered within real projective geometry, which yields a mathematically well-defined framework for their deformations and limits without the appearance of singularities.…
A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators $x_1, ..., x_n$ and the cyclic relations: $x_{i_k}x_{i_{k-1}} ... x_{i_1} = x_{i_{k-1}} ... x_{i_1} x_{i_k} = ... =…
A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power…
We exhibit cocycles representing certain classes in the rational cohomology of of the general linear group with coefficients in the divided powers of a Frobenius twist of the adjoint representation. These classes' existence was anticipated…
We generalize Exel's notion of partial group action to monoids. For partial monoid actions that can be defined by means of suitably well-behaved systems of generators and relations, we employ classical rewriting theory in order to describe…
We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a…
Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations…
Building from ideas of hypercomplex analysis on the quaternionic unit ball, we introduce Hermitian, Riemannian and K\"ahler-like structures on the latter. These are built from the so-called regular M\"obius transformations. Such geometric…
This paper concerns our earlier conjecture about the equivalence of a derived completion construction applied to the representation spectrum of the absolute Galois group of a geometric field is equivalent to the algebraic K-theory of the…
We introduce a family of maps parametrised by certain ribbon graphs. It is based on a connection in non-commutative geometry and contains the double divergence as a special case. Applying the construction to the case of the group algebra of…
The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory…
We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type $(I_\sigma)$ and the absolute geometry is a generalized hexagon, the moving absolute geometry…