Related papers: On Families in Differential Geometry

This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…

For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the…

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We…

This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy…

We provide a framework which generalizes algebraic models of a homotopy theory of spaces to the genuine equivariant case for a discrete group. We explain how this applies to commutative differential graded algebra (cdga) models and complete…

Let $\Theta$ be a variety of algebras. In every $\Theta$ and every algebra $H$ from $\Theta$ one can consider algebraic geometry in $\Theta$ over $H$. We consider also a special categorical invariant $K_\Theta (H)$ of this geometry. The…

We give a natural definition of a Poisson Differential Algebra. Consistence conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on differential calculus in a simple canonical form…

A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject…

In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$, $\{-1, 1\}$ labels are assigned to the edges, and we consider a family $\mathcal{S}_G$ of (spanning) subgraphs of…

We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…

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…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…

A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…

We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…

Given a family of rational curves depending on a real parameter, defined by its parametric equations, we provide an algorithm to compute a finite partition of the parameter space (${\Bbb R}$, in general) so that the shape of the family…

Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic…

We suggest a geometrical framework to discuss periodic layered structures in the unit disk. The band gaps appear when the point representing the system approaches the unit circle. We show that the trace of the matrix describing the basic…

DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the…

We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformation theory.