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We reconstruct all (2+1)D quantum double models of finite groups from their boundary symmetries through the repeated application of a gauging procedure, extending the existing construction for abelian groups. We employ the recently proposed…
We introduce the concept of braided alternative bialgebra. The theory of cocycle bicrossproducts for alternative bialgebras is developed. As an application, the extending problem for alternative bialgebra is solved by using some non-abelian…
Chow varieties are a parameter space for cycles of a given variety of a given codimension and degree. We construct their analog for differential algebraic varieties with differential algebraic subvarieties, answering a question of Gao, Li…
We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a…
Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel…
This article is devoted to the geometric construction which states a natural correspondence between topological coverings of a foliated manifolds and noncommutative coverings of the operator algebras. However this correspondence is not one…
We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian…
An informal introduction to some new geometric partial differential equations motivated by string theories is provided. Some of these equations are also interesting from the point of view of non-K\"ahler geometry and the theory of…
We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and…
We extend the equations of motion that describe non-relativistic elastic collision of two particles in one dimension to an arbitrary associative algebra. Relativistic elastic collision equations turn out to be a particular case of these…
We consider non-relativistic curved geometries and argue that the background structure should be generalized from that considered in previous works. In this approach the derivative operator is defined by a Galilean spin connection valued in…
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…
Necessary and sufficient conditions for some deformation algebras to provide formal Frobenius structures are given. Also, examples of formal Frobenius structures with fundamental tensor that is not of the deformation type and examples of…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
We derive anomaly constraints for Abelian and non-Abelian discrete symmetries using the path integral approach. We survey anomalies of discrete symmetries in heterotic orbifolds and find a new relation between such anomalies and the…
In this paper we study principally polarized complex abelian varieties that admit an automorphism of order 3. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do…
In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form…
We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure…
An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist…
The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this…