Two-Valued Groups, Kummer Varieties, and Integrable Billiards
Victor Matveevich Buchstaber,Vladimir Dragović +1 more
- Vol. 4, Iss: 1, pp 27-57
TLDR
In this paper, the authors studied algebraic two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems and introduced a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid.Abstract:
A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties. One of our approaches is based on the theory of $$\sigma $$
-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid. Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics. The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were ”virtual”, while in the present case the resulting bundles are effectively realizable.read more
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The Sokolov case, integrable Kirchhoff elasticae, and genus 2 theta functions via discriminantly separable polynomials
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Discriminantly separable polynomials and quad-equations
Vladimir Dragović,Katarina Kukić +1 more
TL;DR: In this article, the discriminantly separable polynomials of degree two in each of three variables are defined by a property that all discriminants of two variables are factorized as products of two polynomial components of one variable each.
References
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Multivalued groups, their representations and Hopf algebras
TL;DR: In this paper, the authors introduce the concept of n-valued groups and study their algebraic and topological properties, and show that ann-Hopf algebras do not admit the structure of ann-valued group and that certain commutativenvalued groups do not arise by applying then-coset construction to any commutative group.
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Geometry of integrable billiards and pencils of quadrics
TL;DR: In this article, the authors studied the deep interplay between geometry of quadrics in d-dimensional space and the dynamics of related integrable billiard systems and derived corresponding analytic conditions of Cayley's type giving the full description of periodical billiard trajectories.
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Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: Beyond Poncelet porisms
TL;DR: In this article, a group structure on the set T of lines simultaneously tangent to d-1 quadrics from a given confocal family in the d-dimensional Euclidean space is constructed, and the related results of Reid, Donagi and Knorrer are further developed.
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Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances
TL;DR: In this article, a review of very recent results related to the Poncelet Theorem, on the occasion of its bicentennial, is presented, focusing on the three main issues: A) The case of Pseudo-Euclidean spaces, presenting a recent notion of relativistic quadrics, and applying it to the description of periodic trajectories of billiards within quadrics.