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Variational calculus on Lie algebroids

Eduardo Martínez (2008)

ESAIM: Control, Optimisation and Calculus of Variations

It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.

Varieties of modules over tubular algebras

Christof Geiss, Jan Schröer (2003)

Colloquium Mathematicae

We classify the irreducible components of varieties of modules over tubular algebras. Our results are stated in terms of root combinatorics. They can be applied to understand the varieties of modules over the preprojective algebras of Dynkin type 𝔸₅ and 𝔻₄.

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

Igor Burban, Thilo Henrich (2015)

Journal of the European Mathematical Society

In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical r -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic r -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication and prove...

Vector form brackets in Lie algebroids

Albert Nijenhuis (1996)

Archivum Mathematicum

A brief exposition of Lie algebroids, followed by a discussion of vector forms and their brackets in this context - and a formula for these brackets in “deformed” Lie algebroids.

Veränderungen über einen Satz von Timmesfeld – I. Quadratic Actions

Adrien Deloro (2013)

Confluentes Mathematici

We classify quadratic SL 2 ( 𝕂 ) - and 𝔰𝔩 2 ( 𝕂 ) -modules by crude computation, generalising in the first case a Theorem proved independently by F.G. Timmesfeld and S. Smith. The paper is the first of a series dealing with linearisation results for abstract modules of algebraic groups and associated Lie rings.

Vertex algebras and the formal loop space

Mikhail Kapranov, Eric Vasserot (2004)

Publications Mathématiques de l'IHÉS

We construct a certain algebro-geometric version ℒ ( X ) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme ℒ 0 ( X ) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on ℒ ( X ) supported in ℒ 0 ( X ) . We also show that ℒ ( X ) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic...

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