Related papers: Variational Lie derivative and cohomology classes

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be…

In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie…

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these…

Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth…

We study higher--order variational derivatives of a generic second--order Lagrangian ${\cal L}={\cal L}(x,\phi,\partial\phi,\partial^2\phi)$ and in this context we discuss the Jacobi equation ensuing from the second variation of the action.…

We prove a comparison theorem between locally analytic group cohomology and Lie algebra cohomology for locally analytic representations of a Lie group over a nonarchimedean field of characteristic 0. The proof is similar to that of…

We use a new variational method --based on the theory of anti-selfdual Lagrangians developed in [2] and [3]-- to establish the existence of solutions of convex Hamiltonian systems that connect two given Lagrangian submanifolds in $\R^{2N}$.…

In the present paper we study local and 2-local derivations of locally finite split simple Lie algebras. Namely, we show that every local and 2-local derivation on such Lie algebra is a derivation.

We show that under certain technical assumptions any weakly nonlocal Hamiltonian structure compatible with a given nondegenerate weakly nonlocal symplectic structure $J$ can be written as the Lie derivative of $J^{-1}$ along a suitably…

A general theorem on conservation laws for arbitrary difference equations is proved. The theorem is based on an introduction of an adjoint system related with a given difference system, and it does not require the existence of a difference…

For a field $\mathbb{F}$, let $L_k(\mathbb{F})$ be the Lie algebra of derivations $f(t)\frac{d}{dt}$ of the polynomial ring $\mathbb{F}[t]$, where $f(t)$ is a polynomial of degree $\geqslant k$. For any $k\geqslant -1$, we present a basis…

The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group,…

This paper investigates cohomology and support varieties for Lie superalgebras and restricted Lie superalgebras over a field of characteristic 2. The existence of an underlying ordinary Lie algebra allows us to obtain results that are still…

This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…

A theory of local convexity for a second order differential equation (SODE) on a Lie algebroid is developed. The particular case when the SODE is homogeneous quadratic is extensively discussed.

We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new…

The paper proposes and motivates a conjecture on the invariance of cohomological support loci under derived equivalence. It contains a proof in the case of surfaces, and explains further developments and consequences.

The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge from the corresponding differential equations find their most general expression in the theorems formulated in…

We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.