Last multipliers on Lie algebroids

Abstract

In this paper we extend the theory of last multipliers as solutions of the Liouville's transport equation to Lie algebroids with their top exterior power as trivial line bundle (previously developed for vector fields and multivectors). We define the notion of exact section and the Liouville equation on Lie algebroids. The aim of the present work is to develop the theory of this extension from the tangent bundle algebroid to a general Lie algebroid (e.g. the set of sections with a prescribed last multiplier is still a Gerstenhaber subalgebra). We present some characterizations of this extension in terms of Witten and Marsden differentials.

Holomorphic last multipliers on complex manifolds

“…Since If we consider the particular case n = 2 with c 11 1 = c 11 2 = c 22 1 = c 22 2 = 0 and c 12 1 , c 12 2 ∈ C, then the general solution of (3.30) is α = A/(c 12 1 z 1 + c 12 2 z 2 ), which also follows from Example 3.4.…”

“…The first named author of this work initiated their study on manifolds in [8] and [9], where he pointed out their relationship with the Liouville equation of transport. Since then, the last multipliers have been considered in various (non-flat) settings such as: Riemannian and Poisson geometry in [9], Lie-Poisson structures in [10], weighted manifolds in [11], Lie algebroids in [12].…”

Last Multipliers for Multivectors With Applications to Poisson Geometry

“…3) From the two previous remarks it results that a natural extension of our theory seems to work on Lie algebroids using the tools of [12] and [14]. Hence, a sequel paper [4] is forthcoming.…”

The geometry of Lie algebroids and its applications to optimal control

“…An important Lie algebroid is the cotangent bundle of a Poisson manifold [60]. Being related to many areas of geometry, as connections theory [75,40,24,31,49,76,111,95] cohomology [73,75] foliations and pseudogroups, symplectic and Poisson geometry [66,124,120,39,121,61,33,36,97,99,107] the Lie algebroids are today the object of extensive studies. More precisely, Lie algebroids have applications in mechanical systems and optimal control theory [30,77,78,49,29,44,4,92,96,98,16] (distributional systems) and are a natural framework in which one can be developed the theory of differential operators (exterior derivative and Lie derivative) and differential equations.…”