We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.

2021-08-22math.DG

Higher Haantjes Brackets and Integrability

We propose a new, infinite class of brackets generalizing the Frölicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the ope…

2021-09-02math.DG

Crossed modules and the integrability of Lie brackets

We show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions of integrable transitive Lie algebroids by Lie algebra bundles. Such a lifting obstruction is directly related with the classification of extensions of transitive Lie groupoids. We also give a classification of such extensions which differentiates to the classification of transitive Lie algebroids discussed in \cite{KCHM:new}.

2006-09-25math.DG

Compatibility, multi-brackets and integrability of systems of PDEs

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer delta-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are applied to establish new integration methods and solve several differential-geometric problems.

2008-02-20math.APmath.DG

Integrability of Poisson brackets

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas. As an application of these results, we show that a Poisson manifold admits a complete symplectic realization if, and only if, it is integrable. We discuss also the integration of submanifolds and Morita equivalence of Poi…

2003-12-23math.DGmath.GTmath.SG

Integrability of Lie brackets

In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.

2004-02-17math-phmath.DGmath.MP

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Extensions of Lie Brackets

We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequence. We also give a description of the groupoid integrating an extension in case a complete connection can be fixed. The problem of integrability is also studied.

2010-01-18math.DG

Lax pairs for the equations describing compatible nonlocal Poisson brackets of hydrodynamic type, and integrable reductions of the Lame equations

The nonlinear equations for the general nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived and the integrability of these equations by the method of inverse scattering problem is proved. For these equations, the Lax pairs with a spectral parameter are presented. Moreover, we demonstrate the integrability of the equations for some especially important partial classes of compatible nonlocal Poisson brackets of hydrodynamic type, in particular, for the most important case when one of the compatible Poisson brackets is local and also for the case when one of…

2002-02-05math-phmath.DGmath.MP

通过 bi-Poisson 约化研究 bi-Hamiltonian 系统的局部双可积性

Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction

本文证明 bi-Hamiltonian 系统在实光滑与复解析情形下的局部双可积性，并利用 Poisson 括号 Casimir、铅笔特征值和 Hamiltonians 构造完整 bi-involution 函数集。

2024-10-27math.DGmath.SG

双哈密顿系统的局部双可积性，第二部分：实光滑情形

Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case

该研究证明实光滑流形上的双哈密顿系统具有局部双可积性，通过扩展标准积分集构造双对合完备函数集，并验证扩展族微分可实现包含标准积分微分的任意双拉格朗日子空间，推进了双哈密顿系统的可积性理论研究。

2024-10-29math.DGmath.SG

Prequantization and Lie brackets

We start by describing the relationship between the classical prequantization condition and the integrability of a certain Lie algebroid associated to the problem and use this to give a global construction of the prequantizing bundle in terms of path spaces (Introduction), then we rephrase the problem in terms of groupoids (second section), and then we study the more general problem of prequantizing groupoids endowed with a multiplicative 2-forms (the rest of the paper).

2004-03-16math-phmath.DGmath.MP

偏微分方程的不变性缩减。III：泊松括号

Invariant Reduction for Partial Differential Equations. III: Poisson Brackets

Department of Mathematics and Statistics, University of Saskatchewan（萨斯喀彻温大学数学与统计系）

本文研究了偏微分方程有限维系统在不变性缩减下的局部哈密顿算子继承机制，展示了其在点、接触及更高对称性下的统一应用，并通过实例证明继承的泊松括号与守恒量之间的关系。

2026-04-29math.DGnlin.SI

Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator

The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the corresponding Gauss-Codazzi equations and draw parallels between this geometrical problem and the theory of compatible Poisson brackets of hydrodynamic type. It turns out that coordinate hypersurfaces of the n-orthogonal systems arising in the theory of compatible Poisson brackets…

2001-07-17math.DG

李变换群——微分方程对称群分析导论

Lie Transformation Groups -- An Introduction to Symmetry Group Analysis of Differential Equations

本文是一份微分方程对称群分析课程讲义，以Olver的著作为基础，从Stefan-Sussman理论出发系统介绍局部变换群理论，内容自足，仅需微分几何和李群基础知识。

2024-12-31math.DG

Flux homomorphism on symplectic groupoids

An anologue of the Calabi invariant for Poisson manifolds is considered. For any Poisson manifold $P$, the Poisson bracket on $C^{\infty}(P)$ extends to a Lie bracket on the space $Ω^{1}(P)$ of all differential one-forms, under which the space $Z^{1}(P)$ of closed one-forms and the space $B^{1}(P)$ of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: $$0\lon \reals \lon C^{\infty}(P)\stackrel{d}{\lon} Z^{1}(P)\stackrel{f}{\lon} H^{1}(P, \reals)\lon 0, $$ where $H^{1}(P,\reals)$ is considered as a trivial Lie algebra, and $f$ is the map sending each clos…

1996-05-05dg-gamath.DG