Related papers: Quasi-derivations and QD-algebroids

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the…

We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a…

The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector…

Natural analogs of Lie brackets on affine bundles are studied, based on natural examples from differential geometry and analytical mechanics. In particular, a close relation to Lie algebroids and, by a sort of duality, to affine analogs of…

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this…

Take a holomorphic Lie algebroid $(V,\phi)$ over a rationally connected smooth complex projective variety $X$. We show that, under certain conditions, a vector bundle $E$ over $X$ admits a $(V,\phi)$-connection if and only if $E$ is…

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of…

Let $X$ be an irreducible smooth complex projective variety. Let $G$ be a linear algebraic group over $\mathbb{C}$. We define the notion of Lie algebroid valued connection on holomorphic principal $G$--bundles on $X$, and study their basic…

Lie algebras of smooth sections are Lie algebras obtained from bundles of Lie algebras, where the latter are vector bundles of which the fibers are Lie algebras. We also consider the $\operatorname{C}^k$-sections for $k \in \mathbb{N}$.…

We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their…

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic,…

A VB-algebroid is a vector bundle object in the category of Lie algebroids. We attach to every VB-algebroid a differential graded Lie algebra and we show that it controls deformations of the VB-algebroid structure. Several examples and…

If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of…

We extend to Poisson manifolds the theory of hamiltonian Lie algebroids originally developed by two of the authors for presymplectic manifolds. As in the presymplectic case, our definition, involving a vector bundle connection on the Lie…

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that…

Let $\mathcal{L}=(L,[\cdot\,,\cdot],\delta)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove…

Lie quasi-states on a real Lie algebra are functionals which are linear on any abelian subalgebra. We show that on the symplectic Lie algebra of rank at least 3 there is only one continuous non-linear Lie quasi-state (up to a scalar factor,…

In this paper, we consider Lie algebroids over commutative ringed spaces. Lie algebroids over ringed spaces unify the existing notion of Lie algebroids over smooth manifolds, complex manifolds, analytic spaces, algebraic varieties, and…

A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric…

We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in…