Related papers: Quasi-derivations and QD-algebroids

We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If $\mathbb{L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X) = K_q(\mathbb{L})$ for all $q\le\dim(X)+1$.

We prove that the Lie algebra $\mathcal{S}(\mathcal{P}(E,M))$ of symbols of linear operators acting on smooth sections of a vector bundle $E\to M,$ characterizes it. To obtain this, we assume that $\mathcal{S}(\mathcal{P}(E,M))$ is seen as…

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and…

Two Lie algebroids are presented that are linked to the construction of the linearizing output of an affine in the input nonlinear system. The algorithmic construction of the linearizing output proceeds inductively, and each stage has two…

Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…

From a Lie algebra $\mathfrak{g}$ satisfying $\mathcal{Z}(\mathfrak{g})=0$ and $\Lambda^2(\mathfrak{g})^\mathfrak{g}=0$ (in particular, for $\g$ semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form…

We prove that for a vector bundle $ E \to M$, the Lie algebra $\mathcal{D}_{\mathcal{E}}(E)$ generated by all differential operators on $E$ which are eigenvectors of $L_{\mathcal{E}},$ the Lie derivative in the direction of the Euler vector…

We introduce a representation theory of q-Lie algebras defined earlier in \cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in…

Let $\mathfrak q=Lie Q$ be an algebraic Lie algebra of index 1, i.e., a generic $Q$-orbit on $\mathfrak q^*$ has codimension 1. We show that the following conditions are equivalent: $\mathfrak q$ is contact; a generic $Q$-orbit on…

We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie…

Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related…

We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant algebroids, e.g. from twisted Poisson structures, as well as from twisted actions of a Lie algebra. We moreover define a cohomology for them,…

Let $E_1,\dots ,E_k$ and $E$ be natural vector bundles defined over the category $\Cal Mf_m^+$ of smooth oriented $m$--dimensional manifolds and orientation preserving local diffeomorphisms, with $m\geq 2$. Let $M$ be an object of $\Cal…

We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on…

A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a…

We reprove the results of Jordan [18] and Siebert [31] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not…

We discuss various old and new definitions of the notion of a vector field on a convenient manifold that can be proved to give rise to Lie algebras, and are in finite dimensions equivalent to the standard notion of a vector field.

We consider Lie algebroids over an algebraic space (or topological ringed space) as quasicoherent sheaves of Lie-Rinehart algebras. We express hypercohomology for a locally free Lie algebroid (not necessarily of finite rank) as a derived…

Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are…

We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid ($ \mathcal{L} $) connections on a compact Riemann surface $X$ of genus $g \,\geq\, 3$. A smooth compactification of the moduli space of…