This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator K-theory). These dualities are expressed in terms of categories of modules. In this paper, we develop a general framework needed to describe these dualities. In various geometric contexts, e.g. complex geometry, gener...

We study Lie bialgebroid crossed modules which are pairs of algebroid in duality that canonically give rise to bialgebroids. A one-one correspondence between such and co-quadratic Manin triples [Formula: see text] is established, where a pair transverse Dirac structures text].

For a Lie groupoid G over smooth manifold M we construct the adjoint action of étale # germs local bisections on algebroid g . With this action, form associated convolution C c ∞ ( ) / R -bialgebra , We represent in algebra transversal distributions This construction extends Cartier-Gabriel decomposition Hopf with finite support group.

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type deformation called unfoldings algebroids, which generalizes the theory foliations developed by T. Suwa. show that one to correspondence between transversal flat connections natural algebroid bases exists.

In this paper, we construct a homotopy Poisson algebra of degree 3 associated to split Lie 2-algebroid, by which give new approach characterize 2-bialgebroid. We develop the differential calculus 2-algebroid and establish Manin triple theory for 2-algebroids. More precisely, notion strict Dirac structure define 2-algebroids be CLWX with two transversal structures. show that there is one-to-one ...

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbol k[x_1,x_2,\dots, x_n]/I$ where k$ is field characteristic $0$ $I\subseteq \boldsymbol x_n]$ an ideal. Assume that there Poisson bracket $\{\:,\:\}$ on $S$ such $\{I,S\}\subseteq I$ let us denote induced by as well. It well-known $[\mathrm d x_i,\mathrm x_j]:=\mathrm d\{x_i,x_j\}$ defines Lie $A$-module $\Omega_{A|\boldsymbol...

An important result for regular foliations is their formal semi-local triviality near simply connected leaves. We extend this to singular all 2-connected leaves and a wide class of 1-connected by proving Levi-Malcev theorem the semi-simple part holonomy Lie algebroid.

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎wit...

This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable...