This is a condensed report from the ongoing project aimed on higher principal connections and their relation with differential cohomology theories generalised short exact sequences of $L_\infty $ algebroids. A historical stem for our paper sir M. Atiyah who observed bijective correspondence between data horizontal distribution fibre bundle set sections certain splitting sequence Lie algebroids,...

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundn...

We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stability results for foliations and group actions.

We give a natural obstruction theoretic interpretation to the first Pontryagin class in terms of Courant algebroids. As an application we calculate the class of the stack of algebras of chiral differential operators. In particular, we establish the existence and uniqueness of the chiral de Rham complex. Sverhu molot, snizu serp, Зto – nax, Sovetski gerb. Hoqexь жni, a hoqexь ku ...

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17...

We study free dg-Lie algebroids over arbitrary derived schemes, and compute their universal enveloping jet algebras. also introduce twisted connections, relate them to lifts on square zero extensions. This construction allows us provide new conceptual approaches existing results concerning the deformation theory of subschemes.

We present two classes of examples Hopf algebroids associated with non-commutative principal bundles. The first comes from deforming the bundle while leaving unchanged structure algebra. second is related to a quantum homogeneous space; this needs careful deformation algebra in order preserve compatibilities between operations.