Starting with a Lie algebroid A over a space M we lift its action to the canonical transformations on the affine bundle R over the cotangent bundle T * M. Such lifts are classified by the first cohomology H 1 (A). The resulting object is a Hamiltonian algebroid A H over R with the anchor map from Γ(A H) to Hamiltonians of canonical transformations. Hamiltonian algebroids generalize Lie algebras...

In this paper, using the equivalence between category of crossed modules algebras and algebra-algebroids, we will explore notions ideality factors for these algebraic structures. We give structure a two sided ideal an algebra-algebroid notion quotient algebra-algebroid. By considering algebra-algebroid, show that module corresponding to is Conversely, by taking module, also module.

We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give a new example of a Lie bialgebroid on Poisson manifolds.

For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evens, J.-H. Lu, and A. Weinstein.

We study Galois extensions M (co-)H ⊂ M for H-(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in Hopf-Galois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of Kreimer-Takeuchi, Doi-Takeuchi and Cohen-FischmanMontgomery. We find that the ...

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by m...

It is shown that a quantum groupoid (or a QUE algebroid, i.e., deformation of the universal enveloping algebra of a Lie algebroid) naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e., the quantization problem, is raised, and it is proved for all regular triangular Lie bialgebroids. For a Poisson manifold P , the existence of a star-product is shown to be ...

We compute the 5-local cohomology of a particular Hopf algebroid related to algebraic topology and algebraic geometry. This computation arises in a number of settings, and it is a 5-local analogue of the Weierstrass Hopf algebroid used to compute tmf homology. We finish the paper by computing Adams-Novikov differentials in the cohomology, giving the homotopy, V (0)homology, and V (1)-homology o...

A double field theory algebroid (DFT algebroid) is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding symmetries theory. In particular, DFT structure defined on vector bundle over doubled spacetime equipped with C-bracket this paper, we give definition as curved L∞-algebra and show how implementation strong constraint can formulated an morphism. Our resul...