Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics. MSC (2000): 37J05, 53A40, 53D99, 55R10, 70H99.

This paper presents an effective method for computing Standard bases for the local ring of an algebroid branch and for its module of Kähler differentials. This allows us to determine the semigroup of values of the ring and the values of its Kähler differentials, which in the case of complex analytic branches are, respectively, important topological and analytical invariants. c © 2006 Elsevier L...

Given a pair of Lie algebroid structures on a vector bundle A (over M) and its dual A∗, and provided the A∗-module L = (∧A ⊗ ∧T ∗M) 1 2 exists, there exists a canonically defined differential operator D̆ on Γ(∧A ⊗ L ). We prove that the pair (A,A∗) constitutes a Lie bialgebroid if, and only if, D̆ is a Dirac generating operator as defined by Alekseev & Xu [1].

We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel’d double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (w...

It is shown that the Lagrange’s equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.

In the context of the variational bi-complex, we re-explain that irreducible gauge systems define a particular example of a Lie algebroid. This is used to review some recent and not so recent results on gauge, global and asymptotic symmetries. Research Director of the Fund for Scientific Research-FNRS. E-mail: [email protected]

In this paper we discuss the relation between the unimodularity of a Lie algebroid τA : A → Q and the existence of invariant volume forms for the hamiltonian dynamics on the dual bundle A. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.

We study the conditions that an operator, defined on the spaces of degree 0 and 1 of a Gerstenhaber or BV algebra, has to satisfy so that we can find an extension that generates the whole algebra. When applied to the Gerstenhaber or BV algebra associated to a Lie algebroid, it gives a global proof of the correspondence between BV-generators and flat connections.

This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on Lie group so that its differentiation gives corresponding algebra. Direct products groups, including decompositions Iwasawa and Langlands, carry natural operators. Formal inverse is precisely crossed homomorphism group, whose tangent map differential $1$ A factorization theorem groups proved, deriving directly l...