We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this s...

‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

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.

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]

We introduce an n-ary Lie algebroid canonically associated to a NambuPoisson manifold. We also prove that every Nambu-Poisson bracket defined on functions is induced by some differential operator on the exterior algebra, and characterize such operators. Some physical examples are presented.

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.