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...

Let Γ be a finite group acting linearly on a vector space V . We compute the Lie algebra cohomology of the Lie algebra of Γ-invariant formal vector fields on V . We use this computation to define characteristic classes for foliations on orbifolds.

Let (g, k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a ⊂ p. The restriction map S(p) → S(a) where W = W (g0 : a) is the Weyl group, is injective. We determine its image explicitly. In particular, our theorem applies to the case of a symmetric superpair of group type, i.e. (k ⊕ k, k) with the flip involution where k is a classical Lie supe...

Bloch and Okounkov’s correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of ĝl ∞ modules of level one. Recent works have calculated these character functions for higher levels for ĝl ∞ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type D of h...

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.

2 Kac coordinates 5 2.1 Based automorphisms and affine root systems . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Torsion points, Kac coordinates and the normalization algorithm . . . . . . . . . . . . 9 2.3 μm-actions on Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Principal μm-actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We define reduced zeta functions of Lie algebras, which can be derived from motivic zeta functions using the Euler characteristic. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to analyse. We prove that reduced zeta functions are multiplicative under certain conditions and investigate which reduced zeta functions have functional equations.

Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray are studied. The basic observation is that the Jacobi endomorphism and the Berwald curvature are invariant under these symmetries; this implies the invariance of the further curvature data. Our main technical tool is an appropriate Lie derivative operator along the tangent bundle projection. M.S.C....

55 words MAIN TEXT: 924 words REFERENCES: 103 words ENTIRE TEXT: 1100 words Reasoning as a lie detection device Jean-Louis Dessalles Telecom ParisTech 46 rue Barrault – F-75013 Paris – France Tel. +33145817529 [email protected] http://www.dessalles.fr

The integrable hierarchy of commuting vector fields for the localized induction equation of 3D hydrodynamics, and its associated recursion operator, are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept-out surface.