We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of the Batalin–Vilkovisky algebra. While such an operator of order 2 defines a Lie algebra structure on A, an operator of an order higher than 2 (Koszul–Akman definition) leads to the structure of a strongly homotopy Lie algebra (L∞–algebra) on A. This allows us to give a defin...

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian Ld : Q × Q → R, where Q is the configuration manifold, and a (generally nonintegrable) distribution D ⊂ TQ. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the...

We show the space of expanding Blaschke products on S is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure-preserving topological covering maps of S.

Let M0 = G0/H be a (n+1)-dimensional Cahen-Wallach Lorentzian symmetric space associated with a symmetric decomposition g0 = h + m and let S(M0) be the spin bundle defined by the spin representation ρ : H → GLR(S) of the stabilizer H . This article studies the superizations of M0, i.e. its extensions to a homogeneous supermanifold M = G/H whose sheaf of superfunctions is isomorphic to Λ(S∗(M0))...

The phase space of an integrable volume-preserving map with one action is foliated by a oneparameter family of invariant tori. Perturbations lead to chaotic dynamics with interesting transport properties. We show that near a rank-one resonant torus the mapping can be reduced to a volume-preserving standard map. This map is a twist map only when the frequency map crosses the resonance curve tran...

0.1. Let g be a complex finite-dimensional contragredient Lie superalgebra. These algebras were classified by V. Kac in [K1] and the list (excluding Lie algebras) consists of four series: A(m|n), B(m|n), C(m), D(m|n) and the exceptional algebrasD(2, 1, a), F (4), G(3). The finite-dimensional contragredient Lie superalgebras with zero Killing form (or, equivalently, with dual Coxeter number equa...

Volume-preserving algorithms (VPAs) for the charged particles dynamics is preferred because of their long-term accuracy and conservativeness for phase space volume. Lie algebra and the Baker-Campbell-Hausdorff (BCH) formula can be used as a fundamental theoretical tool to construct VPAs. Using the Lie algebra structure of vector fields, we split the volume-preserving vector field for charged pa...