In this paper the authors introduce an analogue of nilpotent cone, N {\mathcal N}</mml:annotatio...

Definition 1.1. [Ol] An m-dimensional manifold M is a topological space covered by a collection of open subsets Wα ⊂M (coordinate charts) and maps Xα : Wα → Vα ⊂ R one-to-one and onto, where Vα is an open, connected subset of R. (Wα,Xα) define coordinates on M. M is a smooth manifold if the maps Xαβ = Xβ ◦X−1 α , are smooth where they are defined, i.e. on Xα(Wα ∩Wβ) to Xβ(Wα ∩Wβ). Example 1.2. ...

We use the power-counting formalism of effective field theory to study the size of loop corrections in theories of slow-roll inflation, with the aim of more precisely identifying the limits of validity of the usual classical inflationary treatments. We keep our analysis as general as possible in order to systematically identify the most important corrections to the classical inflaton dynamics. ...

We present a general framework for the optimal control of driftless nonlinear systems defined by means of distributions of smooth vector fields that generate nilpotent Lie algebras. A smooth varying inner product on the planes of the distribution, yields the energy functional that allows to approach the optimal control problem as a sub-Riemannian geodesic problem. This class of systems is relev...

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuper-algebra. Then we classify the cuspidal Levi sub...

A formalism describing the dynamics of classical and quantum systems from a group theoretical point of view is presented. We apply it to the simple example of the classical free particle. The Galileo group G is the symmetry group of the free equations of motion. Consideration of the free particle Lagrangian semi-invariance under G leads to a larger symmetry group, which is a central extension o...

We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie alge...

These notes introduce SU (2) as an example of a compact Lie group. The definition of SU (2) is SU (2) = A A a 2 × 2 complex matrix, det A = 1, AA * = A * A = 1l In the name SU (2), the " S " stands for " special " and refers to the condition det A = 1 and the " U " stands for " unitary " and refers to the conditions AA * = A * A = 1l. The adjoint matrix A * is the complex conjugate of the trans...