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Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations. [ABSTRACT FROM AUTHOR]

Let G be a unipotent algebraic group defined over a p -adic field of characteristic zero. We denote by G the set of rational points of G. It is a p -adic Lie group with Lie algebra denoted by . Let π be an irreducible unitary representation of G in a Hilbert space ℋ π , f be a linear form on and be a polarization at f. We denote by χ f a character of H = exp () related to f. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors (ℋ π − ∞ ) H , χ f of  π. [ABSTRACT FROM AUTHOR]

Let F be a nonarchimedean local field of characteristic zero and G be a unipotent algebraic group defined over F. The set of rational points of G , denoted by G , is a p -adic Lie group. Let g be the Lie algebra of G. Now let H be a normal closed subgroup of G , χ be a unitary character of H and π be an irreducible unitary representation of G in a Hilbert space H π. The aim of this paper is the determination of the space formed by the χ -semi-invariant vectors of π. [ABSTRACT FROM AUTHOR]

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. [ABSTRACT FROM AUTHOR]

The coadjoint orbits and the Casimir functions are described for free nilpotent Lie groups of step . The symplectic foliation consists of affine subspaces in the Lie coalgebra. Left-invariant time-optimal problems are considered on Carnot groups of step for which the set of admissible velocities is a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. The first integrals of the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle are described. For two-dimensional coadjoint orbits, the constancy and periodicity properties of the solutions of this subsystem, as well as the phase flow, are described. [ABSTRACT FROM AUTHOR]

We consider the semidirect product G = K × V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product h, i. We denote by Ĝ the unitary dual of G (note that we identify each representation π ∈ Ĝ to its classes [π]) and by g‡/G the space of admissible coadjoint orbits, where g is the Lie algebra of G. It was pointed out by Lipsman that the correspondence between g‡/G and Ĝ is bijective. Under some assumption on G, we prove that the Lipsman mapping Θ : g‡/G -→ Ĝ O -→ πO is a homeomorphism. [ABSTRACT FROM AUTHOR]

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits. • We study the Toda flow on Hessenberg elements of real, split simple Lie algebras. • Nontrivial invariants of the maximal compact subgroup K in involution are constructed. • The formula for the number ν of such K-invariants involves only Lie-algebraic data. • Dimension of generic Hessenberg coadjoint orbits = 2 ν. • Toda is integrable on generic Hessenberg orbits for the classical simple Lie algebras. [ABSTRACT FROM AUTHOR]

We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant. [ABSTRACT FROM AUTHOR]

We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown. [ABSTRACT FROM AUTHOR]

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K = U n , SO n. Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie (K) ⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G = K C. We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K , we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K -spaces. An essential tool in our study is the dual Poisson-Lie group K ⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K ⁎ , and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K ⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie (K) ⁎ → K ⁎. [ABSTRACT FROM AUTHOR]

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine closures of orbits of all heights except p - 1 (in which case we only obtain a conditional statement) in the dual space W* under the induced action of G. As a corollary we prove that the algebra of invariants k[W*]G is trivial. [ABSTRACT FROM AUTHOR]

We prove a simple formula for the transverse Poisson structure to a coadjoint orbit (in the dual of a Lie algebra $\{\backslash frak\; g\}$) and use it in examples such as $\{\backslash frak\; s\}\{\backslash frak\; o\}(4)^*$ and $\{\backslash frak\; s\}\{\backslash frak\; p\}(4)^*$. We also give a sufficient condition on the isotropy subalgebra of $\backslash mu\; \backslash in\; \{\backslash frak\; g\}^*$ so that the transverse Poisson structureto the coadjoint orbit of μ is linear. [ABSTRACT FROM AUTHOR]

In 1969, Jean-Marie Souriau introduced a "Lie Groups Thermodynamics" in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau's model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo's group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau's moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. [ABSTRACT FROM AUTHOR]