A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra e(n) = so(n)⋉Rn. We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable cases of these systems: the Lagrange top and the Clebsch case, respectively. The constructio...

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

This paper discusses several algorithmic ways of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two component peakon type dual systems from their two component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group...

Abstract. In his 2005 paper, S.T. Smith proposed an intrinsic Cramér-Rao bound on the variance of estimators of a parameter defined on a Riemannian manifold. In the present technical note, we consider the special case where the parameter lives in a Lie group. In this case, by choosing ,e.g., the right invariant metric, parallel transport becomes very simple, which allows a more straightforward ...

On the Automorphism Tower of Free Nilpotent Groups Martin Dimitrov Kassabov 2003 In this thesis I study the automorphism tower of free nilpo-tent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. Th...

The motion of a rigid body in a quadratic potential is an important example of an integrable Hamiltonian system on a dual to a semidirect product Lie algebra so(n)⋉Symm(n). We give a Lagrangian derivation of the corresponding equations of motion, and introduce a discrete time analog of this system. The construction is based on the discrete time Lagrangian mechanics on Lie groups, accompanied wi...

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...

in this paper we construct the category of coverings of fundamental generalized lie group-groupoid associatedwith a connected generalized lie group. we show that this category is equivalent to the category of coverings of aconnected generalized lie group. in addition, we prove the category of coverings of generalized lie groupgroupoidand the category of actions of this generalized lie group-gro...

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of elliptic type defined on SL (2, ℝ). Our method uses symmetry reduction which translates problem into Riemannian two dimensional quotient space, projections can be easily visualized. As byproduct, we obtain an alternative derivation characterization cut-locus. use classification results three right invariant...