Starting from the classification of real Manin triples we look for those that are isomorphic as 6-dimensional Drinfeld doubles i.e. Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the non-isomorphic structures and give explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of 6-dimensional real Drinfeld doubles. It consists of 22 classes of non-isomorphic Drinfeld doubles.
A nilpotent Lie algebra n n,1 with an (n − 1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n n,1 as their nilradical are obtained. Their dimension is at most n + 2. The generalized Casimir invariants of n n,1 and of its solvable extensions are calculated. For n = 4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.
We consider three dimensional superintegrable systems in a magnetic field. We study the class of such systems which separate in Cartesian coordinates in the limit when the magnetic field vanishes, i.e. possess two second order integrals of motion of the ‘Cartesian type’. For such systems we look for additional integrals up to second order in momenta which make these systems minimally or maximally superintegrable and construct their polynomial algebras of integrals and their trajectories. We observe that the structure of the leading order terms of the Cartesian type integrals should be considered in a more general form than for the case without magnetic field.
We consider a charged particle moving in a static electromagnetic field described by the vector potential A( x) and the electrostatic potential V ( x). We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely (P 1 , P 2 ), (L 3 , P 3 ) and (L 1 , L 2 , L 3 ). For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.