Cylindrical first order superintegrability with complex magnetic fields

This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space $\mathbb{E}_3$ in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [X. Peng et al., Phys. Rev. Lett. 114 (2015)], they were so far not addressed in the growing literature on superintegrability. Here we venture into this field by searching for additional first order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.Comment: The following article has been submitted to the Journal of Mathematical Physic

Classification of Poisson-Lie T-dual models with two-dimensional targets

Four-dimensional Manin triples and Drinfeld doubles are classified and corresponding two-dimensional Poisson-Lie T-dual sigma models on them are constructed. The simplest example of a Drinfeld double allowing decomposition into two nontrivially different Manin triples is presented.Comment: 6 pages, LaTeX; correction: two Manin triples originally considered separately are shown to lead to the same Drinfeld doubl

Poisson-Lie T-plurality as canonical transformation

We generalize the prescription realizing classical Poisson-Lie T-duality as canonical transformation to Poisson-Lie T-plurality. The key ingredient is the transformation of left-invariant fields under Poisson-Lie T-plurality. Explicit formulae realizing canonical transformation are presented and the preservation of canonical Poisson brackets and Hamiltonian density is shown.Comment: 11 pages. Details of calculations added, version accepted for publicatio

Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

In this paper we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of Theorem 1 in [A Marchesiello and L \v{S}nobl 2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203]. We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of [A Marchesiello and L \v{S}nobl 2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203].Comment: 18 page

Classification of 6-dimensional real Drinfeld doubles

Starting from the classification of real Manin triples done in a previous paper we look for those that are isomorphic as 6-dimensional 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.Comment: 27 pages, corrected minor mistakes and typos, added reference

On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

We extend the investigation of three-dimensional (3D) Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry. Three different integrable cases are considered as starting points for superintegrability: the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. These integrable cases were introduced in [1]. In this paper, we focus on linear and some special cases of quadratic superintegrability. We investigated all possible additional linear integrals of motion for the oblate and the prolate spheroidal cases, separately. For both cases, no previously unknown superintegrable system arises. In addition, we looked for special quadratic integrals of motion for all three cases. We found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases. We also found one new minimally superintegrable system that lies at the intersection of the cylindrical, spherical, oblate spheroidal and prolate spheroidal cases. By imposing additional conditions on these systems, we found for each quadratically minimally superintegrable system a new infinite family of higher-order maximally superintegrable systems linked with the caged and harmonic oscillator without magnetic fields, respectively, through a time-dependent canonical transformation.Comment: 12 figures, preprin