Quantum cylindrical integrability in magnetic fields

We present the classification of quadratically integrable systems of the cylindrical type with magnetic fields in quantum mechanics. Following the direct method used in classical mechanics by [F Fournier et al 2020 J. Phys. A: Math. Theor. 53 085203] to facilitate the comparison, the cases which may a priori differ yield 2 systems without any correction and 2 with it. In all of them, the magnetic field $B$ coincides with the classical one, only the scalar potential $W$ may contain a $\hbar^2$-dependent correction. Two of the systems have both cylindrical integrals quadratic in momenta and are therefore not separable. These results form a basis for a prospective study of superintegrability.Comment: 8 pages. Submission to SciPost, proceedings of 34th International Colloquium on Group Theoretical Methods in Physic

On the structure of maximal solvable extensions and of Levi extensions of nilpotent algebras

We establish an improved upper estimate on dimension of any solvable algebra s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we consider Levi decomposable algebras with a given nilradical n and investigate restrictions on possible Levi factors originating from the structure of characteristic ideals of n. We present a new perspective on Turkowski's classification of Levi decomposable algebras up to dimension 9.Comment: 21 pages; major revision - one section added, another erased; author's version of the published pape

Description of surfaces associated with CPN−1CP^{N-1} sigma models on Minkowski space

The objective of this paper is to construct and investigate smooth orientable surfaces in $R^{N^2-1}$ by analytical methods. The structural equations of surfaces in connection with $CP^{N-1}$ sigma models on Minkowski space are studied in detail. This is carried out using moving frames adapted to surfaces immersed in the $su(N)$ algebra. The first and second fundamental forms of this surface as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The Gaussian curvature, the mean curvature vector and the Willmore functional expressed in terms of a solution of $CP^{N-1}$ sigma model are obtained. An example of a surface associated with the $CP^1$ model is included as an illustration of the theoretical results.Comment: 19 pages, 1 figure; shorter version, some typos and minor mistakes correcte

All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional maximal Abelian ideal). We find that for given n such a solvable algebra is unique up to isomorphisms. Using the method of moving frames we construct a basis for the Casimir invariants of the nilradical n_(n,2). We also construct a basis for the generalized Casimir invariants of its solvable extension s_(n+1) consisting entirely of rational functions of the chosen invariants of the nilradical.Comment: 19 pages; added references, changes mainly in introduction and conclusions, typos corrected; submitted to J. Phys. A, version to be publishe

Invariant solutions of the supersymmetric sine-Gordon equation

A comprehensive symmetry analysis of the N=1 supersymmetric sine-Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables. Next, we consider the super-sine-Gordon equation expressed in terms of a bosonic superfield involving anticommuting independent variables. In each case, a Lie (super)algebra of symmetries is determined and a classification of all subgroups having generic orbits of codimension 1 in the space of independent variables is performed. The method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model. Several types of algebraic, hyperbolic and doubly periodic solutions are obtained in explicit form.Comment: 27 pages, major revision, the published versio

A class of solvable Lie algebras and their Casimir Invariants

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.Comment: 16 page