When regular regions are discovered in the parameter space of a system, where the parameter might be an external field applied to the Hydrogen atom, it usually indicates the existence of new (approximate) integrals of motion, and consequently quantum numbers [1]. In studies of many–body systems, the large number of degrees of freedom generally precludes one from using methods introduced in simp...

All possible Drinfel’d double structures for the anti-de Sitter Lie algebra so(2, 2) and de Sitter Lie algebra so(3, 1) in (2+1)-dimensions are explicitly constructed and analysed in terms of a kinematical basis adapted to (2+1)-gravity. Each of these structures provides in a canonical way a pairing among the (anti-)de Sitter generators, as well as a specific classical r-matrix, and the cosmolo...

some lie algebra analogues of schur's theorem and its converses are presented. as a result, it is shown that for a capable lie algebra l we always have dim l=z(l)  2(dim(l2))2. we also give give some examples sup- porting our results.

We prove that the extended Toda hierarchy of [1] admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination...

Study of the normalizer of the MAD-group corresponding to a fine grading offers the most important tool for describing symmetries in the system of non-linear equations connected with contraction of a Lie algebra. One fine grading that is always present in any Lie algebra sl(n, C) is the Pauli grading. The MAD-group corresponding to it is generated by generalized Pauli matrices. For such MAD-gro...

We prove that the extended Toda hierarchy of [1] admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination...

Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is...

A bstract We present a Sugawara-type construction for boundary charges in 4d BF theory and general family of related TQFTs. Starting from the underlying current Lie algebra symmetries, this gives rise to well-defined quadratic forming an vector fields. In case 3d (i.e. gravity), it was shown [1] that leads two-dimensional diffeomorphism which satisfy certain modular duality. Here we show adapti...

We shall outline two ways of introducing the modification of Einstein’s relativistic symmetries of special relativity theory the Poincaré symmetries. The most complete way of introducing the modifications is via the noncocommutative Hopfalgebraic structure describing quantum symmetries. Two types of quantum relativistic symmetries are described, one with constant commutator of quantum Minkowski...

The Lie symmetry analysis for the study of a 1+n fourth-order Schrödinger equation inspired by modification deformation algebra in presence minimum length is applied. Specifically, we perform detailed classification scalar field potential function where non-trivial symmetries exist and simplify equation. Then, qualitative allows reduced ordinary differential to be analysed understand asymptotic...