Related papers: Bilinear Systems Induced by Proper Lie Group Actio…
Infinitesimal symmetries of a classical mechanical system are usually described by a Lie algebra acting on the phase space, preserving the Poisson brackets. We propose that a quantum analogue is the action of a Lie bi-algebra on the…
Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, defined by invariance under the…
We give a complete classification of the class of Lie algebras of simply connected real Lie groups whose nontrivial coadjoint orbits are of codimension 1. Such a Lie group belongs to a well-known class, called the class of MD-groups. The…
After recalling standard nonlinear port-Hamiltonian systems and their algebraic constraint equations, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the…
In the framework of operator theory, we investigate a close Lie theoretic relationship between all operator ideals and certain classical groups of invertible operators that can be described as the solution sets of certain algebraic…
Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are…
We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical $r$-matrices and matched pairs induced by Poisson…
In this paper, we investigate the controllability of bilinear control systems of the form $\dot{s} = As + uBs$, where $s \in \mathbb{S}^2$ and $A, B \in gl(3, \mathbb{R})$ are skew-symmetric matrices. First, we prove that the algebraic…
Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms.…
In the study of NIL-affine actions on nilpotent Lie groups we introduced so called LR-structures on Lie algebras. The aim of this paper is to consider the existence question of LR-structures, and to start a structure theory of LR-algebras.…
We study integrable systems on double Lie algebras in absence of Ad-invariant bilinear form by passing to the semidirect product with the $\tau $-representation. We show that in this stage a natural Ad-invariant bilinear form does exist,…
Necessary or sufficient conditions are presented for the existence of various types of actions of Lie groups and Lie algebras on manifolds.
We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the…
The Lie algebra of the Poincar\'e-Maxwell group is derived in a manner that provides the interpretation of the equations of motion. It is clarified that the dynamics obtained from the orbit method is exactly equivalent to the classical…
We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the…
We develop the Lie theory of Lie-admissible algebras whose product is enriched with higher operations modeled on directed graphs with a view to apply it to the deformation theories controlled by this kind of Lie algebras. We produce…
The "coquecigrue" problem for Leibniz algebras is that of finding an appropriate generalization of Lie's third theorem, that is, of finding a generalization of the notion of group such that Leibniz algebras are the corresponding tangent…
The Lie product and the order relation are viewed as defining structures for Hamiltonian dynamical systems. Their admissible combinations are singled out by the requirement that the group of the Lie automorphisms be contained in the group…
Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for…
We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a…