Solvability of a Lie algebra of vector fields implies their integrability by quadratures

@article{Cariena2016SolvabilityOA,
  title={Solvability of a Lie algebra of vector fields implies their integrability by quadratures},
  author={Jos{\'e} F. Cari{\~n}ena and Fernando Falceto and Janusz Grabowski},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2016},
  volume={49}
}
We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures. 

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References

SHOWING 1-10 OF 14 REFERENCES

Geometry of Lie integrability by quadratures

In this paper, we extend the Lie theory of integration by quadratures of systems of ordinary differential equations in two different ways. First, we consider a finite-dimensional Lie algebra of

The Euler-Jacobi-Lie integrability theorem

This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor

Introduction to Lie Algebras and Representation Theory

Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-

Generalized Liouville method of integration of Hamiltonian systems

In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of level

Remarks on a Lie Theorem on the Integrability of Differential Equations in Closed Form

is integrable by quadratures [1] (for details, see also [2, 3]). More precisely, all of its solutions can be found by “algebraic operations” (including inversion of functions) and “quadratures,” that

Remarks on nilpotent Lie algebras of vector fields.

In [1] a local description of analytic vector fields finitely generating a transitive nilpotent Lie algebra L on a manifold is given. Our aim is to generalize this result by (i) omitting the

An extension of a theorem of Nagano on transitive Lie algebras

Let M be a real analytic manifold, and let L be a transitive Lie algebra of real analytic vector fields on M. A concept of completeness is introduced for such Lie algebras. Roughly speaking, L is

On the structure of transitively differential algebras

We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain the vector fields ${\partial}/{\partial x_i} \; (i=1,\ldots, n)$ and $x_1{\partial}/{\partial x_1}+

Mathematical Methods of Classical Mechanics

Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid

INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY