Related papers: Revisiting Lie integrability by quadratures from a…
Every Lie algebra over a field $E$ gives rise to new Lie algebras over any subfield $F \subseteq E$ by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of…
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a…
We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually based on.
We review the developments of a recently proposed approach to study integrable theories in any dimension. The basic idea consists in generalizing the zero curvature representation for two-dimensional integrable models to space-times of…
In this paper, simplicity of quadratic Lie conformal algebras are investigated. From the point view of the corresponding Gel'fand-Dorfman bialgebras, some sufficient conditions and necessary conditions to ensure simplicity of quadratic Lie…
In this paper, the author gives two methods to construct complete Lie algebras. Both methods show that the derivation algebras of some Lie algebras are complete.
In this paper, we introduce the commutativity degree of a finite-dimensional Lie algebra over a finite field and determine upper and lower bounds for it. Moreover, we study some relations between the notion of commutativity degree and known…
We study integrable Euler equations on the Lie algebra $\mathfrak{gl}(3,\mathbb{R})$ by interpreting them as evolutions on the space of hexagons inscribed in a real cubic curve.
We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…
The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.
We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.
In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2…
Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in…
Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article…
Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
Lie theory is, beyond any doubt, an absolutely essential part of differential geometry. It is therefore necessary to seek its generalization to $\mathbb{Z}$-graded geometry. In particular, it is vital to construct non-trivial and explicit…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
In this paper, we study the generalized derivation of a Lie sub-algebra of the Lie algebra of polynomial vector fields on $\mathbb{R}^n$ where $n\geq1$, containing all constant vector fields and the Euler vector field, under some conditions…