Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

@article{Arzhantsev2010FiniteDimensionalSI,
  title={Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One},
  author={Ivan Arzhantsev and E. A. Makedonskii and Anatoliy P. Petravchuk},
  journal={Ukrainian Mathematical Journal},
  year={2010},
  volume={63},
  pages={827-832}
}
Let Wn($$ {\mathbb K} $$) be the Lie algebra of derivations of the polynomial algebra $$ {\mathbb K} $$[X] :=$$ {\mathbb K} $$[x1,…,xn]over an algebraically closed field $$ {\mathbb K} $$ of characteristic zero. A subalgebra $$ L \subseteq {W_n}(\mathbb{K}) $$ is called polynomial if it is a submodule of the $$ {\mathbb K} $$[X]-module Wn($$ {\mathbb K} $$). We prove that the centralizer of every nonzero element in L is abelian, provided that L is of rank one. This fact allows one to classify… 

On finite-dimensional subalgebras of derivation Lie algebras

Let $\mathbb{K}$ be a field, $R$ be an associative and commutative $\mathbb{K}$-algebra and $L$ be a Lie algebra over $\mathbb{K}$. We give some descriptions of injections from $L$ to Lie algebra of

Residual and fixed modules

The article presents some sufficient conditions for the commutativity of transvections with elements of linear groups over division ring in the language of residual and fixed submodules. The residual

Solvable Lie Algebras of Derivations of Rank One

Let K be a field of characteristic zero, A = K[x1, . . . , xn] the polynomial ring and R = K(x1, . . . , xn) the field of rational functions in n variables over K. The Lie algebra Wn(K) of all

On Noncommutative Bases of Free Modules of Derivations over Polynomial Rings

Let 𝕂 be an algebraically closed field of characteristic zero and W n be the Lie algebra of all 𝕂-derivations of the polynomial ring R in n variables over 𝕂. It is proved that every Lie algebra of

Lie algebras with Abelian centralizers

In the paper, finite-dimensional real Lie algebras for which the centralizers of all nonzero element are Abelian are studied. These Lie algebras are also characterized by the transitivity condition

Faithful representations of Lie algebras and Homogeneous Spaces

The faithful representations of real Lie algebras g with dimensions $n\leq 4$ acting on the dual space of the universal covering algebra of an ideal s of g, are calculated. With the help of left and

On finite dimensional Lie algebras of planar vector fields with rational coefficients

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional

References

SHOWING 1-10 OF 10 REFERENCES

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial.

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.-

Lie algebras of vector fields in the real plane

Finite-dimensional real analytic Lie algebras of vector fields on R 2 are completely classified up to changes of local coordinates

Theorie der Transformationsgruppen I

Einleitung Erster Abschnitt: Allgemeine Eigenschaften der endlichen continuirlichen Transformationsgruppen Zweiter Abschnitt: Theorie der Beruhrungstransformationen und der Gruppen von

INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY

On the Ideals of a Lie Algebra of Derivations

Theorie Der Transformationsgruppen (3 vols)

Arzhantsev: Department of Higher Algebra, Faculty of Mechanics and Mathematics

  • Leninskie Gory