19 research outputs found
On nilpotent Lie algebras of derivations of fraction fields
Let be an arbitrary field of characteristic zero and a commutative
associative -algebra which is an integral domain. Denote by the
fraction field of and by the Lie algebra of
-derivations of obtained from via
multiplication by elements of If is a subalgebra of
denote by the dimension of the vector space over the
field and by the field of constants of in Let be a
nilpotent subalgebra with . It is proven that
the Lie algebra (as a Lie algebra over the field ) is isomorphic to a
finite dimensional subalgebra of the triangular Lie subalgebra of
the Lie algebra where with , In particular, a characterization of nilpotent Lie algebras
of vector fields with polynomial coefficients in three variables is obtained.Comment: Corrected typos. Revised formulation of Theorem 1, results unchange
On one-sided Lie nilpotent ideals of associative rings
We prove that a Lie nilpotent one-sided ideal of an associative ring is
contained in a Lie solvable two-sided ideal of . An estimation of derived
length of such Lie solvable ideal is obtained depending on the class of Lie
nilpotency of the Lie nilpotent one-sided ideal of One-sided Lie nilpotent
ideals contained in ideals generated by commutators of the form are also studied.Comment: 5 page
On closed rational functions in several variables
Let k be an algebraically closed field of characteristic zero. An element F
from k(x_1,...,x_n) is called a closed rational function if the subfield k(F)
is algebraically closed in the field k(x_1,...,x_n). We prove that a rational
function F=f/g is closed if f and g are algebraically independent and at least
one of them is irreducible. We also show that the rational function F=f/g is
closed if and only if the pencil af+bg contains only finitely many reducible
hypersurfaces. Some sufficient conditions for a polynomial to be irreducible
are given.Comment: Added references, corrected some typo
Finite-dimensional subalgebras in polynomial Lie algebras of rank one
Let W_n(K) be the Lie algebra of derivations of the polynomial algebra
K[X]:=K[x_1,...,x_n] over an algebraically closed field K of characteristic
zero. A subalgebra L of W_n(K) is called polynomial if it is a submodule of the
K[X]-module W_n(K). We prove that the centralizer of every nonzero element in L
is abelian provided L has rank one. This allows to classify finite-dimensional
subalgebras in polynomial Lie algebras of rank one.Comment: 5 page
Centralizers of linear and locally nilpotent derivations
Let be an algebraically closed field of characteristic zero, the polynomial ring, the field of
rational functions, and let W_n(K) = \Der_{K}A be the Lie algebra of all
-derivations on . If is linear (i.e. of the
form ) we give a
description of the centralizer of in and point out an algorithm
for finding generators of as a module over the ring of
constants in case when is the basic Weitzenboeck derivation. In more
general case when the ring is a finitely generated domain over and
is a locally nilpotent derivation on we prove that the centralizer
is a "large" \ subalgebra in , namely
\rk_A C_{\Der A}(D) := \dim_R RC_{\Der A}(D) equals
where is the field of fraction of the ring $A.Comment: 10 page
Lie algebras of derivations with large abelian ideals
We study subalgebras L of rank m over R of the Lie algebra Wn(K) with an abelian ideal I ⊂ L of the same rank m over R