12 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
A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable
This is an old paper put here for archeological purposes. It is proved that a
finite-dimensional Lie algebra over a field of characteristic p>5, that can be
written as a vector space (not necessarily direct) sum of two nilpotent
subalgebras, is solvable. The same result (but covering also the cases of low
characteristics) was established independently by V. Panyukov (Russ. Math.
Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the
proof were developed later in arXiv:math/0204004. Many inaccuracies in the
English translation are corrected, otherwise the text is identical to the
published version.Comment: v2: minor change