367 research outputs found
Normal and normally outer automorphisms of free metabelian nilpotent Lie algebras
Let L be the free m-generated metabelian nilpotent of class c Lie algebra
over a field of characteristic 0. An automorphism f of L is called normal if
f(I)=I for every ideal I of the algebra L. Such automorphisms form a normal
subgroup N(L) of Aut(L) containing the group of inner automorphisms. We
describe the group of normal automorphisms of L and the quotient group of
Aut(L) modulo N(L).Comment: to appear in Serdica Mathematical Journa
Constants of Weitzenb\"ock derivations and invariants of unipotent transformations acting on relatively free algebras
In commutative algebra, a Weitzenb\"ock derivation is a nonzero triangular
linear derivation of the polynomial algebra in several
variables over a field of characteristic 0. The classical theorem of
Weitzenb\"ock states that the algebra of constants is finitely generated. (This
algebra coincides with the algebra of invariants of a single unipotent
transformation.) In this paper we study the problem of finite generation of the
algebras of constants of triangular linear derivations of finitely generated
(not necessarily commutative or associative) algebras over assuming that
the algebras are free in some sense (in most of the cases relatively free
algebras in varieties of associative or Lie algebras). In this case the algebra
of constants also coincides with the algebra of invariants of some unipotent
transformation. \par The main results are the following: 1. We show that the
subalgebra of constants of a factor algebra can be lifted to the subalgebra of
constants. 2. For all varieties of associative algebras which are not nilpotent
in Lie sense the subalgebras of constants of the relatively free algebras of
rank are not finitely generated. 3. We describe the generators of the
subalgebra of constants for all factor algebras modulo a
-invariant ideal . 4. Applying known results from commutative
algebra, we construct classes of automorphisms of the algebra generated by two
generic matrices. We obtain also some partial results on relatively
free Lie algebras.Comment: 31 page
The Anick automorphism of free associative algebras
We prove that the well-known Anick automorphism of the free associative
algebra F over an arbitrary field F of characteristic 0 is wild.Comment: 15 page
Gr\"{o}bner-Shirshov bases for metabelian Lie algebras
In this paper, we establish the Gr\"{o}bner-Shirshov bases theory for
metabelian Lie algebras. As applications, we find the Gr\"{o}bner-Shirshov
bases for partial commutative metabelian Lie algebras related to circuits,
trees and some cubes.Comment: 20 page
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