939 research outputs found
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
Equations in simple Lie algebras
Given an element of the finitely generated free Lie algebra,
for any Lie algebra we can consider the induced polynomial map . Assuming that is an arbitrary field of characteristic , we prove
that if is not an identity in , then this map is dominant for any
Chevalley algebra . This result can be viewed as a weak infinitesimal
counterpart of Borel's theorem on the dominancy of the word map on connected
semisimple algebraic groups.
We prove that for the Engel monomials and, more
generally, for their linear combinations, this map is, moreover, surjective
onto the set of noncentral elements of provided that the ground field
is big enough, and show that for monomials of large degree the image of this
map contains no nonzero central elements.
We also discuss consequences of these results for polynomial maps of
associative matrix algebras.Comment: 22 page
Finite-dimensional Lie subalgebras of algebras with continuous inversion
We investigate the finite-dimensional Lie groups whose points are separated
by the continuous homomorphisms into groups of invertible elements of locally
convex algebras with continuous inversion that satisfy an appropriate
completeness condition. We find that these are precisely the linear Lie groups,
that is, the Lie groups which can be faithfully represented as matrix groups.
Our method relies on proving that certain finite-dimensional Lie subalgebras of
algebras with continuous inversion commute modulo the Jacobson radical.Comment: 9 pages; to appear in the journal Studia Mathematic
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
- …