60 research outputs found
Geometry of word equations in simple algebraic groups over special fields
This paper contains a survey of recent developments in investigation of word
equations in simple matrix groups and polynomial equations in simple
(associative and Lie) matrix algebras along with some new results on the image
of word maps on algebraic groups defined over special fields: complex, real,
p-adic (or close to such), or finite.Comment: 44 page
Word maps in Kac-Moody setting
The paper is a short survey of recent developments in the area of word maps
evaluated on groups and algebras. It is aimed to pose questions relevant to
Kac--Moody theory.Comment: 16 pag
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
From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical
We prove that an element of prime order belongs to the solvable
radical of a finite (or, more generally, a linear) group if and only if
for every the subgroup generated by is solvable. This
theorem implies that a finite (or a linear) group is solvable if and only
if in each conjugacy class of every two elements generate a solvable
subgroup.Comment: 28 page
Elementary equivalence of Kac-Moody groups
The paper is devoted to model-theoretic properties of Kac-Moody groups with
the focus on elementary equivalence of Kac-Moody groups. We show that
elementary equivalence of (untwisted) affine Kac-Moody groups implies
coincidence of their generalized Cartan matrices and the elementary equivalence
of their ground fields. We also show that elementary equivalence of arbitrary
Kac-Moody groups over finite fields implies coincidence of these fields and an
isomorphism of their twin root data. The similar result is established for
Kac-Moody groups defined over infinite subfields of the algebraic closures of
finite fields.Comment: 10 page
On first order rigidity for linear groups
The paper is a short survey of recent developments in the area of first order descriptions of linear groups. It is aimed to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac–Moody groups
The Diophantine problem in Chevalley groups
In this paper we study the Diophantine problem in Chevalley groups , where is an indecomposable root system of rank , is
an arbitrary commutative ring with . We establish a variant of double
centralizer theorem for elementary unipotents . This theorem is
valid for arbitrary commutative rings with . The result is principle to show
that any one-parametric subgroup , , is Diophantine
in . Then we prove that the Diophantine problem in is
polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine
problem in . This fact gives rise to a number of model-theoretic corollaries
for specific types of rings.Comment: 44 page
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