102 research outputs found
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
Tits type alternative for groups acting on toric affine varieties
Given a toric affine algebraic variety and a collection of one-parameter
unipotent subgroups of which are
normalized by the torus acting on , we show that the group generated by
verifies the following alternative of Tits' type: either
is a unipotent algebraic group, or it contains a non-abelian free subgroup. We
deduce that if is -transitive on a -orbit in , then contains a
non-abelian free subgroup, and so, is of exponential growth.Comment: 24 pages. The main result strengthened, the proof of Proposition 4.8
written in more detail; some references added; the referee remarks taken into
account; the title change
Flexible varieties and automorphism groups
Given an affine algebraic variety X of dimension at least 2, we let SAut (X)
denote the special automorphism group of X i.e., the subgroup of the full
automorphism group Aut (X) generated by all one-parameter unipotent subgroups.
We show that if SAut (X) is transitive on the smooth locus of X then it is
infinitely transitive on this locus. In turn, the transitivity is equivalent to
the flexibility of X. The latter means that for every smooth point x of X the
tangent space at x is spanned by the velocity vectors of one-parameter
unipotent subgroups of Aut (X). We provide also different variations and
applications.Comment: Final version; to appear in Duke Math.
Varieties covered by affine spaces and their cones
It was shown in arXiv:2303.02036 that the affine cones over flag manifolds
and rational smooth projective surfaces are elliptic in the sense of Gromov.
The latter remains true after successive blowups of points on these varieties.
In the present note we extend this to smooth projective spherical varieties (in
particular, toric varieties) successively blown up along linear subvarieties.
The same also holds, more generally, for projective varieties covered by affine
spaces.Comment: 10 page
Infinite transitivity, finite generation, and Demazure roots
An affine algebraic variety X of dimension at least 2 is called flexible if
the subgroup SAut(X) in Aut(X) generated by the one-parameter unipotent
subgroups acts m-transitively on reg(X) for any m 1. In the previous
paper we proved that any nondegenerate toric affine variety X is flexible. In
the present paper we show that one can find a subgroup of SAut(X) generated by
a finite number of one-parameter unipotent subgroups which has the same
transitivity property, provided the toric variety X is smooth in codimension 2.
For X= with n2, three such subgroups suffice.Comment: 25 page
On orbits of the automorphism group on a complete toric variety
Let X be a complete toric variety and Aut(X) be the automorphism group. We
give an explit description of Aut(X)-orbits on X. In particular, we show that
Aut(X) acts on X transitively if and only if X is a product of projective
spaces.Comment: 10 pages, 4 figure
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
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