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When is the automorphism group of an affine variety nested?
Full text linkFor an affine algebraic variety , we study the subgroup
of the group of regular automorphisms
of generated by all the connected algebraic subgroups. We
prove that is nested, i.e., is a direct limit of
algebraic subgroups of , if and only if all the
-actions on commute. Moreover, we describe the structure of
such a group .Comment: 10 pages; minor correction
Is the Affine Space Determined by Its Automorphism Group?
No full textIn this note we study the problem of characterizing the complex affine space An via its automorphism group. We prove the following. Let X be an irreducible quasi-projective n-dimensional variety such that Aut(X) and Aut(An) are isomorphic as abstract groups. If X is either quasi-Affine and toric or X is smooth with Euler characteristic (X) = 0 and finite Picard group Pic(X), then X is isomorphic to An. The main ingredient is the following result. Let X be a smooth irreducible quasiprojective variety of dimension n with finite Pic(X). If X admits a faithful (Z/pZ)naction for a prime p and (X) is not divisible by p, then the identity component of the centralizer CentAut(X)((Z/pZ)n) is a torus
Small G-varieties
Full text linkAnaffine varietywithanactionof a semisimple groupGis called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group K∗ commuting with the G-action.We show that X is determined by the K∗-variety XU of fixed points under a maximal unipotent subgroup U ⊂ G. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient X//G. If G is of type An (n ≥ 2), Cn, E6, E7, or E8, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If n ≥ 5, every smooth affine SLn-variety of dimension < 2n − 2 is an SLn-vector bundle over the smooth quotient X// SLn, with fiber isomorphic to the natural representation or its dual
Automorphisms of the Lie algebra of vector fields on affine -space
Full text linkInternational audienc
Automorphism groups of affine varieties without non-algebraic elements
No full textGiven an affine algebraic variety X, we prove that if the neutral component Aut◦(X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H ⊂ G, and for any g ∈ G some positive power of g belongs to H, then G = H
Vector Fields and Automorphism Groups of Danielewski Surfaces
No full textIn this paper, we study the Lie algebra of vector fields {\operatorname{Vec}}(\textrm{D}p) of a smooth Danielewski surface \textrm{D}p. We prove that the Lie subalgebra \langle{\operatorname{LNV}}(\textrm{D}p) \rangle of {\operatorname{Vec}}(\textrm{D}p) generated by locally nilpotent vector fields is simple. Moreover, if the two Lie algebras \langle{\operatorname{LNV}}(\textrm{D}p) \rangle and \langle{\operatorname{LNV}}(\textrm{D}q) \rangle of two Danielewski surfaces \textrm{D}p and \textrm{D}q are isomorphic, then the surfaces \textrm{D}p and \textrm{D}q are isomorphic. As an application we prove that the ind-groups {\operatorname{Aut}}(\textrm{D}p) and {\operatorname{Aut}}(\textrm{D}q) are isomorphic if and only if \textrm{D}p \simeq \textrm{D}q as a variety. We also show that any automorphism of the ind-group {\operatorname{Aut}}\circ (\textrm{D}p) is inner
Characterizing smooth affine spherical varieties via the automorphism group
Full text linkLet G be a connected reductive algebraic group. We prove that for a quasi-affine G-spherical variety the weight monoid is determined by the weights of its non-trivial G_a-actions that are homogeneous with respect to a Borel subgroup of G. As an application we get that a smooth affine spherical variety that is non-isomorphic to a torus is determined by its automorphism group (considered as an ind-group) inside the category of smooth affine irreducible varieties
Maximal commutative unipotent subgroups and a characterization of affine spherical varieties
No full textBracket width of current Lie algebras
Full text linkThe length of an element z of a Lie algebra L is defined as the smallest number s needed to represent z as a sum of s brackets. The bracket width of L is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra g over an algebraically closed field k of characteristic zero, we study the bracket width of current Lie algebras L=g⊗A. We show that for an arbitrary A the bracket width is at most 2. For A=k[[t]] and A=k[t] we compute the bracket width for algebras isomorphic to sln and sp2n
Groups of automorphisms of some affine varieties
Full text linkIn 1966 Shafarevich introduced the notion of an ind-variety. It turns out that Aut(X) has a natural structure of an ind-variety for any affine algebraic variety X. In this thesis we study the structure of Aut(X) viewed as an ind-group
and a structure of a Lie algebra Lie Aut(X). We compute the automorphism group of the Lie algebra of the group of automorphisms of an affine n-space (jointly with Hanspeter Kraft). We also prove that Lie subalgebras of Lie Aut(A^2) isomorphic to
the Lie algebra of the group of affine transformations of an affine plane A^2 are isomorphic if and only if Jacobian Conjecture holds in dimension 2. In the second part of the thesis we consider an n-dimensional affine variety X endowed with a non-trivial regular SL(n,C)-action. We prove that if Aut(X) is isomorphic to Aut(Y) as an ind-group for some irreducible affine normal variety Y, then Y is isomorphic to X as a variety. At the end of the thesis we present an example found with Matthias Leuenberger of two affine surfaces such that their so-called special automorphism groups are isomorphic as abstract groups, but not isomorphic as ind-groups
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