Regeta, Andriy. Groups of automorphisms of some affine varieties. 2015, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_12194
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Abstract
In 1966 Shafarevich introduced the notion of an indvariety. It turns out that Aut(X) has a natural structure of an indvariety for any affine algebraic variety X. In this thesis we study the structure of Aut(X) viewed as an indgroup
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 nspace (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 ndimensional affine variety X endowed with a nontrivial regular SL(n,C)action. We prove that if Aut(X) is isomorphic to Aut(Y) as an indgroup 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 socalled special automorphism groups are isomorphic as abstract groups, but not isomorphic as indgroups.
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 nspace (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 ndimensional affine variety X endowed with a nontrivial regular SL(n,C)action. We prove that if Aut(X) is isomorphic to Aut(Y) as an indgroup 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 socalled special automorphism groups are isomorphic as abstract groups, but not isomorphic as indgroups.
Advisors:  Kraft, Hanspeter and Furter, JeanPhilippe 

Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft) 
UniBasel Contributors:  Regeta, Andriy and Kraft, Hanspeter 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  12194 
Thesis status:  Complete 
Number of Pages:  1 OnlineRessource (1 Band (verschiedene Seitenzählungen)) 
Language:  English 
Identification Number: 

edoc DOI:  
Last Modified:  22 Jan 2018 15:52 
Deposited On:  24 Jul 2017 13:55 
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