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Lie Subalgebras of Vector Fields on Affine 2-Space and the Jacobian Conjecture

Regeta, Andriy. (2013) Lie Subalgebras of Vector Fields on Affine 2-Space and the Jacobian Conjecture. Preprints Fachbereich Mathematik, 2013 (27).

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Abstract

We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the group $\mathrm{Aff}_{2}(K)$ of affine transformations of ${\mathbb A}^{2}$. We then show that the following three statements are equivalent:
(a) The Jacobian Conjecture holds in dimension 2;
(b) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are conjugate under $\mathrm{Aut}({\mathbb A}^{2})$;
(c) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are algebraic.
Finally, we use these results to show that the automorphism groups of the Lie algebras $\mathrm{Vec}({\mathbb A}^{2})$, $\mathrm{Vec}^{0}({\mathbb A}^{2})$ and $\mathrm{Vec}^{c}({\mathbb A}^{2})$ are all isomorphic to $\mathrm{Aut}({\mathbb A}^{2})$.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Regeta, Andriy
Item Type:Preprint
Publisher:Universität Basel
Language:English
Last Modified:13 May 2019 05:48
Deposited On:28 Mar 2019 09:52

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