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Algebraically generated groups and their Lie algebras

  • Kraft, Hanspeter
  • Zaidenberg, M
Publication Date
Mar 21, 2022
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The automorphism group Aut(X) of an affine variety X is an ind-group. Its Lie algebra is canonically embedded into the Lie algebra Vec(X) of vector fields on X. We study the relations between closed subgroups G of Aut(X) and the corresponding Lie subalgebras of Vec(X).We show that a subgroup G ⊆ Aut(X) generated by a family of connected algebraic subgroups G_i of Aut(X) is algebraic if and only if the Lie G_i ⊆ Vec(X ) generate a finite dimensional Lie subalgebra of Vec(X).Extending a result by Cohen-Draisma (2003) we prove that a locally finite Lie algebra L ⊆ Vec(X) generated by locally nilpotent vector fields is algebraic, i.e. L = Lie G of some algebraic subgroup G ⊆ Aut(X).Along the same lines we prove that if a subgroup G ⊆ Aut(X) generated by finitely many unipotent algebraic groups is solvable, then it is a unipotent algebraic group.Furthermore, we give an example of a free subgroup F ⊆ Aut(A^2) generated by two algebraic elements such that the Zariski closure F is a free product of two nested commutative closed unipotent ind-subgroups.To any ind-group G one can associate a canonical ideal L_G ⊆ Lie G generated by the Lie algebras of algebraic subgroups of G, and another one associated to the set of all algebraic subvarieties of G. We study functorial properties of these ideals, ind-subgroups of finite codimension in G and the corresponding Lie subalgebras of Lie G.

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