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- W4221139437 abstract "The automorphism group Aut(X) of an affine variety X is an ind-group. Its Lie algebra is canonically embedded into the Lie algebra VF(X) of vector fields on X. We study the relations between subgroups of Aut(X) and Lie subalgebras of VF(X). We show that a subgroup G of Aut(X) generated by a family of connected algebraic subgroups G_i of Aut(X) is algebraic if and only if the Lie algebras Lie G_i generate a finite dimensional Lie subalgebra of VF(X). Extending a result by Cohen-Draisma we prove that a locally finite Lie algebra L of VF(X) generated by locally nilpotent vector fields is algebraic, i.e. L = Lie G for an algebraic subgroup G of Aut(X). Along the same lines we prove that if a subgroup G of Aut(X) generated by finitely many connected algebraic groups is solvable, then it is a solvable algebraic group. We also show that the derived length a unipotent algebraic subgroup U of Aut(X) is bounded above by dim X. This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of Aut(A^n) with a dense orbit in A^n is conjugate to a subgroup of the de Jonqui`eres subgroup. Furthermore, we give an example of a free subgroup F of Aut(A^2) generated by two algebraic elements such that the Zariski closure of 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 of 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." @default.
- W4221139437 created "2022-04-03" @default.
- W4221139437 creator A5042135883 @default.
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- W4221139437 date "2022-03-21" @default.
- W4221139437 modified "2023-09-28" @default.
- W4221139437 title "Algebraically generated groups and their lie algebras" @default.
- W4221139437 doi "https://doi.org/10.48550/arxiv.2203.11356" @default.
- W4221139437 hasPublicationYear "2022" @default.
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