Algebraically generated groups and their Lie algebras
Résumé
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 affine ind-group G one can associate a canonical ideal L_G ⊆ Lie G. It is linearly generated by the tangent spaces T_e X for all algebraic subsets X ⊆ G which are smooth in e. It has the important property that for a surjective homomorphism φ: G → H the induced homomorphism dφ_e : L_G → L_H is surjective as well. Moreover, if H ⊆ G is a subnormal closed ind-subgroup of finite codimension, then L_H has finite codimension in L_G.
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