Algebraically generated groups and their lie algebras

TL;DR

This paper explores the structure of automorphism groups of affine varieties, their Lie algebras, and the conditions under which subgroups are algebraic, providing new insights into their algebraic and geometric properties.

Contribution

It establishes criteria for when subgroups of automorphism groups are algebraic based on their Lie algebras and extends known results to broader classes of ind-groups and Lie algebras.

Findings

A subgroup generated by connected algebraic subgroups is algebraic iff its Lie algebra is finite dimensional.

Locally finite Lie algebras generated by locally nilpotent vector fields are algebraic.

Unipotent algebraic subgroups have derived length bounded by the dimension of the variety.

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 affine ind-group G one can associate a canonical ideal L_G \subset Lie G. It is linearly generated by the tangent spaces T_e X for all algebraic subsets X \subset G which are smooth in e. It has the important property that for a surjective homomorphism {\phi}: G \to H the induced homomorphism d{\phi}_e : L_G \to L_H is surjective as well. Moreover, if H \subset G is a subnormal closed ind-subgroup of finite codimension, then L_H has finite codimension in L_G.