Algebraically generated groups and their Lie algebras

Abstract

AbstractThe automorphism group of an affine variety is an ind‐group. Its Lie algebra is canonically embedded into the Lie algebra of vector fields on . We study the relations between subgroups of and Lie subalgebras of . We show that a subgroup generated by a family of connected algebraic subgroups of is algebraic if and only if the Lie algebras generate a finite‐dimensional Lie subalgebra of . Extending a result by Cohen–Draisma (Transform. Groups 8 (2003), no. 1, 51–68), we prove that a locally finite Lie algebra generated by locally nilpotent vector fields is algebraic, that is, for an algebraic subgroup . Along the same lines, we prove that if a subgroup generated by finitely many connected algebraic groups is solvable, then it is an algebraic group. We also show that a unipotent algebraic subgroup has derived length . This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of with a dense orbit in is conjugate to a subgroup of the de Jonquières subgroup. Furthermore, we give an example of a free subgroup generated by two algebraic elements such that the Zariski closure is a free product of two nested commutative closed unipotent ind‐subgroups. To any affine ind‐group , one can associate a canonical ideal . It is linearly generated by the tangent spaces for all algebraic subsets that are smooth in . It has the important property that for a surjective homomorphism , the induced homomorphism is surjective as well. Moreover, if is a subnormal closed ind‐subgroup of finite codimension, then has finite codimension in .