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On limits of algebraic subgroups


We shall consider complex algebraic varieties and groups. Both analytic and Zariski topology will be used. Unless stated otherwise, the analytic topology is meant. For elements g1, …, gp of an algebraic group G, we denote by (g1, …, gp) the Zariski closure of the subgroup generated by g1, …, gp. If (g1, …, gp)=G, we say that G is Zariski generated by g1, …, gp. Any algebraic group is Zariski generated by finitely many elements. In particular, any connected reductive group is Zariski generated by two elements [Vi]. Closeness of algebraic subgroups can be evaluated by the closeness of their Zariski generating sets. As usually, we denote the tangent Lie algebras of Lie groups G, H, … by the corresponding gothic letters g, h, … We set Gp=GxxG Let H be an algebraic subgroup G, and g1, g2,G. Suppose that tere exists a limit l=lim Ad(gn)h in the relevant Grassmanian, and set L=lim gnHgn-1={limgnhngn-1 : h1, h2,H}. (Here h1, h2,.. are supposed to be chosen in such a way that limgnhngn-1 should exist.) Obviously, L is a subgroup of G.

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