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On limits of algebraic subgroups

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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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420-427

References

[CR] Ch. W. Curtis, I. Reiner, Linear representations of finite groups and associates algebras, New York: Interscience Publ. (1962). [MN] K. Morinaga, T. Nono, On the logarithmic functions of matrices, I., J. Sci. Hiroshima Univ. Ser. A 14 (1950), 107–114. [Ne] Yu.A. Neretin, An estimate of the number of parameters defining an n–dimensional algebra, Izv. Akad. Nauk SSSR Ser. Mat. 51(1987), 306–318(Russian); English transl.: Math. USSR Izvestiya 30 (1988), 283-294. [PR] S. Page, R.W. Richardson, Stable subalgebras of Lie algebras of Lie algebras and associate algebras, Trans. Amer. Math. Soc. 127 (1967), 302–312. [Ri] R.W. Richardson, Conjugacy classes of n–tuples in Lie algebras and algebraic groups, Duke Math. J. 57 (1988), 1–35. [Vi] E.B. Vinberg, On invariants of a set of matrices, J. Lie Theory 6 (1996), 249–169. [VO] E.B. Vinberg, A.L. Onishchik, Seminar on Lie groups and algebraic groups, Moskva, Nauka, (1988) (Russian); English transl.: A.L. Onishchik, E.B. Vinberg, Lie Groups and Algebraic Groups, Springer–Verlag (1990). [VP] E.B. Vinberg, V.L. Popov, Invariant theory, in: Itogi nauki i tekhniki. Sovr. probl. mat. Fundam. napr. 55, 137–309, Moskva, VINITI (1989) (Russian); English transl.: V.L. Popov, E.B. Vinberg, Invariant theory, in: Algebraic Geometry IV (Encyclopedia Math. Sci. 55), 123–284, Springer–Verlag (1994).

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