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A characterization of orbit closure and applications. (English) Zbl 0646.17002
T. Skjelbred and T. Sund [C. R. Acad. Sci., Paris, Sér. A 286, 241-242 (1978; Zbl 0375.17006)] studied the central extensions with kernel $$k^ r$$ of n-r dimensional nilpotent Lie algebras $${\mathfrak g}$$ over an algebraically closed field. Let $${\mathbb{B}}$$ be the open set consisting of 2-cocycles $$B\in Z^ 2({\mathfrak g},k^ r)$$ for which $$B^{\perp}$$ meets trivially the center of $${\mathfrak g}$$, $${\mathfrak g}(B)$$ be the algebra obtained by central extension from $${\mathfrak g}$$ by $$k^ r$$ via B, and G be the automorphism group of the category of central extensions from $${\mathfrak g}$$ by $$k^ r$$. The mapping $$B\to {\mathfrak g}(B)$$ induces a one-to-one mapping from $${\mathbb{B}}/G$$ into the space $$N_ n/GL_ n(k)$$ of isomorphic classes of nilpotent Lie algebras on $$k^ n$$. They obtained in this way an interesting principle of classification for nilpotent Lie algebras.
In this paper the authors improve that result by showing that the mapping $$B\to {\mathfrak g}(B)$$ preserves orbit closure. This work allows in particular to obtain contractions of $${\mathfrak g}(B)$$ (when $${\mathfrak g}$$ is fixed) from the contractions of B. A counter-example is given at the end of the article. The method uses a result of A. Lubotzky and A. R. Magid [Mem. Am. Math. Soc. 336 (1985; Zbl 0598.14042)] for the varieties of representations of groups, applied here to varieties of Lie algebras.
Reviewer: R.Carles

##### MSC:
 17B30 Solvable, nilpotent (super)algebras
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##### References:
 [1] Grunewald, F; O’Halloran, J, Varieties of nilpotent Lie algebras of dimension less than six, J. algebra, 112, 315-325, (1988) · Zbl 0638.17005 [2] Lubotsky, A; Magid, A, Varieties of representations of finitely generated groups, () [3] Atiyah, M.F; MacDonald, I.G, Introduction to commutative algebra, (1969), Addison-Wesley Reading, MA · Zbl 0175.03601 [4] Santharoubane, L.J, Infinite families of nilpotent Lie algebras, J. math. soc. Japan, 35, 3, 515-519, (1983) · Zbl 0489.17003 [5] Skjelbred, T; Sund, T, On the classification of nilpotent Lie algebras, () · Zbl 0422.17002
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