A characterization of orbit closure and applications.

*(English)*Zbl 0646.17002T. 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.

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 |

##### Keywords:

representation varieties; classification for nilpotent Lie algebras; orbit closure; contractions; varieties of Lie algebras
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\textit{F. Grunewald} and \textit{J. O'Halloran}, J. Algebra 116, No. 1, 163--175 (1988; Zbl 0646.17002)

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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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