Bjar, Harald; Laudal, Olav Arnfinn Deformation of Lie algebras and Lie algebras of deformations. (English) Zbl 0708.14005 Compos. Math. 75, No. 1, 69-111 (1990). The paper is based on the book of O. A. Laudal and G. Pfister [“Local moduli and singularities”, Lect. Notes Math. 1310 (1988; Zbl 0657.14005)]. The cohomology groups of a finite dimensional Lie algebra are defined in the sense of O. A. Laudal [“Formal moduli of algebraic structures”, Lect. Notes Math. 754 (1979; Zbl 0438.14007)]. Let \(\pi: \mathbb X\to\mathbb H=\mathrm{Spec}(H)\) be a miniversal deformation of the \(k\)-scheme \(X\) and assume that the sub-Lie algebra \(\mathbb V\) of \(\mathrm{Der}_k(H)\) is the kernel of the Kodaira-Spencer map associated to \(\pi\). Then \(\mathbb H_0=\mathrm{Spec}(H_0)\) is the complement of the support of \(\mathbb V\), and \(\Lambda^0=\mathbb V\otimes_H H_0\) is a flat \(H_0\)-Lie algebra which defines a deformation of the Lie algebra \(L^0(X) = H^0(X,\theta_X)/A_{\pi}\). Here \(A_{\pi}\) is the Lie ideal of those infinitesimal automorphisms of \(X\) that lift to \(\mathbb X{\hat{\;}}=\mathbb X\otimes_HH{\hat{\;}}.\) The main result of the paper states that for the quasihomogeneous isolated plane curve singularity \(f: x^k_1+x^k_2\), the map \(\mathbf{1}: H_0\to\) moduli space of Lie algebras of \(\dim(\tau (f))\) is locally an immersion. Reviewer: Alice Fialowski (Budapest) Cited in 1 ReviewCited in 5 Documents MSC: 14D15 Formal methods and deformations in algebraic geometry 14H20 Singularities of curves, local rings 17B56 Cohomology of Lie (super)algebras Keywords:miniversal deformation; deformation of the Lie algebra; quasihomogeneous isolated plane curve singularity Citations:Zbl 0657.14005; Zbl 0438.14007 PDFBibTeX XMLCite \textit{H. Bjar} and \textit{O. A. Laudal}, Compos. 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Zeitschrift 191 (1986), pp. 489-506. · Zbl 0589.17012 · doi:10.1007/BF01162338 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.