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Deformation of Lie algebras and Lie algebras of deformations. (English) Zbl 0708.14005

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.

MSC:

14D15 Formal methods and deformations in algebraic geometry
14H20 Singularities of curves, local rings
17B56 Cohomology of Lie (super)algebras
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References:

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