Different methods for the study of obstructions in the schemes of Jacobi
Let \(\mathfrak g = R \oplus \mathfrak n\) be a Lie algebra of dimension \(m\) with completely reducible part \(R\) and nilpotent part \(\mathfrak n\) with dimension \(n\). The authors consider the variety of Lie subalgebras which are \(R\)-invariant by \(L_n^R(\mathbb{C})\). The classical deformation theory is clearly valid for this subvariety. They introduce for a 2-cocycle the notion of strongly integrable up to order \(p\), if from the lower level Mauer-Cartan equations it follows the \(p\)-th equation. They illustrate the method on some filiform Lie algebras. Let \(T\) be a torus and consider \(L_n^T(\mathbb{C})\). In Section 2 the authors study such nilpotent algebras with an extra condition. They associate a Lie algebra \(\Phi_0\) from this class with an admissible part \(\mathcal A\) and define the slice to \((\Phi_0,\mathcal A)\). The local ring of the slice at \(\Phi_0\) gives the universal infinitesimal deformation of \(\Phi_0\) in the slice. The local study of Lie algebras from \(L_n^T\) is based on central extensions of nilpotent algebras. Here also they give examples to illustrate the method. In Section 3 the authors develop a global method for studying Lie algebras from \(L_n^T = \mathrm{Spec}(A_n)\) with \(A_n=\mathbb{C}[X_{ij}^k]/\mathrm{Jac}_n\). They reduce the dimension and use factorizations which helps in rigidity problems, finding non-reduced points etc. In Section 4 they view a Lie algebra \(\mathfrak g \) with a finite number of \(r\) generators as a quotient of a free Lie algebra \( \mathfrak L_r\) with \(r\) generators by an ideal \(\mathfrak J\): \(0 \to \mathfrak J \to \mathfrak{ L}_r \to \mathfrak g \to 0\). Denote \(\mathfrak n = \mathfrak {L}_r/\mathfrak J\). Let \(T_r\) be a maximal torus on \(\mathfrak {L}_r\) and \(T\) a subtorus of \(T_r\). The authors give a geometrization of \(\mathfrak n\) with the help of a specific Grassmannian. They show that the Zariski tangent spaces at \(\mathfrak n\) are given by \(H^2(\mathfrak n,\mathfrak n)^T\). In Section 5 the rigidity of \(\mathfrak g = T \oplus \mathfrak n\) Lie algebras is considered where \(n(\alpha_i)=1\) for \(1 \leq i \leq r\). They also give some examples.
scientific article; zbMATH DE number 5973452
scientific article