Graded Lie algebras and quasihomogeneous complete intersection singularities
The main result of the paper is one of those determining geometrical objects by associated algebraic structures. Let \(X\subset{\mathbb{C}}^n\) be the germ of an isolated singularity determined by the ideal \(I\). There are several complex Lie algebras associated with \(X\): the Lie algebra \[ \text{Der}_I({\mathcal O}_n)=\{\delta\in \text{Der}({\mathcal O}_n)\mid\;\delta (I)\subset I\} \] of germs of vector fields on \({\mathbb{C}}^n\) that are tangent to \(X\), the Lie algebra \[ \Theta (X)=\text{Der}({\mathcal O}_X)=\text{Der}_I({\mathcal O}_n)\otimes{\mathcal O}_X \] of all vector fields on \(X\), and the reduced tangent Lie algebra \[ {\mathcal T}(X)=\Theta(X)/{\mathcal H}(X), \] where \({\mathcal H}_X\) is the Lie algebra of so-called Hamiltonian vector fields. The type of a quasihomogeneous isolated complete intersection singularity (qh icis) \(X\) is given by \((\underline{w},\underline {d})\), where \(w_i\) are the weights of the variables and \(d_j\) are the degrees of the defining polynomials. It is proved that if two qh icis \(X\) and \(X'\) of types \((\underline{w},\underline{d})\) and \((\underline{w},\underline {d'})\), respectively, fulfil the condition that the degree of each Hamiltonian derivation is greater than \(\max\{ 5w_i,d_j,d'_k\}\), then \(X\simeq X'\) if and only if \({\mathcal T}(X) \simeq{\mathcal T}(X')\) as Lie algebras. The proof depends on investigations of graded Lie algebras with Euler generators.
scientific article; zbMATH DE number 954754