Lie algebras with finite-dimensional polynomial centralizer (Q1614679)

This interesting paper deals with the following setting: Let \(V\) be a finite-dimensional vector space over \(K=\mathbb{R}\) or \(\mathbb{C}\), \(M\subset gl(V)\) be a Lie algebra and \(P(V)\) be the Lie algebra of all polynomial vector fields on \(V\). The main working tool in the paper is the centralizer \(C(M)= \{f\in P(V): [f,M]=0\}\). The conditions for finite or infinite dimensionality of \(C(M)\) are given using the space \(I_0(M)= \{\varphi\in \mathbb{K}[x_1, \dots,x_n]: X_B (\varphi)=0\) for all \(B\in M\}\) of all polynomial invariants of \(M\), where \(X_B\) denotes the derivation operator in the direction of the vector field \(B\). In the case of finite dimensionality of \(C(M)\), two nice applications to normal forms are given and also one application concerning the elementary solutions of some ODE's.