Some Lie algebra structures on symmetric powers

Let k be a field of any characteristic, V a finite-dimensional vector space over k, and Sᵈ (V^*) be the d-th symmetric power of the dual space V^*. Given a linear map on V and an eigenvector w of, we prove that the pair (, w) can be used to construct a new Lie algebra structure on Sᵈ (V^*). We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if is a nilpotent map. We also classify the Lie algebras for all possible pairs (, w), when k=C and V is two-dimensional.