The Sharp Upper Estimate Conjecture for the Dimension δk(V) of New Derivation Lie Algebra

Abstract

Hussain, Yau, and Zuo introduced the Lie algebra Lk(V) from the derivation of the local algebra Mk(V):=On/(g+J1(g)+⋯+Jk(g)). To find the dimension of a newly defined algebra is an important task in order to study its properties. In this regard, we compute the dimension of Lie algebra L5(V) and justify the sharp upper estimate conjecture for fewnomial isolated singularities. We also verify the inequality conjecture: δ5(V)<δ4(V) for a general class of singularities. Our findings are novel and an addition to the study of Lie algebra.