Hussain, N.; Al-Kenani, A.N.; Asif, M. On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities. Axioms2024, 13, 216.
Hussain, N.; Al-Kenani, A.N.; Asif, M. On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities. Axioms 2024, 13, 216.
Hussain, N.; Al-Kenani, A.N.; Asif, M. On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities. Axioms2024, 13, 216.
Hussain, N.; Al-Kenani, A.N.; Asif, M. On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities. Axioms 2024, 13, 216.
Abstract
Lie algebra plays an important role in the study of singularity theory and other field of sciences. Finding numerous invariants linked with isolated singularities is always a main interest in the classification theory of isolated singularities.
Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work.
Hussain, Yau and Zuo \cite{HYZ10} have been found a new class of Lie algebra \texorpdfstring{ $ \mathcal{L}_ k (V)$}{LG}, \texorpdfstring{$k\geq 1$}{LG} i.e., Der ($M_{k}(V), M_{k}(V)$) and purposed a conjecture over its dimension $\delta_{k}(V)$ for $k\geq0$. Later they proved it true for $k$ up to $k=1,2,3,4,5$. In this work, the main concern is whether it's true for a higher value of $k$. According to this, we calculate first, the dimension of Lie algebra \texorpdfstring{$\mathcal{L}_ k (V)$}{LG} for $k=6$ and then compute the upper estimate conjecture of fewnomial isolated singularities. Along with, we also justify the inequality conjecture: \texorpdfstring{$\delta_{k+1}(V) < \delta_{k}(V)$}{LG} for $k=6$. Our calculated results are innovative and a new addition to the study of singularity theory.
Copyright:
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