An estimate of the number of parameters defining an $n$-dimensional algebra

Consider an arbitrary family of nonisomorphic $n$-dimensional complex Lie algebras (respectively, associative algebras, commutative algebras) that depends continuously on a certain set of parameters $t_1,\dots,t_N\in\mathbf C$. The asymptotics is obtained for the largest number $N$ of parameters possible when $n$ is fixed: $\frac 2{27}n^3+O(n^{8/3})$, $\frac 4{27}n^3+O(n^{8/3})$, $\frac 2{27}n^3+O(n^{8/3})$ respectively. A decomposition into irreducible components is also studied for the algebraic variety $\text{Lie}_n$ of all possible Lie algebra structures on the linear space $\mathbf C^n$.
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