Integrable Generators of Polynomial Vector Fields for Complex Classical Lie Groups

Abstract

We consider in this paper complex Lie groups ${\mathrm{SL}}_n(\mathbb{C})$ for $n\ge 2$, ${\mathrm{SO}}_n(\mathbb{C})$ for $n\ge 5$, and ${\mathrm{Sp}}_{2n}(\mathbb{C})$ for $n\ge 1$. We show that for any such Lie group, the Lie algebra of polynomial vector fields can be generated by three $\mathbb{C}$-complete polynomial vector fields.

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