AI 中文总结
本文确定了Aut($\boldsymbol{\reals}^2$)李代数的所有正则Borel子代数，明确其导出长度为2或3，还给出了正则Abel与可解子代数的组合描述及同构类。
AI 中文摘要
本文中，我们找到了Lie(Aut($\boldsymbol{\reals}^2$))的所有正则Borel子代数，即关于标准$\boldsymbol{\reals}^2$-分次由齐次导子生成的极大可解子代数。由此可知，正则Borel子代数的导出长度为2或3。我们还描述了这类子代数的同构类，在此过程中，我们给出了Lie(Aut($\boldsymbol{\reals}^2$))的正则Abel子代数和正则可解子代数的组合描述。
英文摘要
In this paper, we find all regular Borel subalgebras of Lie(Aut($\mathbb{A}^2$)), i.e., maximal solvable subalgebras generated by homogeneous derivations with respect to the standard $\mathbb{Z}^2$-grading. It follows that a regular Borel subalgebra has derived length $2$ or $3$. We also describe isomorphism classes of such subalgebras. On the way, we give a combinatorial description of the regular Abelian subalgebras and the regular solvable subalgebras of Lie(Aut($\mathbb{A}^2$)).
Comments28 pages
