有限域上有限维李代数的互极大图：三角形计数与结构不变量
Comaximal Graphs of finite-dimensional Lie algebras over finite fields: Triangle counts and structural invariants
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中文总结 AI 辅助
该文研究有限域上李代数的互极大图，得到三维李代数的三角形计数公式，拓展四维李代数的图结构分类，关联图论与李代数结构性质，提供新组合不变量。
中文摘要 AI 辅助
设$L$是域$F$上的有限维李代数，互极大图$\boldsymbol{\rm \u0393(L)}$的顶点为$L$的非真非零子代数，当两个子代数生成$L$时相邻；此前已对有限域上维数不超过3的李代数的该图结构完成分类。本文从两方面拓展该工作：其一，对$\boldsymbol{\rm \u0394_q}$上所有三维李代数，得到三角形数量$\boldsymbol{\rm t(\u0393(L))}$的显式公式；其二，将分类拓展到$\boldsymbol{\rm \u0394_q}$上的多个四维李代数族，包括阿贝尔代数、海森伯代数、幂零形代数及$\boldsymbol{\rm \u0394_q}$上的$\boldsymbol{\rm \u0393\u039B_2}$。还关联了$\boldsymbol{\rm \u0393(L)}$的图论性质（如完备性、弗拉蒂尼子代数的作用）与$L$的结构性质（包括超可解性），所得结果为有限域上有限维李代数提供了新的组合不变量。
英文摘要
Let $L$ be a finite-dimensional Lie algebra over a field $F$. The comaximal graph $Γ(L)$ has as vertices the proper nonzero subalgebras of $L$, two of them adjacent whenever they generate $L$; its structure was previously classified for Lie algebras of dimension at most 3 over finite fields. Here we extend that work in two directions. First, we obtain explicit formulas for the number of triangles $t(Γ(L))$ for every three-dimensional Lie algebra over $\F_q$. Second, we extend the classification to several four-dimensional families over $\mathbb{F}_q$, the abelian, Heisenberg, and filiform algebras, and $\mathfrak{gl}_2(\F_q)$. We also relate graph-theoretic properties of $Γ(L)$, such as completeness and the role of the Frattini subalgebra, to structural properties of $L$, including supersolvability. These results yield new combinatorial invariants for finite-dimensional Lie algebras over finite fields.
