AI 中文总结
本文研究上三角矩阵代数的微分恒等式，为微分指数为3的极小$L$-簇分类奠基，证明相关作用可约化，确定$UT_3$的微分恒等式与余维序列，且$k \geq 3$时的$UT_k$生成的$L$-簇必包含带对应作用的$UT_3$。
AI 中文摘要
设$L$是特征为零的域$F$上，通过导子作用在结合代数$A$上的李代数。$A$关于该作用满足的多项式恒等式称为微分恒等式，或称$L$-恒等式。本文研究$k \times k$上三角矩阵代数$UT_k$的微分恒等式，并为微分指数为3的极小$L$-簇的分类迈出第一步。我们首先证明，只要$UT_k$生成具有导子的极小代数簇，$L$-作用就可替换为其半单部分，更准确地说，只需考虑由对角元诱导的内导子。随后，我们将该约化应用于$UT_3$，并对$UT_3$上的每一类此类作用，明确确定微分恒等式的$T_L$-理想及对应的微分余维数序列。最后，我们证明，由$UT_k$（$k \geq 3$）生成的每一个$L$-簇，都包含带有上述某类$L$-作用的$UT_3$。
英文摘要
Let $L$ be a Lie algebra acting by derivations on an associative algebra $A$ over a field $F$ of characteristic zero. The polynomial identities satisfied by $A$ with respect to this action are called differential identities, or $L$-identities. In this paper, we study the differential identities of the algebra $UT_k$ of $k\times k$ upper triangular matrices and take a first step toward the classification of minimal $L$-varieties of differential exponent $3$. We first prove that, whenever $UT_k$ generates a minimal variety of algebras with derivations, the $L$-action can be replaced by its semisimple part. More precisely, it is enough to consider inner derivations induced by diagonal elements. We then apply this reduction to $UT_3$ and explicitly determine the $T_L$-ideal of differential identities and the corresponding differential codimension sequence for every such action on $UT_3$. Finally, we show that every $L$-variety generated by $UT_k$, with $k\geq 3$, contains $UT_3$ endowed with one of these $L$-actions.
Comments28 pages
