AI 中文总结
本文研究了在trace零矩阵上的Engel映射的满射性，证明了其在不同环上的满射条件与剩余域上的满射性相关，并展示了如何通过特定条件将元素表示为Engel映射的结果。
AI 中文摘要
在不同特征的字段上，非交换多项式的满射性已在李代数上被广泛研究。本文研究了在M₂(𝑂)上的Engel映射的满射性，其中𝑂是一个局部主理想环，其最大理想是完备的，并且其剩余域k的特征不等于2。我们证明了在M₂(𝑂)上由Engel多项式e_{m+1}(x, y) = [⋯[[x, y], y], ⋯, y]_{m+1次}诱导的(m+1)次Engel映射的像可以通过在M₂(k)上对应的Engel映射的像确定，对于m≥1。此外，我们证明了在sl₂(𝑂)上(m+1)次Engel映射是满射的当且仅当对应的sl₂(k)上的Engel映射是满射的。在某些温和的条件下，我们的结果表明，每个g∈sl₂(𝑂)都可以表示为g = e_{m+1}(h₁, h₂)，其中h₁, h₂都在sl₂(𝑂)中，且m≥1。
英文摘要
The surjectivity of various noncommutative polynomials has been studied extensively on Lie algebras over fields of different characteristics. In this article, we study the surjectivity of Engel Maps over trace zero matrices in $\mathrm{M}_2(\mathcal{O})$, where $\mathcal{O}$ is a local principal ideal ring complete with respect to its maximal ideal and has a residue field $k$ of characteristic $\neq 2$. We show that the image of $(m+1)$-th Engel map induced by the Engel polynomial $e_{m+1}(x, y) = [\cdots[[x, \underbrace{y], y], \dots, y]}_{m+1 \text{ times}}$ over $\mathrm{M}_2(\mathcal{O})$ can be determined by the image of the corresponding Engel map over $\mathrm{M}_2(k)$, for $m\geq 1$. Moreover, we prove that the $(m+1)$-th Engel map on $\mathfrak{sl}_2^{\circ}(\mathcal{O})=\left\{\ A\in \mathrm{M}_2(\mathcal{O})\ \mid\ \mathrm{tr}(A)=0\right\}$ is surjective if and only if the corresponding Engel map on $\mathfrak{sl}_2(k)$ is surjective. Under some mild condition on the residue field $k$, our results shows that every $g\in \mathfrak{sl}_2^{\circ}(\mathcal{O})$ can be expressed as $g = e_{m+1}(h_1, h_2)$ for $m\geq 1$, where both $h_1, h_2$ are in $\mathfrak{sl}_2^{\circ}(\mathcal{O})$.
CommentsVersion 2, 14 pages, Comments are welcome
