单李代数中的方程
Equations in simple Lie algebras
摘要：
给定一个生成的自由李代数的元素$P(X_1,...,X_d)$，对于任何李代数$g$，我们可以考虑导出多项式映射$P: g^d\to g$ .假设$K$是特征$\ne 2$的任意域，证明了如果$P$不是$sl(2,K)$中的恒等式，则该映射对任意Chevalley代数$g$都是占优的.这个结果可以看作是Borel关于连通半单代数群上词映射的支配性定理的一个弱无穷小对应。我们证明了恩格尔单项式$[[[X,Y],Y],...,Y]$，更一般地说，为他们的线性组合，这个地图是，此外，满射到一组非中心元素的$g$提供的地面场$K$是足够大的，并表明，为单单项式的大程度的图像，这个地图包含没有非零中心元素。我们还讨论了这些结果的后果结合矩阵代数的多项式映射。
英文摘要：
Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra, for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove that if $P$ is not an identity in $sl(2,K)$, then this map is dominant for any Chevalley algebra $g$. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominancy of the word map on connected semisimple algebraic groups. We prove that for the Engel monomials $[[[X,Y],Y],...,Y]$ and, more generally, for their linear combinations, this map is, moreover, surjective onto the set of noncentral elements of $g$ provided that the ground field $K$ is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
