AI 中文总结
本文确定了特征 p≥5 的 Cartan 型受限单李代数受限单模的一阶扩张群，回答了 Benkart-Feldvoss 问题，并通过有限受限上同调复形计算扩张，给出了例外模的闭式公式，且确定了特征七下 S(3;1)^(1) 的箭图。
AI 中文摘要
我们确定了在特征 $p\ge5$ 的代数闭域上，Cartan 型受限单李代数的受限单模之间的一阶扩张群，这回答了 Benkart 和 Feldvoss 问题中的受限情形。由抛物诱导模产生的扩张通过分级抛物线的有限受限上同调复形计算，对于例外模，我们在所有四个族中获得了闭式公式。平凡模和任何非平凡例外单模都没有自扩张。在特征七中，我们确定了 $S(3;1)^{(1)}$ 的箭图，它有 49 个顶点和 566 条箭头。
英文摘要
We determine the first extension groups between simple restricted modules for the restricted simple Lie algebras of Cartan type over an algebraically closed field of characteristic $p\ge5$, which answers the restricted case of a problem of Benkart and Feldvoss. Extensions from parabolically induced modules are computed by finite restricted-cohomology complexes of the grading parabolic, and for the exceptional modules we obtain closed formulas in all four families. Neither the trivial module nor any nontrivial exceptional simple has self-extensions. In characteristic seven we determine the quiver of $S(3;1)^{(1)}$, which has 49 vertices and 566 arrows.
Comments67 pages
