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arXiv · 2610.08184

Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type

Chao Ma
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Abstract

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.

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Chao Ma. 2026-10-06. Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type. https://arxiv.org/abs/2610.08184

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