On the $1$-cohomology of $\mathrm{SL}(n,{\mathbb K})$ on the dual of its adjoint module

TL;DR

This paper provides a clear proof that the first cohomology group of the special linear group over the dual of its adjoint module is isomorphic to the derivations of the field, covering all cases including previously unresolved ones.

Contribution

It offers a straightforward proof of the isomorphism for all n > 3, including the exceptional cases where the field size is 2 or 4 and n is even.

Findings

Established the isomorphism for all n > 3.

Resolved the cases where |K| ∈ {2, 4} and n is even.

Unified the proof approach for different field sizes.

Abstract

Given a field $\mathbb K$, for any $n\geq 3$ the first cohomology group $H^1(G_n,A^*_n)$ of the special linear group $G_n = \mathrm{SL}(n,{\mathbb K})$ over the dual $A^*_n$ of its adjoint module $A_n$ is isomorphic to the space $\mathrm{Der}({\mathbb K})$ of the derivations of $\mathbb K$, except possibly when $|{\mathbb K}| \in \{2, 4\}$ and $n$ is even. This fact is stated by S. Smith and H. V\"{o}lklein in their paper "A geometric presentation for the adjont module of $\mathrm{SL}_3(k)$" (J. Algebra 127 (1989), 127--138). They claim that when $|{\mathbb K}| > 9$ this fact follows from the main result of V\"{o}lklein's paper "The 1-cohomology of the adjoint module of a Chevalley group" (Forum Math. 1 (1989), 1--13), but say nothing that can help the reader to deduce it from that result. When $|{\mathbb K}| \leq 9$ they obtain the isomorphism $H^1(G_n,A^*_n) \cong \mathrm{Der}({\mathbb K})$ by means of other results from homological algebra, which however miss the case $|{\mathbb K}| \in\{2, 4\}$ with $n $ even. In the present paper we shall provide a straightforward proof of the isomorphism $H^1(G_n,A^*_n) \cong \mathrm{Der}({\mathbb K})$ under the hypothesis $n > 3$. Our proof also covers the above mentioned missing case.