Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Semilinear automorphisms of reductive algebraic groups   Let $ G $ be a connected reductive algebraic group over a field $ k $. We
study the group of semilinear automorphisms Aut($ G\to $Spec $k$) consisting of
algebraic automorphisms of $ G $ over automorphisms of $ k $. We focus on the
exact sequence $ 1\to $Aut $G\to $Aut ($ G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1
$. When $G$ is quasi-split, we show that Aut$_{G}(k)$ is isomorphic to
Aut$_{\mathcal{R}(G)}(k)$, where $\mathcal{R}(G)$ denotes the scheme of based
root datum of $G$. Furthermore, the exact sequence $ 1\to $Aut $G\to $Aut ($
G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1 $ splits if and only if the exact
sequence $ 1\to \text{Aut }\mathcal{R}(G) \to \text{Aut }(\mathcal{R}(G) \to
\text{Spec } k)\to \text{Aut}_{\mathcal{R}(G)}(k)\to 1 $ splits. As a
corollary, we get many examples of algebraic groups $ G $ over $ k $ whose
group of abstract automorphisms does not decompose as the semidirect product of
$ \text{Aut } G $ with $\text{Aut}_G(k) $. We also study the same questions for
inner forms of SL$_n $ over a local field.

26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat