[Paper Review] Automorphisms of Chevalley groups of type $B_l$ over local rings with 1/2

E. I. Bunina|ArXiv.org|Nov 22, 2009

Finite Group Theory Research34 references3 citations

TL;DR

This paper proves that every automorphism of a Chevalley group of type $B_l$ ($l \geq 2$) over a commutative local ring containing $1/2$ is standard—i.e., a composition of ring automorphisms, inner automorphisms, and central automorphisms. The proof relies on extending the ring to include square roots, constructing torus elements that normalize the elementary subgroup, and analyzing the action of automorphisms on the group and its quotient by the center, ultimately showing that any automorphism is induced by a combination of ring, inner, and central automorphisms.

ABSTRACT

In the given paper we prove that every automorphism of a Chevalley group of type $B_l$, $l\geqslant 2$, over a commutative local ring with 1/2 is standard, i. e., it is a composition of ring, inner and central automorphisms.

Motivation & Objective

• To classify all automorphisms of Chevalley groups of type $B_l$ over commutative local rings containing $1/2$.
• To extend previous results on automorphisms of Chevalley groups over local rings to the $B_l$ root system.
• To establish that every automorphism is standard, i.e., a composition of ring, inner, and central automorphisms.
• To generalize methods from earlier works on $A_l$, $D_l$, $E_l$, $F_4$, and $G_2$ to the $B_l$ case.
• To prove that automorphisms of the full Chevalley group $G_{\pi}(\Phi,R)$ are induced by automorphisms of the elementary subgroup $E_{\pi}(\Phi,R)$, extended via torus elements and ring automorphisms.

Proposed method

• Construct an extension $S$ of the local ring $R$ by adjoining square roots of specific elements $r_1$ and $r_l$, enabling the definition of torus elements in $G_{\pi}(\Phi,S)$.
• Define torus elements $t_1$ and $t_l$ in $G_{\pi}(\Phi,S)$ using the Chevalley generators $h_{\alpha_i}(s)$, ensuring they normalize the elementary subgroup $E_{\pi}(\Phi,R)$.
• Use the fact that the quotient of the elementary Chevalley group by its center is isomorphic to the adjoint elementary group $E_{\mathrm{ad}}(\Phi,R)$, allowing reduction to known results on automorphisms of $E_{\mathrm{ad}}(\Phi,R)$.
• Lift ring automorphisms $\overline{\rho}$ from the quotient group to the full group $E_{\pi}(\Phi,R)$, preserving their action on the center.
• Construct a conjugation automorphism $\varphi_g$ via an element $g = te \in G_{\pi}(\Phi,S)$, where $e \in E_{\pi}(\Phi,R)$ and $t \in T_{\pi}(\Phi,S)$, to realize inner-like automorphisms.
• Define a composite automorphism $\psi = \varphi_{g^{-1}} \circ \rho^{-1} \circ \varphi$ that acts trivially on the elementary subgroup, proving it is central, hence trivial in the case of perfect groups.

Experimental results

Research questions

• RQ1 Are all automorphisms of Chevalley groups of type $B_l$ over local rings with $1/2$ standard, i.e., compositions of ring, inner, and central automorphisms?
• RQ2 Can the automorphism group of $E_{\pi}(\Phi,R)$ for $\Phi = B_l$ be fully described using lifts from the adjoint group $E_{\mathrm{ad}}(\Phi,R)$?
• RQ3 How can torus elements in $G_{\pi}(\Phi,S)$ be explicitly constructed over an extended ring $S$ to normalize the elementary subgroup and realize automorphisms?
• RQ4 To what extent do automorphisms of the full Chevalley group $G_{\pi}(\Phi,R)$ descend from automorphisms of the elementary subgroup $E_{\pi}(\Phi,R)$?
• RQ5 Is every automorphism of $G_{\pi}(\Phi,R)$ central when it acts trivially on the elementary subgroup $E_{\pi}(\Phi,R)$?

Key findings

• Every automorphism of a Chevalley group of type $B_l$ over a commutative local ring with $1/2$ is standard, i.e., a composition of ring, inner, and central automorphisms.
• The automorphism group of the elementary Chevalley group $E_{\pi}(\Phi,R)$ is generated by ring automorphisms and inner automorphisms induced by elements of $G_{\pi}(\Phi,S)$ for a suitable extension $S$ of $R$.
• The quotient of $E_{\pi}(\Phi,R)$ by its center is isomorphic to the adjoint elementary group $E_{\mathrm{ad}}(\Phi,R)$, and automorphisms of the latter lift to the former.
• Torus elements $t_1$ and $t_l$ in $G_{\pi}(\Phi,S)$ are explicitly constructed using square roots of $r_1$ and $r_l$, ensuring correct conjugation action on root elements.
• Any automorphism of $G_{\pi}(\Phi,R)$ that acts trivially on $E_{\pi}(\Phi,R)$ is central, and since $E_{\pi}(\Phi,R)$ is perfect, such automorphisms are trivial.
• The full Chevalley group $G_{\pi}(\Phi,R)$ is generated by its torus and elementary subgroup, so the automorphism structure is fully determined by its action on $E_{\pi}(\Phi,R)$ and the torus.