A short construction of the Lie algebra $G_2(K)$ over fields $K$ of characteristic $2$

TL;DR

The paper addresses the explicit construction of the Lie algebra of type $G_2(K)$ over fields of characteristic $2$. It uses a geometric framework based on the generalized quadrangle $O_6^-(2)$, root bases $ Delta$, and Lie roots $R_ Delta$, together with a Weyl-group element $d$ of order $3$, to realize $G_2(K)$ as the centralizer $C_{D_L}(d)$ inside a $D_4$-type subalgebra of dimension $28$ that lies within an $E_6$-driven structure. The main contribution is an explicit, modular construction: identify 6 fixed Lie roots and 6 orbits under $d$, form two 12-dimensional spans $ igl\uon{S_X}igr}$, and show that $G=C_{H_L}(d)igoplusigl\uon{S_X}igr}$ is a closed Lie algebra of type $G_2$ with $ ext{dim}=14$, embedded in $D_L$. This provides a concrete, elementary realization of the exceptional type $G_2$ in characteristic $2$ and clarifies its embedding in $D_4$ and $E_6$-related structures, with potential implications for understanding Lie algebras in bad characteristics.

Abstract

The purpose of this paper is to give an explicit and elementary construction for the Lie algebras of type $G_2(K)$ of dimension 14, over the field K of characteristic 2. We say an elementary construction on the account that we use not more than little naive linear algebra notions.

Theorems & Definitions (14)

• Remark 1
• Definition 2
• Remark 3
• Proposition 4
• Remark 5
• Proposition 6
• Proposition 7
• Remark 8
• Theorem 9
• Proposition 10