Automorphism group schemes of special simple Jordan pairs of types I and IV
Diego Aranda-Orna, Alberto Daza-García
TL;DR
This work determines the automorphism group schemes of finite-dimensional simple Jordan pairs of types I and IV, along with related Jordan triple systems, over fields with $\mathrm{char}\,\mathbb{F}\neq 2$. The authors explicitly identify the automorphism group schemes: for type IV, ${\mathbf{Aut}}(\mathcal{V}^{(IV)}(W,b))\simeq {\mathbf{GO}}(W,b)$ and ${\mathbf{Aut}}(\mathcal{T}^{(IV)}(V,b))\simeq {\mathbf O}(V,b)\times {\boldsymbol{\mu}}_2$, leading to ${\mathbf{Aut}}(\mathcal{T})\simeq {\mathbf{Aut}}(J)\times {\boldsymbol{\mu}}_2$; for type I, the automorphism group schemes are described by the central-product of general linear groups, ${\mathbf{Aut}}(\mathcal{V}^{(I)}_n)\simeq ( {\mathbf{GL}}_n \otimes_{ {\mathbf{G}}_m } {\mathbf{GL}}_n ) \rtimes {\boldsymbol{\mu}}_2^{(+)}$, with corresponding expressions for the rectangular and square cases. The paper also shows that certain triple systems, though inducing isomorphic Jordan pairs, have non-isomorphic automorphism group schemes, providing an independent proof of nonisomorphism results in S85 via group-scheme data. The results refine the understanding of gradings and symmetries in these algebraic structures and connect automorphism groups to familiar classical groups via explicit central-product and orthogonal-similarity constructions.
Abstract
In this work, the automorphism group schemes of finite-dimensional simple Jordan pairs of types I and IV, and of some Jordan triple systems related to them, are determined. We assume $\mathrm{char}(\mathbb{F}) \neq 2$ for the base field $\mathbb{F}$.