Albert Algebras over Rings and Related Torsors

Abstract

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to non-isomorphic structures. We begin by showing that isotopes of Albert algebras are obtained as twists by a certain ${\mathrm{F}}_4$-torsor with total space a group of type ${\mathrm{E}}_6$, and using this, that Albert algebras over rings in general admit non-isomorphic isotopes, even in the split case as opposed to the situation over fields. We then consider certain ${\mathrm{D}}_4$-torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is non-trivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are non-isomorphic and have non-isometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

arXiv commentIn the second version, some theorems have been strengthened

References in corpus (1)

• Octonion algebras over rings are not determined by their norms

Cited by in corpus (1)

• Albert algebras over Z and other rings