Orthogonal involution on algebras of degree 16 and the Killing form of E8

Details Orthogonal involution on algebras of degree 16 and the Killing form of E8 Orthogonal involution on algebras of degree 16 and the Killing form of E8

2007-12-26

arxiv.org/abs/0712.4133v2

in "Quadratic forms--algebra, arithmetic, and geometry", Contemporary Math., vol. 493 (2009), 131-162

Mathematics

Rings and Algebras

Small revisions since v1

Scientific paper

We exploit various inclusions of algebraic groups to give a new construction of groups of type E8, determine the Killing forms of the resulting E8's, and define an invariant of central simple algebras of degree 16 with orthogonal involution "in I^3", equivalently, groups of type D8 with a half-spin representation defined over the base field. The determination of the Killing form is done by restricting the adjoint representation to various twisted forms of PGL2 and requires very little computation. An appendix by Kirill Zainoulline contains a type of "index reduction" result for groups of type D.

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