Groups of type ${\mathrm{E}}_8$ over rings via TKK-algebras and their extremal elements

Abstract

Over any commutative ring containing $\frac{1}{6}$, we study Lie algebras $L$ of type ${\mathrm{E}}_8$ that arise from the Tits--Kantor--Koecher (TKK) construction on a Brown algebra, and their twisted forms. We construct a smooth scheme $\mathbf{Y}$ of pairs of extremal elements in $L$. When $L$ arises from the TKK-construction, we express the automorphism group, of type ${\mathrm{E}}_8$, as an ${\mathrm{E}}_7$-torsor over $\mathbf{Y}$. We show that twisting by this torsor produces the graded isomorphism classes of those algebras isomorphic to $L$, and parametrize these classes by using $\mathbf{Y}$. We show that this torsor is non-trivial, yielding isomorphic Lie algebras of type ${\mathrm{E}}_8$ that are not graded isomorphic, as opposed to the behaviour over fields.

27 pages. Comments welcome!