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arXiv · 2606.26791

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

Seidon Alsaody·Jari Desmet
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Abstract

Over any commutative ring containing $\tfrac16$, 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.

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Seidon Alsaody, Jari Desmet. 2026-06-25. Groups of type $\mathrm{E}_8$ over rings via TKK-algebras and their extremal elements. https://arxiv.org/abs/2606.26791

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