[Paper Review] The Kneser-Tits conjecture for groups with Tits-index E_{8,2}^{66} over an arbitrary field

R. Parimala, Jean-Pierre Tignol+1 more|arXiv (Cornell University)|Aug 11, 2010

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper proves the Kneser-Tits conjecture for absolutely simple algebraic groups with Tits index $E_{8,2}^{66}$ over an arbitrary field by establishing that the group of similitude multipliers of a 12-dimensional anisotropic quadratic form with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions where the form becomes hyperbolic. The key result confirms that such groups are generated by their root groups, verifying the conjecture in this case via Clifford algebras, R-equivalence, and quadrangular algebras.

ABSTRACT

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is E_{8,2}^{66}, then G is generated by its root groups, as predicted by the Kneser-Tits conjecture.

Motivation & Objective

• To verify the Kneser-Tits conjecture for algebraic groups with Tits index $E_{8,2}^{66}$ over an arbitrary field.
• To establish that the group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions where the form becomes hyperbolic.
• To show that such groups are generated by their root groups, as predicted by the Kneser-Tits conjecture.
• To extend the proof using R-triviality of $\mathrm{PGO}_+^+(q)$ in characteristic 0 and generalize to arbitrary characteristic.

Proposed method

• Uses the theory of Clifford algebras and the classification of quadratic forms with trivial discriminant and split Clifford invariants to analyze the structure of similitude multipliers.
• Applies the notion of quadrangular algebras from [22, 23] to relate the geometry of Moufang quadrangles to the algebraic structure of exceptional groups.
• Employs the norm principle for spinor norms and Knebusch’s norm theorem in the context of purely inseparable extensions to relate norms from extensions to the spinor norm group.
• Leverages R-triviality of $\mathrm{PGO}_+^+(q)$ in characteristic 0, proven via triality correspondence and hermitian forms over even Clifford algebras, to deduce the main result in arbitrary characteristic.
• Uses the correspondence between 8-dimensional quadratic forms of trivial discriminant and hermitian forms over the simple components of their even Clifford algebra to prove the case for $\dim q = 8$.
• Applies the theory of $R$-equivalence and the structure of $\mathrm{Hyp}_2(q)$ to show equality $G(q) = \mathrm{Hyp}_2(q)$ for anisotropic forms of dimension 8 and 12 under the given conditions.

Experimental results

Research questions

• RQ1 Does the group of $K$-rational points of an absolutely simple algebraic group with Tits index $E_{8,2}^{66}$ get generated by its root groups, as predicted by the Kneser-Tits conjecture?
• RQ2 Is the group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form with trivial discriminant and split Clifford invariant generated by norms from quadratic extensions where the form becomes hyperbolic?
• RQ3 Can the $R$-triviality of $\mathrm{PGO}_+^+(q)$ for such forms in characteristic 0 be used to deduce the Kneser-Tits conjecture in arbitrary characteristic?
• RQ4 How do quadrangular algebras and the structure of Clifford algebras contribute to proving the Kneser-Tits conjecture for $E_{8,2}^{66}$ groups?
• RQ5 What is the precise relationship between the spinor norm group $\mathrm{Sn}(h)$ of a skew-hermitian form and the group $\mathrm{Hyp}_2(q)$ of norms from quadratic extensions?

Key findings

• The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions $E/K$ such that $q_E$ is hyperbolic.
• For anisotropic 8-dimensional quadratic forms with trivial discriminant and Clifford invariant of index 2, the group $G(q)$ is equal to $\mathrm{Hyp}_2(q)$, the subgroup generated by $K^{\times 2}$ and norms from quadratic extensions where $q_E$ is hyperbolic.
• The group $\mathrm{PGO}_+^+(q)$ is $R$-trivial when $\mathrm{char}(K) \neq 2$, which implies that $G(q) = \mathrm{Hyp}_2(q)$ in this case.
• The Kneser-Tits conjecture holds for groups with Tits index $E_{8,2}^{66}$ over an arbitrary field, as such groups are generated by their root groups.
• The proof via $R$-triviality in characteristic 0 extends to arbitrary characteristic, providing a uniform verification of the conjecture.
• The correspondence between 8-dimensional quadratic forms and hermitian forms over the even Clifford algebra allows the reduction of the 12-dimensional case to the 8-dimensional case, confirming $G(q) = \mathrm{Hyp}_2(q)$ for $\dim q = 12$.