[Paper Review] Classification of Certain Subgroups of G2

Andrew Fiori|arXiv (Cornell University)|Jan 14, 2015

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper provides a concrete classification of rational conjugacy classes of maximal tori and simply connected A2 subgroups in groups of type G2 over number fields and p-adic fields, using explicit geometric and algebraic structures of octonions. It establishes a bridge between concrete constructions and Galois cohomology, offering a systematic description that aligns with abstract cohomological classifications.

ABSTRACT

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type G2, focusing on the case of number fields and p-adic fields. In the same context we characterize the rational conjugacy classes of A2 subgroups of G2. Having obtained the concrete characterization, we then relate it to the more abstract characterization which can be given in terms of Galois cohomology. We note that these results on A2 subgroups were simultaneously and independently developed in the work of Hooda whereas the results on tori were simultaneously and independently developed in the work of Beli-Gille-Lee.

Motivation & Objective

• To provide a concrete, explicit classification of rational conjugacy classes of maximal tori in groups of type G2 over number fields and p-adic fields.
• To classify rational conjugacy classes of simply connected A2 subgroups in G2 over the same fields.
• To relate these concrete classifications to the abstract Galois cohomology framework already established in the literature.
• To offer a self-contained, accessible treatment with background material for non-experts, complementing existing advanced treatments.
• To lay foundational groundwork for extending such classifications to other exceptional groups like F4, E6, E7, and E8.

Proposed method

• Utilizes the algebraic structure of octonions over a field k to define the group G2 as the automorphism group of a Cayley algebra.
• Constructs maximal tori and A2 subgroups as stabilizers of specific elements in the octonion algebra over the algebraic closure.
• Applies Galois cohomology to classify forms of these subgroups, using the exact sequences involving normalizers and automorphism groups.
• Relies on Hilbert’s Theorem 90 to compute cohomology groups, particularly H¹(Gal(k̄/k), T̄) for tori T.
• Establishes bijections between isomorphism classes of forms and cohomology classes via the image and kernel conditions in Galois cohomology.
• Connects the geometric construction to the abstract cohomological classification via explicit cocycle lifting and splitting conditions.

Experimental results

Research questions

• RQ1 How can the rational conjugacy classes of maximal tori in G2 over number fields and p-adic fields be explicitly classified?
• RQ2 What is the concrete geometric and algebraic description of the rational conjugacy classes of A2 subgroups in G2?
• RQ3 How do these concrete classifications relate to the abstract Galois cohomology classification of such subgroups?
• RQ4 What role do étale algebras play in classifying forms of tori and their embeddings in G2?
• RQ5 Why do rational conjugacy and rational isomorphism of tori not coincide in this setting, and how is this reflected in cohomology?

Key findings

• The rational conjugacy classes of maximal tori in G2 over a number field or p-adic field are in natural bijection with isomorphism classes of certain quadratic and cubic étale algebras over the base field.
• The rational conjugacy classes of simply connected A2 subgroups in G2 are parametrized by the choice of a non-zero, norm-zero element in the octonion algebra over the algebraic closure, up to scalar multiplication.
• The classification of tori embedding in G2 is given by the image and kernel of a map in Galois cohomology: H¹(Gal(k̄/k), N_k( k̄ )) → H¹(Gal(k̄/k), Aut(T)) with kernel condition.
• The failure of rational conjugacy to coincide with rational isomorphism for tori is explained by the non-triviality of H¹(Gal(k̄/k), T̄), which is computed via Hilbert’s Theorem 90 and exact sequences.
• The results on A2 subgroups are shown to be equivalent to those in independent work by [8], and the torus results match those in [3], confirming consistency with existing cohomological frameworks.
• The cohomological description of forms of tori and A2 subgroups is fully aligned with the abstract classification via Galois cohomology, with explicit cocycle lifting conditions verified.