From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

We construct Lie algebras arising from cubic norm pairs over arbitrary commutative base rings. Such Lie algebras admit a grading by a root system of type $G_2$, and when the cubic norm pair is a cubic Jordan matrix algebra, the $G_2$-grading can be further refined to an $F_4$-grading. We then use these Lie algebras and their gradings to construct corresponding root graded groups. Along the way, we produce many results providing detailed information about the structure of these Lie algebras and groups.

💡 Research Summary

The paper develops a comprehensive theory linking cubic norm pairs over arbitrary commutative rings to Lie algebras and groups graded by the root systems G₂ and F₄. The authors begin by recalling the notion of root‑graded Lie algebras and root‑graded groups, emphasizing that their definitions work over general rings rather than just fields. They then introduce cubic norm pairs (J, J′), a “twin” version of the more familiar cubic norm structures, and systematically develop their basic identities, traces, norms, and multiplication rules, largely following the recent monograph GPR24 and earlier work of Faulkner and Smet.

Using a cubic norm pair, the authors first construct the classical Tits–Kantor–Koecher (TKK) Lie algebra and then modify its 0‑graded component to obtain a larger Lie algebra L(J, J′) that carries a G₂‑grading. The grading is explicit: for each root α of G₂ the corresponding root space Lα is isomorphic either to the base ring k, to J, or to J′. The main structural results (Theorems 4.30, 5.15, 6.25) show that this construction is functorial with respect to surjective homotopies of cubic norm pairs, and that L is simple precisely when k is a field and the trace form of (J, J′) is non‑degenerate.

The next major step is to pass from the Lie algebra to a group. The authors define, for each root α, an α‑parameterization and an α‑exponential automorphism expα : Lα → Aut(L). Because the base ring may not invert 2 or 3, they introduce “higher Leibniz rules” to control the combinatorics of repeated brackets, and they verify the required identities with extensive computer assistance (Appendix A). They also construct α‑Weyl elements (reflections) and prove that these satisfy the usual Weyl group action on the root groups. Consequently, the subgroups Uα := expα(Lα) generate a G₂‑graded group G(J) inside Aut(L), and the group satisfies all axioms of a root‑graded group (commutator relations, existence of Weyl elements, etc.).

Having established the G₂‑graded theory, the authors turn to the refinement to an F₄‑grading. They focus on the special class of cubic norm pairs that arise from cubic Jordan matrix algebras J = Her(C, Γ), where C is a multiplicative conic alternative algebra (e.g., a composition algebra) over k and Γ is an invertible diagonal matrix. By intersecting three carefully chosen 5‑gradings of the G₂‑graded Lie algebra, they obtain an F₄‑grading on the same underlying Lie algebra L(J, J). In this refined grading, the long roots have root spaces isomorphic to k, while the short roots have root spaces isomorphic to C. The authors then lift the G₂‑graded group G(J) to an F₄‑graded group by defining exponential maps for the short roots via conjugation with suitable Weyl elements. Verifying the Weyl element action on all 48 root groups again requires computer assistance, but the result is a fully fledged F₄‑graded group.

The final main theorem (Theorem 11.15) states that for every multiplicative conic alternative algebra C over any commutative ring k, there exists an F₄‑graded group whose root groups are parametrized by C (short roots) and k (long roots). This provides a converse existence result to the classification of F₄‑graded groups given in earlier work (Wieber 2024). Moreover, the paper shows that every cubic norm structure yields a G₂‑graded group, although groups coming from general cubic norm pairs may lack Weyl elements for the short roots.

In summary, the paper achieves three intertwined goals: (1) it develops the theory of cubic norm pairs over arbitrary rings; (2) it constructs G₂‑graded Lie algebras and groups from these pairs, with explicit formulas for brackets, exponentials, and Weyl elements; (3) it refines the construction to obtain F₄‑graded Lie algebras and groups when the pair comes from a cubic Jordan matrix algebra, thereby establishing the existence of F₄‑graded groups for all multiplicative conic alternative algebras. The work extends the landscape of root‑graded structures beyond fields, introduces new technical tools (higher Leibniz rules, extensive computer verification), and opens avenues for further exploration of non‑classical grading phenomena in algebraic groups and Lie theory.