Theorems · Definition · nonassociative algebras

RootPairing.IsG2.toEmbeddedG2

{ι : Type u_1} →
  {R : Type u_2} →
    {M : Type u_3} →
      {N : Type u_4} →
        [inst : CommRing R] →
          [inst_1 : AddCommGroup M] →
            [inst_2 : Module R M] →
              [inst_3 : AddCommGroup N] → [inst_4 : Module R N] → (P : RootPairing ι R M N) → [P.IsG2] → P.EmbeddedG2

By making an arbitrary choice of roots pairing to -3, we can obtain an embedded 𝔤₂ root system just from the knowledge that such a pairs exists.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.G2

Cited by

1 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleRootPairing.IsG2