Theorems · Definition · nonassociative algebras

RootPairing.EmbeddedG2.ofPairingInThree

{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) →
                    [CharZero R] →
                      [inst_6 : P.IsCrystallographic] →
                        [P.IsReduced] → (long short : ι) → P.pairingIn ℤ long short = 3 → P.EmbeddedG2

A pair of roots which pair to +3 are also sufficient to distinguish an embedded 𝔤₂.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.G2

Cited by

2 results in Mathlib

Foundations

Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCharZeroRootPairing.IsCrystallographicRootPairing.IsReduced