Theorems · Inductive type · nonassociative algebras

RootPairing.IsAnisotropic

{ι : Type u_1} →
  {R : Type u_2} →
    {M : Type u_3} →
      {N : Type u_4} →
        [Fintype ι] →
          [inst : AddCommGroup M] →
            [inst_1 : AddCommGroup N] →
              [inst_2 : CommRing R] → [inst_3 : Module R M] → [inst_4 : Module R N] → RootPairing ι R M N → Prop

We say a finite root pairing is anisotropic if there are no roots / coroots which have length zero w.r.t. the root / coroot forms. Examples include crystallographic pairings in characteristic zero RootPairing.instIsAnisotropicOfIsCrystallographic and pairings over ordered scalars. RootPairing.instIsAnisotropicOfLinearOrderedCommRing.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate

Cited by

31 results in Mathlib

Foundations

Depth 7 from the axioms · uses no axioms

Assumes

FintypeAddCommGroupAddCommGroupCommRingModuleModule