Theorems · Theorem · nonassociative algebras

RootPairing.rootForm_restrict_nondegenerate_of_isAnisotropic

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Fintype ι] [inst_1 : AddCommGroup M]
  [inst_2 : AddCommGroup N] [inst_3 : Field R] [inst_4 : Module R M] [inst_5 : Module R N] (P : RootPairing ι R M N)
  [P.IsAnisotropic], LinearMap.Nondegenerate (P.RootForm.restrict (P.rootSpan R))

See also RootPairing.rootForm_restrict_nondegenerate_of_ordered. Note that this applies to crystallographic root systems in characteristic zero via RootPairing.instIsAnisotropicOfIsCrystallographic.

Defined in: Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate
Cited by: 0 results in Mathlib
Foundations: Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Fintype AddCommGroup AddCommGroup Field Module Module RootPairing.IsAnisotropic

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Modulestatement and proof · cited by 20,661
RingHom.idstatement · cited by 18,349
AddCommGroupstatement and proof · cited by 12,871
Fintypestatement and proof · cited by 7,736
Fieldstatement and proof · cited by 7,404
Submodulestatement · cited by 7,192
RootPairingstatement and proof · cited by 710
RootPairing.rootSpanstatement · cited by 55
RootPairing.RootFormstatement · cited by 41
LinearMap.Nondegeneratestatement · cited by 35
RootPairing.IsAnisotropicstatement and proof · cited by 31
LinearMap.BilinForm.restrictstatement · cited by 24

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