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Theorems · Theorem · nonassociative algebras

RootPairing.GeckConstruction.lie_h_f

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Finite ι] [inst_1 : CommRing R]
  [inst_2 : IsDomain R] [inst_3 : CharZero R] [inst_4 : AddCommGroup M] [inst_5 : Module R M] [inst_6 : AddCommGroup N]
  [inst_7 : Module R N] {P : RootPairing ι R M N} [inst_8 : P.IsCrystallographic] {b : P.Base} [inst_9 : Fintype ι]
  (i j : ↥b.support),
  ⁅RootPairing.GeckConstruction.h j, RootPairing.GeckConstruction.f i⁆ =
    -b.cartanMatrix i j • RootPairing.GeckConstruction.f i

Lemma 3.3 (b) from [Geck](Geck2017).

Defined in
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations
Cited by
1 results in Mathlib
Foundations
Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
FiniteCommRingIsDomainCharZeroAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographicFintype
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

RootPairing.GeckConstruction.isSl2Triple · cited by 0GeckConstruction.isSl2Tri…Module · cited by 20661ModuleCommRing · cited by 17173CommRingFinset · cited by 13712FinsetAddCommGroup · cited by 12871AddCommGroupFintype · cited by 7736FintypeMatrix · cited by 4303MatrixFinite · cited by 3029Finiteone_mul · cited by 2841one_mulIsDomain · cited by 2196IsDomainmul_assoc · cited by 1667mul_assocCharZero · cited by 932CharZeroRootPairing · cited by 710RootPairingneg_mul · cited by 654neg_mulBracket.bracket · cited by 642Bracket.bracketmul_neg · cited by 590mul_negGeckConstruction.lie_h_fCITED BYCITES

Cites34

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

  • Modulestatement and proof · cited by 20,661
  • CommRingstatement and proof · cited by 17,173
  • Finsetstatement · cited by 13,712
  • AddCommGroupstatement and proof · cited by 12,871
  • Fintypestatement and proof · cited by 7,736
  • Matrixstatement and proof · cited by 4,303
  • Finitestatement and proof · cited by 3,029
  • one_mulproof · cited by 2,841
  • IsDomainstatement and proof · cited by 2,196
  • mul_assocproof · cited by 1,667
  • CharZerostatement and proof · cited by 932
  • RootPairingstatement and proof · cited by 710

Cited by1

Results whose statement or proof uses this declaration.