Theorems · Theorem · nonassociative algebras

RootPairing.GeckConstruction.lie_h_e

∀ {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.e i⁆ =
    b.cartanMatrix i j • RootPairing.GeckConstruction.e i

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

Defined in

Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations

Cited by

2 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FiniteCommRingIsDomainCharZeroAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographicFintype