Theorems · Theorem · nonassociative algebras

RootPairing.GeckConstruction.trace_toEnd_eq_zero

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : IsDomain R]
  [inst_2 : CharZero R] [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : AddCommGroup N] [inst_6 : Module R N]
  {P : RootPairing ι R M N} [inst_7 : P.IsCrystallographic] [P.IsReduced] {b : P.Base} [inst_9 : Fintype ι]
  [inst_10 : DecidableEq ι] (x : ↥(RootPairing.GeckConstruction.lieAlgebra b)),
  (LinearMap.trace R (↥b.support ⊕ ι → R))
      ((LieModule.toEnd R (↥(RootPairing.GeckConstruction.lieAlgebra b)) (↥b.support ⊕ ι → R)) x) =
    0

This is the main result of lemma 4.1 from [Geck](Geck2017).

Defined in

Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple

Cited by

0 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainCharZeroAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographicRootPairing.IsReducedFintypeDecidableEq