Theorems · Theorem · nonassociative algebras

LieModule.trace_toEnd_genWeightSpace

∀ (R : Type u_2) (L : Type u_3) (M : Type u_4) [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [inst_6 : LieModule R L M]
  [inst_7 : LieRing.IsNilpotent L] [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M]
  (χ : L → R) (x : L),
  (LinearMap.trace R ↥(LieModule.genWeightSpace M χ)) ((LieModule.toEnd R L ↥(LieModule.genWeightSpace M χ)) x) =
    Module.finrank R ↥(LieModule.genWeightSpace M χ) • χ x

Defined in

Mathlib.Algebra.Lie.Weights.Basic

Cited by

3 results in Mathlib

Foundations

Depth 123 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRing.IsNilpotentIsDomainIsPrincipalIdealRingModule.FreeModule.Finite