Theorems · Theorem · nonassociative algebras

LieModule.shiftedGenWeightSpace.toEnd_eq

∀ (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] (χ : L → R) [inst_8 : LieModule.LinearWeights R L M] (x : L),
  (LieModule.toEnd R L ↥(LieModule.shiftedGenWeightSpace R L M χ)) x =
    (LieModule.shiftedGenWeightSpace.shift R L M χ).conj
      ((LieModule.toEnd R L ↥(LieModule.genWeightSpace M χ)) x - χ x • LinearMap.id)

Defined in

Mathlib.Algebra.Lie.Weights.Linear

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModuleLieRing.IsNilpotentLieModule.LinearWeights