Theorems · Theorem · nonassociative algebras

LieModule.toEnd_pow_lie

∀ (R : Type u) {L : Type v} {M : Type w} [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] (x y : L)
  (z : M) (n : ℕ),
  ((LieModule.toEnd R L M) x ^ n) ⁅y, z⁆ =
    ∑ ij ∈ Finset.HasAntidiagonal.antidiagonal n,
      n.choose ij.1 • ⁅((LieAlgebra.ad R L) x ^ ij.1) y, ((LieModule.toEnd R L M) x ^ ij.2) z⁆

Defined in

Mathlib.Algebra.Lie.OfAssociative

Cited by

1 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModule