Theorems · Theorem · nonassociative algebras

LieSubmodule.lieIdeal_oper_eq_tensor_map_range

∀ {R : Type u} [inst : CommRing R] {L : Type v} {M : Type w} [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]
  (I : LieIdeal R L) (N : LieSubmodule R L M),
  ⁅I, N⁆ = ((LieModule.toModuleHom R L M).comp (TensorProduct.LieModule.mapIncl I N)).range

A useful alternative characterisation of Lie ideal operations on Lie submodules. Given a Lie ideal I ⊆ L and a Lie submodule N ⊆ M, by tensoring the inclusion maps and then applying the action of L on M, we obtain morphism of Lie modules f : I ⊗ N → L ⊗ M → M. This lemma states that ⁅I, N⁆ = range f.

Defined in

Mathlib.Algebra.Lie.TensorProduct

Cited by

0 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraAddCommGroupModuleLieRingModuleLieModule