Theorems · Definition · nonassociative algebras

LieAlgebra.rootSpaceWeightSpaceProduct

(R : Type u_1) →
  (L : Type u_2) →
    [inst : CommRing R] →
      [inst_1 : LieRing L] →
        [inst_2 : LieAlgebra R L] →
          (H : LieSubalgebra R L) →
            [inst_3 : LieRing.IsNilpotent ↥H] →
              (M : Type u_3) →
                [inst_4 : AddCommGroup M] →
                  [inst_5 : Module R M] →
                    [inst_6 : LieRingModule L M] →
                      [inst_7 : LieModule R L M] →
                        (χ₁ χ₂ χ₃ : ↥H → R) →
                          χ₁ + χ₂ = χ₃ →
                            TensorProduct R ↥(LieAlgebra.rootSpace H χ₁) ↥(LieModule.genWeightSpace M χ₂) →ₗ⁅R,↥H⁆
                              ↥(LieModule.genWeightSpace M χ₃)

Given a nilpotent Lie subalgebra H ⊆ L together with χ₁ χ₂ : H → R, there is a natural R-bilinear product of root vectors and weight vectors, compatible with the actions of H.

Defined in

Mathlib.Algebra.Lie.Weights.Cartan

Cited by

6 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraLieRing.IsNilpotentAddCommGroupModuleLieRingModuleLieModule