Weights and roots of Lie modules and Lie algebras with respect to Cartan subalgebras

Given a Lie algebra L which is not necessarily nilpotent, it may be useful to study its representations by restricting them to a nilpotent subalgebra (e.g., a Cartan subalgebra). In the particular case when we view L as a module over itself via the adjoint action, the weight spaces of L restricted to a nilpotent subalgebra are known as root spaces.

Basic definitions and properties of the above ideas are provided in this file.

Main definitions

• LieModule.IsWeight
• LieAlgebra.rootSpace
• LieAlgebra.IsRoot
• LieAlgebra.rootSpaceWeightSpaceProduct
• LieAlgebra.rootSpaceProduct
• LieAlgebra.zeroRootSubalgebra_eq_iff_is_cartan

noncomputable def LieModule.IsWeight {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ : LieAlgebra.LieCharacter R ↥H) : Prop

Given a Lie module M of a Lie algebra L, a weight of M with respect to a nilpotent subalgebra H ⊆ L is a Lie character whose corresponding weight space is non-empty.

Equations

• LieModule.IsWeight H M χ = (LieModule.weightSpace M ⇑χ ≠ ⊥)

theorem LieModule.isWeight_zero_of_nilpotent {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Nontrivial M] [LieAlgebra.IsNilpotent R L] [LieModule.IsNilpotent R L M] : LieModule.IsWeight ⊤ M 0

For a non-trivial nilpotent Lie module over a nilpotent Lie algebra, the zero character is a weight with respect to the ⊤ Lie subalgebra.

@[inline, reducible]

noncomputable abbrev LieAlgebra.rootSpace {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ : ↥H → R) : LieSubmodule R (↥H) L

Given a nilpotent Lie subalgebra H ⊆ L, the root space of a map χ : H → R is the weight space of L regarded as a module of H via the adjoint action.

Equations

• LieAlgebra.rootSpace H χ = LieModule.weightSpace L χ

theorem LieAlgebra.zero_rootSpace_eq_top_of_nilpotent {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] [LieAlgebra.IsNilpotent R L] : LieAlgebra.rootSpace ⊤ 0 = ⊤

@[inline, reducible]

noncomputable abbrev LieAlgebra.IsRoot {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ : LieAlgebra.LieCharacter R ↥H) : Prop

A root of a Lie algebra L with respect to a nilpotent subalgebra H ⊆ L is a weight of L, regarded as a module of H via the adjoint action.

Equations

• LieAlgebra.IsRoot H χ = (χ ≠ 0 ∧ LieModule.IsWeight H L χ)

@[simp]

theorem LieAlgebra.rootSpace_comap_eq_weightSpace {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ : ↥H → R) : LieSubmodule.comap (LieSubalgebra.incl' H) (LieAlgebra.rootSpace H χ) = LieModule.weightSpace (↥H) χ

theorem LieAlgebra.lie_mem_weightSpace_of_mem_weightSpace {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] {H : LieSubalgebra R L} [LieAlgebra.IsNilpotent R ↥H] {M : Type u_3} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ : ↥H → R} {χ₂ : ↥H → R} {x : L} {m : M} (hx : x ∈ LieAlgebra.rootSpace H χ₁) (hm : m ∈ LieModule.weightSpace M χ₂) : ⁅x, m⁆ ∈ LieModule.weightSpace M (χ₁ + χ₂)

noncomputable def LieAlgebra.rootSpaceWeightSpaceProductAux (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] {χ₁ : ↥H → R} {χ₂ : ↥H → R} {χ₃ : ↥H → R} (hχ : χ₁ + χ₂ = χ₃) : ↥↑(LieAlgebra.rootSpace H χ₁) →ₗ[R] ↥↑(LieModule.weightSpace M χ₂) →ₗ[R] ↥↑(LieModule.weightSpace M χ₃)

Auxiliary definition for rootSpaceWeightSpaceProduct, which is close to the deterministic timeout limit.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def LieAlgebra.rootSpaceWeightSpaceProduct (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ₁ : ↥H → R) (χ₂ : ↥H → R) (χ₃ : ↥H → R) (hχ : χ₁ + χ₂ = χ₃) : TensorProduct R ↥↑(LieAlgebra.rootSpace H χ₁) ↥↑(LieModule.weightSpace M χ₂) →ₗ⁅R,↥H⁆ ↥↑(LieModule.weightSpace 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.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LieAlgebra.coe_rootSpaceWeightSpaceProduct_tmul (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (M : Type u_3) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (χ₁ : ↥H → R) (χ₂ : ↥H → R) (χ₃ : ↥H → R) (hχ : χ₁ + χ₂ = χ₃) (x : ↥(LieAlgebra.rootSpace H χ₁)) (m : ↥(LieModule.weightSpace M χ₂)) : ↑((LieAlgebra.rootSpaceWeightSpaceProduct R L H M χ₁ χ₂ χ₃ hχ) (x ⊗ₜ[R] m)) = ⁅↑x, ↑m⁆

noncomputable def LieAlgebra.rootSpaceProduct (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ₁ : ↥H → R) (χ₂ : ↥H → R) (χ₃ : ↥H → R) (hχ : χ₁ + χ₂ = χ₃) : TensorProduct R ↥↑(LieAlgebra.rootSpace H χ₁) ↥↑(LieAlgebra.rootSpace H χ₂) →ₗ⁅R,↥H⁆ ↥↑(LieAlgebra.rootSpace H χ₃)

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

Equations

• LieAlgebra.rootSpaceProduct R L H χ₁ χ₂ χ₃ hχ = LieAlgebra.rootSpaceWeightSpaceProduct R L H L χ₁ χ₂ χ₃ hχ

@[simp]

theorem LieAlgebra.rootSpaceProduct_def (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] : LieAlgebra.rootSpaceProduct R L H = LieAlgebra.rootSpaceWeightSpaceProduct R L H L

theorem LieAlgebra.rootSpaceProduct_tmul (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (χ₁ : ↥H → R) (χ₂ : ↥H → R) (χ₃ : ↥H → R) (hχ : χ₁ + χ₂ = χ₃) (x : ↥(LieAlgebra.rootSpace H χ₁)) (y : ↥(LieAlgebra.rootSpace H χ₂)) : ↑((LieAlgebra.rootSpaceProduct R L H χ₁ χ₂ χ₃ hχ) (x ⊗ₜ[R] y)) = ⁅↑x, ↑y⁆

noncomputable def LieAlgebra.zeroRootSubalgebra (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] : LieSubalgebra R L

Given a nilpotent Lie subalgebra H ⊆ L, the root space of the zero map 0 : H → R is a Lie subalgebra of L.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LieAlgebra.coe_zeroRootSubalgebra (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] : (LieAlgebra.zeroRootSubalgebra R L H).toSubmodule = ↑(LieAlgebra.rootSpace H 0)

theorem LieAlgebra.mem_zeroRootSubalgebra (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (x : L) : x ∈ LieAlgebra.zeroRootSubalgebra R L H ↔ ∀ (y : ↥H), ∃ (k : ℕ), ((LieModule.toEndomorphism R (↥H) L) y ^ k) x = 0

theorem LieAlgebra.toLieSubmodule_le_rootSpace_zero (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] : LieSubalgebra.toLieSubmodule H ≤ LieAlgebra.rootSpace H 0

theorem LieAlgebra.le_zeroRootSubalgebra (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] : H ≤ LieAlgebra.zeroRootSubalgebra R L H

@[simp]

theorem LieAlgebra.zeroRootSubalgebra_normalizer_eq_self (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] : LieSubalgebra.normalizer (LieAlgebra.zeroRootSubalgebra R L H) = LieAlgebra.zeroRootSubalgebra R L H

theorem LieAlgebra.is_cartan_of_zeroRootSubalgebra_eq (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] (h : LieAlgebra.zeroRootSubalgebra R L H = H) : LieSubalgebra.IsCartanSubalgebra H

If the zero root subalgebra of a nilpotent Lie subalgebra H is just H then H is a Cartan subalgebra.

When L is Noetherian, it follows from Engel's theorem that the converse holds. See LieAlgebra.zeroRootSubalgebra_eq_iff_is_cartan

@[simp]

theorem LieAlgebra.zeroRootSubalgebra_eq_of_is_cartan (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieSubalgebra.IsCartanSubalgebra H] [IsNoetherian R L] : LieAlgebra.zeroRootSubalgebra R L H = H

theorem LieAlgebra.zeroRootSubalgebra_eq_iff_is_cartan (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R ↥H] [IsNoetherian R L] : LieAlgebra.zeroRootSubalgebra R L H = H ↔ LieSubalgebra.IsCartanSubalgebra H

@[simp]

theorem LieAlgebra.rootSpace_zero_eq (R : Type u_1) (L : Type u_2) [CommRing R] [LieRing L] [LieAlgebra R L] (H : LieSubalgebra R L) [LieSubalgebra.IsCartanSubalgebra H] [IsNoetherian R L] : LieAlgebra.rootSpace H 0 = LieSubalgebra.toLieSubmodule H