Lie modules with linear weights

Given a Lie module M over a nilpotent Lie algebra L with coefficients in R, one frequently studies M via its weights. These are functions χ : L → R whose corresponding weight space LieModule.genWeightSpace M χ, is non-trivial. If L is Abelian or if R has characteristic zero (and M is finite-dimensional) then such χ are necessarily R-linear. However in general non-linear weights do exist. For example if we take:

• R: the field with two elements (or indeed any perfect field of characteristic two),
• L: sl₂ (this is nilpotent in characteristic two),
• M: the natural two-dimensional representation of L,

then there is a single weight and it is non-linear. (See remark following Proposition 9 of chapter VII, §1.3 in N. Bourbaki, Chapters 7--9.)

We thus introduce a typeclass LieModule.LinearWeights to encode the fact that a Lie module does have linear weights and provide typeclass instances in the two important cases that L is Abelian or R has characteristic zero.

Main definitions

• LieModule.LinearWeights: a typeclass encoding the fact that a given Lie module has linear weights, and furthermore that the weights vanish on the derived ideal.
• LieModule.instLinearWeightsOfCharZero: a typeclass instance encoding the fact that for an Abelian Lie algebra, the weights of any Lie module are linear.
• LieModule.instLinearWeightsOfIsLieAbelian: a typeclass instance encoding the fact that in characteristic zero, the weights of any finite-dimensional Lie module are linear.
• LieModule.exists_forall_lie_eq_smul: existence of simultaneous eigenvectors from existence of simultaneous generalized eigenvectors for Noetherian Lie modules with linear weights.

class LieModule.LinearWeights (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] : Prop

A typeclass encoding the fact that a given Lie module has linear weights, vanishing on the derived ideal.

• map_add (χ : L → R) : genWeightSpace M χ ≠ ⊥ → ∀ (x y : L), χ (x + y) = χ x + χ y
• map_smul (χ : L → R) : genWeightSpace M χ ≠ ⊥ → ∀ (t : R) (x : L), χ (t • x) = t • χ x
• map_lie (χ : L → R) : genWeightSpace M χ ≠ ⊥ → ∀ (x y : L), χ ⁅x, y⁆ = 0

def LieModule.Weight.toLinear (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] (χ : Weight R L M) : L →ₗ[R] R

A weight of a Lie module, bundled as a linear map.

Equations

• LieModule.Weight.toLinear R L M χ = { toFun := ⇑χ, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem LieModule.Weight.toLinear_apply (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] (χ : Weight R L M) (a : L) : (toLinear R L M χ) a = χ a

instance LieModule.Weight.instCoeLinearMap (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] : CoeOut (Weight R L M) (L →ₗ[R] R)

Equations

• LieModule.Weight.instCoeLinearMap R L M = { coe := LieModule.Weight.toLinear R L M }

instance LieModule.Weight.instLinearMapClass (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] : LinearMapClass (Weight R L M) R L R

@[simp]

theorem LieModule.Weight.apply_lie {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] {χ : Weight R L M} (x y : L) : χ ⁅x, y⁆ = 0

@[simp]

theorem LieModule.Weight.coe_coe {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] {χ : Weight R L M} : ⇑(toLinear R L M χ) = ⇑χ

@[simp]

theorem LieModule.Weight.coe_toLinear_eq_zero_iff {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] {χ : Weight R L M} : toLinear R L M χ = 0 ↔ χ.IsZero

theorem LieModule.Weight.coe_toLinear_ne_zero_iff {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] {χ : Weight R L M} : toLinear R L M χ ≠ 0 ↔ χ.IsNonZero

@[reducible, inline]

abbrev LieModule.Weight.ker {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] {χ : Weight R L M} : Submodule R L

The kernel of a weight of a Lie module with linear weights.

Equations

• LieModule.Weight.ker = LinearMap.ker (LieModule.Weight.toLinear R L M χ)

instance LieModule.instLinearWeightsOfIsLieAbelian (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsLieAbelian L] [NoZeroSMulDivisors R M] : LinearWeights R L M

For an Abelian Lie algebra, the weights of any Lie module are linear.

theorem LieModule.trace_comp_toEnd_genWeightSpace_eq (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M] [LieRing.IsNilpotent L] (χ : L → R) : ⇑(LinearMap.trace R ↥(genWeightSpace M χ) ∘ₗ ↑(toEnd R L ↥(genWeightSpace M χ))) = Module.finrank R ↥(genWeightSpace M χ) • χ

theorem LieModule.zero_lt_finrank_genWeightSpace {R : Type u_2} {L : Type u_3} {M : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M] [LieRing.IsNilpotent L] {χ : L → R} (hχ : genWeightSpace M χ ≠ ⊥) : 0 < Module.finrank R ↥(genWeightSpace M χ)

instance LieModule.instLinearWeightsOfCharZero (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M] [LieRing.IsNilpotent L] [CharZero R] : LinearWeights R L M

In characteristic zero, the weights of any finite-dimensional Lie module are linear and vanish on the derived ideal.

def LieModule.shiftedGenWeightSpace (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) : LieSubmodule R L M

A type synonym for the χ-weight space but with the action of x : L on m : genWeightSpace M χ, shifted to act as ⁅x, m⁆ - χ x • m.

Equations

• LieModule.shiftedGenWeightSpace R L M χ = LieModule.genWeightSpace M χ

instance LieModule.shiftedGenWeightSpace.instLieRingModuleSubtypeMemLieSubmodule (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) [LinearWeights R L M] : LieRingModule L ↥(shiftedGenWeightSpace R L M χ)

Equations

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

@[simp]

theorem LieModule.shiftedGenWeightSpace.coe_lie_shiftedGenWeightSpace_apply (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) [LinearWeights R L M] (x : L) (m : ↥(shiftedGenWeightSpace R L M χ)) : ↑⁅x, m⁆ = ⁅x, ↑m⁆ - χ x • ↑m

instance LieModule.shiftedGenWeightSpace.instSubtypeMemLieSubmodule (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) [LinearWeights R L M] : LieModule R L ↥(shiftedGenWeightSpace R L M χ)

def LieModule.shiftedGenWeightSpace.shift (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) : ↥(genWeightSpace M χ) ≃ₗ[R] ↥(shiftedGenWeightSpace R L M χ)

Forgetting the action of L, the spaces genWeightSpace M χ and shiftedGenWeightSpace R L M χ are equivalent.

Equations

• LieModule.shiftedGenWeightSpace.shift R L M χ = LinearEquiv.refl R ↥(LieModule.genWeightSpace M χ)

@[simp]

theorem LieModule.shiftedGenWeightSpace.shift_apply (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) (a : ↥(genWeightSpace M χ)) : (shift R L M χ) a = a

@[simp]

theorem LieModule.shiftedGenWeightSpace.shift_symm_apply (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) (a✝ : ↥(genWeightSpace M χ)) : (shift R L M χ).symm a✝ = a✝

theorem LieModule.shiftedGenWeightSpace.toEnd_eq (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) [LinearWeights R L M] (x : L) : (toEnd R L ↥(shiftedGenWeightSpace R L M χ)) x = (shift R L M χ).conj ((toEnd R L ↥(genWeightSpace M χ)) x - χ x • LinearMap.id)

instance LieModule.shiftedGenWeightSpace.instIsNilpotentSubtypeMemLieSubmoduleOfIsNoetherian (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] (χ : L → R) [LinearWeights R L M] [IsNoetherian R M] : IsNilpotent L ↥(shiftedGenWeightSpace R L M χ)

By Engel's theorem, if M is Noetherian, the shifted action ⁅x, m⁆ - χ x • m makes the χ-weight space into a nilpotent Lie module.

theorem LieModule.exists_forall_lie_eq_smul (R : Type u_2) (L : Type u_3) (M : Type u_4) [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [LieRing.IsNilpotent L] [LinearWeights R L M] [IsNoetherian R M] (χ : Weight R L M) : ∃ (m : M), m ≠ 0 ∧ ∀ (x : L), ⁅x, m⁆ = χ x • m

Given a Lie module M of a nilpotent Lie algebra L with coefficients in R, if a function χ : L → R has a simultaneous generalized eigenvector for the action of L then it has a simultaneous true eigenvector, provided M is Noetherian and has linear weights.

theorem LieModule.exists_nontrivial_weightSpace_of_isNilpotent (k : Type u_1) (L : Type u_3) (M : Type u_4) [LieRing L] [AddCommGroup M] [LieRingModule L M] [LieRing.IsNilpotent L] [Field k] [LieAlgebra k L] [Module k M] [Module.Finite k M] [LieModule k L M] [LinearWeights k L M] [IsTriangularizable k L M] [Nontrivial M] : ∃ (χ : Module.Dual k L), Nontrivial ↥(weightSpace M ⇑χ)

See LieModule.exists_nontrivial_weightSpace_of_isSolvable for the variant that only assumes that L is solvable but additionally requires k to be of characteristic zero.