Quotients of Lie algebras and Lie modules

Given a Lie submodule of a Lie module, the quotient carries a natural Lie module structure. In the special case that the Lie module is the Lie algebra itself via the adjoint action, the submodule is a Lie ideal and the quotient carries a natural Lie algebra structure.

We define these quotient structures here. A notable omission at the time of writing (February 2021) is a statement and proof of the universal property of these quotients.

Main definitions

• LieSubmodule.Quotient.lieQuotientLieModule
• LieSubmodule.Quotient.lieQuotientLieAlgebra

Tags

lie algebra, quotient

instance LieSubmodule.instHasQuotient {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] : HasQuotient M (LieSubmodule R L M)

The quotient of a Lie module by a Lie submodule. It is a Lie module.

instance LieSubmodule.Quotient.addCommGroup {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} : AddCommGroup (M ⧸ N)

instance LieSubmodule.Quotient.module' {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} {S : Type u_1} [Semiring S] [SMul S R] [Module S M] [IsScalarTower S R M] : Module S (M ⧸ N)

instance LieSubmodule.Quotient.module {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} : Module R (M ⧸ N)

instance LieSubmodule.Quotient.isCentralScalar {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} {S : Type u_1} [Semiring S] [SMul S R] [Module S M] [IsScalarTower S R M] [SMul Sᵐᵒᵖ R] [Module Sᵐᵒᵖ M] [IsScalarTower Sᵐᵒᵖ R M] [IsCentralScalar S M] : IsCentralScalar S (M ⧸ N)

instance LieSubmodule.Quotient.inhabited {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} : Inhabited (M ⧸ N)

@[reducible, inline]

abbrev LieSubmodule.Quotient.mk {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} : M → M ⧸ N

Map sending an element of M to the corresponding element of M ⧸ N, when N is a Lie submodule of the Lie module M.

@[simp]

theorem LieSubmodule.Quotient.mk_eq_zero' {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} {m : M} : mk m = 0 ↔ m ∈ N

theorem LieSubmodule.Quotient.is_quotient_mk {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} (m : M) : Quotient.mk'' m = mk m

def LieSubmodule.Quotient.lieSubmoduleInvariant {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} [LieAlgebra R L] [LieModule R L M] : L →ₗ[R] ↥((↑N).compatibleMaps ↑N)

Given a Lie module M over a Lie algebra L, together with a Lie submodule N ⊆ M, there is a natural linear map from L to the endomorphisms of M leaving N invariant.

def LieSubmodule.Quotient.actionAsEndoMap {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : L →ₗ⁅R⁆ Module.End R (M ⧸ N)

Given a Lie module M over a Lie algebra L, together with a Lie submodule N ⊆ M, there is a natural Lie algebra morphism from L to the linear endomorphism of the quotient M/N.

instance LieSubmodule.Quotient.actionAsEndoMapBracket {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : Bracket L (M ⧸ N)

Given a Lie module M over a Lie algebra L, together with a Lie submodule N ⊆ M, there is a natural bracket action of L on the quotient M/N.

instance LieSubmodule.Quotient.lieQuotientLieRingModule {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : LieRingModule L (M ⧸ N)

instance LieSubmodule.Quotient.lieQuotientLieModule {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : LieModule R L (M ⧸ N)

The quotient of a Lie module by a Lie submodule, is a Lie module.

instance LieSubmodule.Quotient.lieQuotientHasBracket {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) : Bracket (L ⧸ I) (L ⧸ I)

@[simp]

theorem LieSubmodule.Quotient.mk_bracket {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) (x y : L) : mk ⁅x, y⁆ = ⁅mk x, mk y⁆

instance LieSubmodule.Quotient.lieQuotientLieRing {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) : LieRing (L ⧸ I)

instance LieSubmodule.Quotient.lieQuotientLieAlgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) : LieAlgebra R (L ⧸ I)

def LieSubmodule.Quotient.mk' {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : M →ₗ⁅R,L⁆ M ⧸ N

LieSubmodule.Quotient.mk as a LieModuleHom.

@[simp]

theorem LieSubmodule.Quotient.mk'_apply {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] (a✝ : M) : (mk' N) a✝ = mk a✝

@[simp]

theorem LieSubmodule.Quotient.surjective_mk' {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : Function.Surjective ⇑(mk' N)

@[simp]

theorem LieSubmodule.Quotient.range_mk' {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : (mk' N).range = ⊤

instance LieSubmodule.Quotient.isNoetherian {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [IsNoetherian R M] : IsNoetherian R (M ⧸ N)

theorem LieSubmodule.Quotient.mk_eq_zero {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] {m : M} : (mk' N) m = 0 ↔ m ∈ N

@[simp]

theorem LieSubmodule.Quotient.mk'_ker {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : (mk' N).ker = N

@[simp]

theorem LieSubmodule.Quotient.map_mk'_eq_bot_le {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N N' : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] : map (mk' N) N' = ⊥ ↔ N' ≤ N

theorem LieSubmodule.Quotient.lieModuleHom_ext {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] ⦃f g : M ⧸ N →ₗ⁅R,L⁆ M⦄ (h : f.comp (mk' N) = g.comp (mk' N)) : f = g

Two LieModuleHoms from a quotient lie module are equal if their compositions with LieSubmodule.Quotient.mk' are equal.

See note [partially-applied ext lemmas].

theorem LieSubmodule.Quotient.lieModuleHom_ext_iff {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] {N : LieSubmodule R L M} [LieAlgebra R L] [LieModule R L M] {f g : M ⧸ N →ₗ⁅R,L⁆ M} : f = g ↔ f.comp (mk' N) = g.comp (mk' N)

theorem LieSubmodule.Quotient.toEnd_comp_mk' {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) [LieAlgebra R L] [LieModule R L M] (x : L) : (LieModule.toEnd R L (M ⧸ N)) x ∘ₗ ↑(mk' N) = ↑(mk' N) ∘ₗ (LieModule.toEnd R L M) x

noncomputable def LieHom.quotKerEquivRange {R : Type u_1} {L : Type u_2} {L' : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') : (L ⧸ f.ker) ≃ₗ⁅R⁆ ↥f.range

The first isomorphism theorem for morphisms of Lie algebras.

@[simp]

theorem LieHom.quotKerEquivRange_invFun {R : Type u_1} {L : Type u_2} {L' : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') (a✝ : ↥(↑f).range) : f.quotKerEquivRange.invFun a✝ = (↑f).quotKerEquivRange.invFun a✝

@[simp]

theorem LieHom.quotKerEquivRange_toFun {R : Type u_1} {L : Type u_2} {L' : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') (a : L ⧸ (↑f).ker) : f.quotKerEquivRange a = (↑f).quotKerEquivRange a