Lie subalgebras

This file defines Lie subalgebras of a Lie algebra and provides basic related definitions and results.

Main definitions

• LieSubalgebra
• LieSubalgebra.incl
• LieSubalgebra.map
• LieHom.range
• LieEquiv.ofInjective
• LieEquiv.ofEq
• LieEquiv.ofSubalgebras

Tags

lie algebra, lie subalgebra

structure LieSubalgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] extends Submodule : Type v

• carrier : Set L
• add_mem' : ∀ {a b : L}, a ∈ s.carrier → b ∈ s.carrier → a + b ∈ s.carrier
• zero_mem' : 0 ∈ s.carrier
• smul_mem' : ∀ (c : R) {x : L}, x ∈ s.carrier → c • x ∈ s.carrier
• lie_mem' : ∀ {x y : L}, x ∈ s.carrier → y ∈ s.carrier → ⁅x, y⁆ ∈ s.carrier

  A Lie subalgebra is closed under Lie bracket.

A Lie subalgebra of a Lie algebra is submodule that is closed under the Lie bracket. This is a sufficient condition for the subset itself to form a Lie algebra.

instance instZeroLieSubalgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : Zero (LieSubalgebra R L)

The zero algebra is a subalgebra of any Lie algebra.

instance instInhabitedLieSubalgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : Inhabited (LieSubalgebra R L)

instance instCoeLieSubalgebraSubmoduleToSemiringToCommSemiringToAddCommMonoidToAddCommGroupToModule (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : Coe (LieSubalgebra R L) (Submodule R L)

instance LieSubalgebra.instSetLikeLieSubalgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : SetLike (LieSubalgebra R L) L

instance LieSubalgebra.instAddSubgroupClassLieSubalgebraToSubNegMonoidToAddGroupToAddCommGroupInstSetLikeLieSubalgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : AddSubgroupClass (LieSubalgebra R L) L

instance LieSubalgebra.lieRing (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : LieRing { x // x ∈ L' }

A Lie subalgebra forms a new Lie ring.

instance LieSubalgebra.instModuleSubtypeMemLieSubalgebraInstMembershipInstSetLikeLieSubalgebraAddCommMonoidToSemiringToCommSemiringToAddCommMonoidToAddCommGroupToModuleToSubmodule (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] {R₁ : Type u_1} [Semiring R₁] [SMul R₁ R] [Module R₁ L] [IsScalarTower R₁ R L] (L' : LieSubalgebra R L) : Module R₁ { x // x ∈ L' }

A Lie subalgebra inherits module structures from L.

instance LieSubalgebra.instIsCentralScalarSubtypeMemLieSubalgebraInstMembershipInstSetLikeLieSubalgebraSmulToSemiringToCommSemiringToAddCommMonoidToAddCommGroupToModuleToSubmoduleToSMulToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToSMulZeroClassToZeroToMonoidWithZeroToSMulWithZeroToMulActionWithZeroMulOppositeZeroMonoidWithZeroSemiring (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] {R₁ : Type u_1} [Semiring R₁] [SMul R₁ R] [SMul R₁ᵐᵒᵖ R] [Module R₁ L] [Module R₁ᵐᵒᵖ L] [IsScalarTower R₁ R L] [IsScalarTower R₁ᵐᵒᵖ R L] [IsCentralScalar R₁ L] (L' : LieSubalgebra R L) : IsCentralScalar R₁ { x // x ∈ L' }

instance LieSubalgebra.instIsScalarTowerSubtypeMemLieSubalgebraInstMembershipInstSetLikeLieSubalgebraSmulToSemiringToCommSemiringToAddCommMonoidToAddCommGroupToModuleToSubmoduleToSMulIdToSMulToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToSMulZeroClassToZeroToCommMonoidWithZeroToSMulWithZeroToMonoidWithZeroToMulActionWithZeroLeftToMonoidToMulActionSmulToSMulToSMulZeroClassToZeroToMonoidWithZeroToSMulWithZeroToMulActionWithZero (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] {R₁ : Type u_1} [Semiring R₁] [SMul R₁ R] [Module R₁ L] [IsScalarTower R₁ R L] (L' : LieSubalgebra R L) : IsScalarTower R₁ R { x // x ∈ L' }

instance LieSubalgebra.instIsNoetherianSubtypeMemLieSubalgebraInstMembershipInstSetLikeLieSubalgebraToSemiringToCommSemiringAddCommMonoidToAddCommMonoidToAddCommGroupToModuleToSubmoduleInstModuleSubtypeMemLieSubalgebraInstMembershipInstSetLikeLieSubalgebraAddCommMonoidToSemiringToCommSemiringToAddCommMonoidToAddCommGroupToModuleToSubmoduleToSMulIdLeftToMonoidToMonoidWithZeroToMulActionToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToMulActionWithZero (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) [IsNoetherian R L] : IsNoetherian R { x // x ∈ L' }

instance LieSubalgebra.lieAlgebra (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : LieAlgebra R { x // x ∈ L' }

A Lie subalgebra forms a new Lie algebra.

@[simp]

theorem LieSubalgebra.zero_mem {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : 0 ∈ L'

theorem LieSubalgebra.add_mem {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} {y : L} : x ∈ L' → y ∈ L' → x + y ∈ L'

theorem LieSubalgebra.sub_mem {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} {y : L} : x ∈ L' → y ∈ L' → x - y ∈ L'

theorem LieSubalgebra.smul_mem {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (t : R) {x : L} (h : x ∈ L') : t • x ∈ L'

theorem LieSubalgebra.lie_mem {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} {y : L} (hx : x ∈ L') (hy : y ∈ L') : ⁅x, y⁆ ∈ L'

theorem LieSubalgebra.mem_carrier {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} : x ∈ L'.carrier ↔ x ∈ ↑L'

@[simp]

theorem LieSubalgebra.mem_mk_iff {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : R) {x : L}, x ∈ { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }.toAddSubsemigroup.carrier → c • x ∈ { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }.toAddSubsemigroup.carrier) (h₄ : ∀ {x y : L}, x ∈ { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }.toAddSubmonoid.toAddSubsemigroup.carrier → y ∈ { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }.toAddSubmonoid.toAddSubsemigroup.carrier → ⁅x, y⁆ ∈ { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }.toAddSubmonoid.toAddSubsemigroup.carrier) {x : L} : x ∈ { toSubmodule := { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }, lie_mem' := h₄ } ↔ x ∈ S

@[simp]

theorem LieSubalgebra.mem_coe_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} : x ∈ L'.toSubmodule ↔ x ∈ L'

theorem LieSubalgebra.mem_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} : x ∈ ↑L' ↔ x ∈ L'

@[simp]

theorem LieSubalgebra.coe_bracket {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (x : { x // x ∈ L' }) (y : { x // x ∈ L' }) : ↑⁅x, y⁆ = ⁅↑x, ↑y⁆

theorem LieSubalgebra.ext_iff {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (x : { x // x ∈ L' }) (y : { x // x ∈ L' }) : x = y ↔ ↑x = ↑y

theorem LieSubalgebra.coe_zero_iff_zero {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (x : { x // x ∈ L' }) : ↑x = 0 ↔ x = 0

theorem LieSubalgebra.ext {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' : LieSubalgebra R L) (L₂' : LieSubalgebra R L) (h : ∀ (x : L), x ∈ L₁' ↔ x ∈ L₂') : L₁' = L₂'

theorem LieSubalgebra.ext_iff' {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' : LieSubalgebra R L) (L₂' : LieSubalgebra R L) : L₁' = L₂' ↔ ∀ (x : L), x ∈ L₁' ↔ x ∈ L₂'

@[simp]

theorem LieSubalgebra.mk_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : R) {x : L}, x ∈ { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }.toAddSubsemigroup.carrier → c • x ∈ { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }.toAddSubsemigroup.carrier) (h₄ : ∀ {x y : L}, x ∈ { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }.toAddSubmonoid.toAddSubsemigroup.carrier → y ∈ { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }.toAddSubmonoid.toAddSubsemigroup.carrier → ⁅x, y⁆ ∈ { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }.toAddSubmonoid.toAddSubsemigroup.carrier) : ↑{ toSubmodule := { toAddSubmonoid := { toAddSubsemigroup := { carrier := S, add_mem' := h₁ }, zero_mem' := h₂ }, smul_mem' := h₃ }, lie_mem' := h₄ } = S

@[simp]

theorem LieSubalgebra.coe_to_submodule_mk {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : Submodule R L) (h : ∀ {x y : L}, x ∈ { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := (_ : ∀ (c : R) {x : L}, x ∈ p.carrier → c • x ∈ p.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier → y ∈ { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := (_ : ∀ (c : R) {x : L}, x ∈ p.carrier → c • x ∈ p.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier → ⁅x, y⁆ ∈ { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := (_ : ∀ (c : R) {x : L}, x ∈ p.carrier → c • x ∈ p.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier) : { toSubmodule := { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := (_ : ∀ (c : R) {x : L}, x ∈ p.carrier → c • x ∈ p.carrier) }, lie_mem' := h }.toSubmodule = p

theorem LieSubalgebra.coe_injective {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Function.Injective SetLike.coe

theorem LieSubalgebra.coe_set_eq {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' : LieSubalgebra R L) (L₂' : LieSubalgebra R L) : ↑L₁' = ↑L₂' ↔ L₁' = L₂'

theorem LieSubalgebra.to_submodule_injective {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Function.Injective LieSubalgebra.toSubmodule

@[simp]

theorem LieSubalgebra.coe_to_submodule_eq_iff {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' : LieSubalgebra R L) (L₂' : LieSubalgebra R L) : L₁'.toSubmodule = L₂'.toSubmodule ↔ L₁' = L₂'

theorem LieSubalgebra.coe_to_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : ↑L'.toSubmodule = ↑L'

instance LieSubalgebra.lieRingModule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {M : Type w} [AddCommGroup M] [LieRingModule L M] : LieRingModule { x // x ∈ L' } M

Given a Lie algebra L containing a Lie subalgebra L' ⊆ L, together with a Lie ring module M of L, we may regard M as a Lie ring module of L' by restriction.

@[simp]

theorem LieSubalgebra.coe_bracket_of_module {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {M : Type w} [AddCommGroup M] [LieRingModule L M] (x : { x // x ∈ L' }) (m : M) : ⁅x, m⁆ = ⁅↑x, m⁆

instance LieSubalgebra.lieModule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {M : Type w} [AddCommGroup M] [LieRingModule L M] [Module R M] [LieModule R L M] : LieModule R { x // x ∈ L' } M

Given a Lie algebra L containing a Lie subalgebra L' ⊆ L, together with a Lie module M of L, we may regard M as a Lie module of L' by restriction.

def LieModuleHom.restrictLie {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {M : Type w} [AddCommGroup M] [LieRingModule L M] {N : Type w₁} [AddCommGroup N] [LieRingModule L N] [Module R N] [Module R M] (f : M →ₗ⁅R,L⁆ N) (L' : LieSubalgebra R L) : M →ₗ⁅R,{ x // x ∈ L' }⁆ N

An L-equivariant map of Lie modules M → N is L'-equivariant for any Lie subalgebra L' ⊆ L.

@[simp]

theorem LieModuleHom.coe_restrictLie {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {M : Type w} [AddCommGroup M] [LieRingModule L M] {N : Type w₁} [AddCommGroup N] [LieRingModule L N] [Module R N] [Module R M] (f : M →ₗ⁅R,L⁆ N) : ↑(LieModuleHom.restrictLie f L') = ↑f

def LieSubalgebra.incl {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : { x // x ∈ L' } →ₗ⁅R⁆ L

The embedding of a Lie subalgebra into the ambient space as a morphism of Lie algebras.

@[simp]

theorem LieSubalgebra.coe_incl {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : ↑(LieSubalgebra.incl L') = Subtype.val

def LieSubalgebra.incl' {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : { x // x ∈ L' } →ₗ⁅R,{ x // x ∈ L' }⁆ L

The embedding of a Lie subalgebra into the ambient space as a morphism of Lie modules.

@[simp]

theorem LieSubalgebra.coe_incl' {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) : ↑(LieSubalgebra.incl' L') = Subtype.val

def LieHom.range {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) : LieSubalgebra R L₂

The range of a morphism of Lie algebras is a Lie subalgebra.

@[simp]

theorem LieHom.range_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) : ↑(LieHom.range f) = Set.range ↑f

@[simp]

theorem LieHom.mem_range {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L₂) : x ∈ LieHom.range f ↔ ∃ y, ↑f y = x

theorem LieHom.mem_range_self {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L) : ↑f x ∈ LieHom.range f

def LieHom.rangeRestrict {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) : L →ₗ⁅R⁆ { x // x ∈ LieHom.range f }

We can restrict a morphism to a (surjective) map to its range.

@[simp]

theorem LieHom.rangeRestrict_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L) : ↑(LieHom.rangeRestrict f) x = { val := ↑f x, property := (_ : ↑f x ∈ LieHom.range f) }

theorem LieHom.surjective_rangeRestrict {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) : Function.Surjective ↑(LieHom.rangeRestrict f)

noncomputable def LieHom.equivRangeOfInjective {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (h : Function.Injective ↑f) : L ≃ₗ⁅R⁆ { x // x ∈ LieHom.range f }

A Lie algebra is equivalent to its range under an injective Lie algebra morphism.

@[simp]

theorem LieHom.equivRangeOfInjective_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (h : Function.Injective ↑f) (x : L) : ↑(LieHom.equivRangeOfInjective f h) x = { val := ↑f x, property := (_ : ↑f x ∈ LieHom.range f) }

theorem Submodule.exists_lieSubalgebra_coe_eq_iff {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : Submodule R L) : (∃ K, K.toSubmodule = p) ↔ ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p

@[simp]

theorem LieSubalgebra.incl_range {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) : LieHom.range (LieSubalgebra.incl K) = K

def LieSubalgebra.map {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (K : LieSubalgebra R L) : LieSubalgebra R L₂

The image of a Lie subalgebra under a Lie algebra morphism is a Lie subalgebra of the codomain.

@[simp]

theorem LieSubalgebra.mem_map {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (K : LieSubalgebra R L) (x : L₂) : x ∈ LieSubalgebra.map f K ↔ ∃ y, y ∈ K ∧ ↑f y = x

theorem LieSubalgebra.mem_map_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (K : LieSubalgebra R L) (e : L ≃ₗ⁅R⁆ L₂) (x : L₂) : x ∈ LieSubalgebra.map e.toLieHom K ↔ x ∈ Submodule.map (↑(LieEquiv.toLinearEquiv e)) K.toSubmodule

def LieSubalgebra.comap {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (K₂ : LieSubalgebra R L₂) : LieSubalgebra R L

The preimage of a Lie subalgebra under a Lie algebra morphism is a Lie subalgebra of the domain.

instance LieSubalgebra.instPartialOrderLieSubalgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : PartialOrder (LieSubalgebra R L)

theorem LieSubalgebra.le_def {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (K' : LieSubalgebra R L) : K ≤ K' ↔ ↑K ⊆ ↑K'

@[simp]

theorem LieSubalgebra.coe_submodule_le_coe_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (K' : LieSubalgebra R L) : K.toSubmodule ≤ K'.toSubmodule ↔ K ≤ K'

instance LieSubalgebra.instBotLieSubalgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Bot (LieSubalgebra R L)

@[simp]

theorem LieSubalgebra.bot_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : ↑⊥ = {0}

@[simp]

theorem LieSubalgebra.bot_coe_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : ⊥.toSubmodule = ⊥

@[simp]

theorem LieSubalgebra.mem_bot {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L) : x ∈ ⊥ ↔ x = 0

instance LieSubalgebra.instTopLieSubalgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Top (LieSubalgebra R L)

@[simp]

theorem LieSubalgebra.top_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : ↑⊤ = Set.univ

@[simp]

theorem LieSubalgebra.top_coe_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : ⊤.toSubmodule = ⊤

@[simp]

theorem LieSubalgebra.mem_top {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L) : x ∈ ⊤

theorem LieHom.range_eq_map {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) : LieHom.range f = LieSubalgebra.map f ⊤

instance LieSubalgebra.instInfLieSubalgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Inf (LieSubalgebra R L)

instance LieSubalgebra.instInfSetLieSubalgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : InfSet (LieSubalgebra R L)

@[simp]

theorem LieSubalgebra.inf_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (K' : LieSubalgebra R L) : ↑(K ⊓ K') = ↑K ∩ ↑K'

@[simp]

theorem LieSubalgebra.sInf_coe_to_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set (LieSubalgebra R L)) : (sInf S).toSubmodule = sInf {x | ∃ s, s ∈ S ∧ s.toSubmodule = x}

@[simp]

theorem LieSubalgebra.sInf_coe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set (LieSubalgebra R L)) : ↑(sInf S) = ⋂ (s : LieSubalgebra R L) (_ : s ∈ S), ↑s

theorem LieSubalgebra.sInf_glb {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set (LieSubalgebra R L)) : IsGLB S (sInf S)

instance LieSubalgebra.completeLattice {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : CompleteLattice (LieSubalgebra R L)

The set of Lie subalgebras of a Lie algebra form a complete lattice.

We provide explicit values for the fields bot, top, inf to get more convenient definitions than we would otherwise obtain from completeLatticeOfInf.

instance LieSubalgebra.addCommMonoid {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : AddCommMonoid (LieSubalgebra R L)

instance LieSubalgebra.instCanonicallyOrderedAddMonoidLieSubalgebra {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : CanonicallyOrderedAddMonoid (LieSubalgebra R L)

@[simp]

theorem LieSubalgebra.add_eq_sup {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (K' : LieSubalgebra R L) : K + K' = K ⊔ K'

@[simp]

theorem LieSubalgebra.inf_coe_to_submodule {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (K' : LieSubalgebra R L) : (K ⊓ K').toSubmodule = K.toSubmodule ⊓ K'.toSubmodule

@[simp]

theorem LieSubalgebra.mem_inf {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) (K' : LieSubalgebra R L) (x : L) : x ∈ K ⊓ K' ↔ x ∈ K ∧ x ∈ K'

theorem LieSubalgebra.eq_bot_iff {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) : K = ⊥ ↔ ∀ (x : L), x ∈ K → x = 0

instance LieSubalgebra.subsingleton_of_bot {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Subsingleton (LieSubalgebra R { x // x ∈ ⊥ })

theorem LieSubalgebra.subsingleton_bot {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] : Subsingleton { x // x ∈ ⊥ }

theorem LieSubalgebra.wellFounded_of_noetherian (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] [IsNoetherian R L] : WellFounded fun x x_1 => x > x_1

def LieSubalgebra.homOfLe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') : { x // x ∈ K } →ₗ⁅R⁆ { x // x ∈ K' }

Given two nested Lie subalgebras K ⊆ K', the inclusion K ↪ K' is a morphism of Lie algebras.

@[simp]

theorem LieSubalgebra.coe_homOfLe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') (x : { x // x ∈ K }) : ↑(↑(LieSubalgebra.homOfLe h) x) = ↑x

theorem LieSubalgebra.homOfLe_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') (x : { x // x ∈ K }) : ↑(LieSubalgebra.homOfLe h) x = { val := ↑x, property := h ↑x (_ : ↑x ∈ K) }

theorem LieSubalgebra.homOfLe_injective {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') : Function.Injective ↑(LieSubalgebra.homOfLe h)

def LieSubalgebra.ofLe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') : LieSubalgebra R { x // x ∈ K' }

Given two nested Lie subalgebras K ⊆ K', we can view K as a Lie subalgebra of K', regarded as Lie algebra in its own right.

@[simp]

theorem LieSubalgebra.mem_ofLe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') (x : { x // x ∈ K' }) : x ∈ LieSubalgebra.ofLe h ↔ ↑x ∈ K

theorem LieSubalgebra.ofLe_eq_comap_incl {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') : LieSubalgebra.ofLe h = LieSubalgebra.comap (LieSubalgebra.incl K') K

@[simp]

theorem LieSubalgebra.coe_ofLe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') : (LieSubalgebra.ofLe h).toSubmodule = LinearMap.range (Submodule.ofLe h)

noncomputable def LieSubalgebra.equivOfLe {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') : { x // x ∈ K } ≃ₗ⁅R⁆ { x // x ∈ LieSubalgebra.ofLe h }

Given nested Lie subalgebras K ⊆ K', there is a natural equivalence from K to its image in K'.

@[simp]

theorem LieSubalgebra.equivOfLe_apply {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {K : LieSubalgebra R L} {K' : LieSubalgebra R L} (h : K ≤ K') (x : { x // x ∈ K }) : ↑(LieSubalgebra.equivOfLe h) x = { val := ↑(LieSubalgebra.homOfLe h) x, property := (_ : ↑(LieSubalgebra.homOfLe h) x ∈ LieHom.range (LieSubalgebra.homOfLe h)) }

theorem LieSubalgebra.map_le_iff_le_comap {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] {f : L →ₗ⁅R⁆ L₂} {K : LieSubalgebra R L} {K' : LieSubalgebra R L₂} : LieSubalgebra.map f K ≤ K' ↔ K ≤ LieSubalgebra.comap f K'

theorem LieSubalgebra.gc_map_comap {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] {f : L →ₗ⁅R⁆ L₂} : GaloisConnection (LieSubalgebra.map f) (LieSubalgebra.comap f)

def LieSubalgebra.lieSpan (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] (s : Set L) : LieSubalgebra R L

The Lie subalgebra of a Lie algebra L generated by a subset s ⊆ L.

theorem LieSubalgebra.mem_lieSpan {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} {x : L} : x ∈ LieSubalgebra.lieSpan R L s ↔ ∀ (K : LieSubalgebra R L), s ⊆ ↑K → x ∈ K

theorem LieSubalgebra.subset_lieSpan {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} : s ⊆ ↑(LieSubalgebra.lieSpan R L s)

theorem LieSubalgebra.submodule_span_le_lieSpan {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} : Submodule.span R s ≤ (LieSubalgebra.lieSpan R L s).toSubmodule

theorem LieSubalgebra.lieSpan_le {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} {K : LieSubalgebra R L} : LieSubalgebra.lieSpan R L s ≤ K ↔ s ⊆ ↑K

theorem LieSubalgebra.lieSpan_mono {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {s : Set L} {t : Set L} (h : s ⊆ t) : LieSubalgebra.lieSpan R L s ≤ LieSubalgebra.lieSpan R L t

theorem LieSubalgebra.lieSpan_eq {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L) : LieSubalgebra.lieSpan R L ↑K = K

theorem LieSubalgebra.coe_lieSpan_submodule_eq_iff {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {p : Submodule R L} : (LieSubalgebra.lieSpan R L ↑p).toSubmodule = p ↔ ∃ K, K.toSubmodule = p

def LieSubalgebra.gi (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : GaloisInsertion (LieSubalgebra.lieSpan R L) SetLike.coe

lieSpan forms a Galois insertion with the coercion from LieSubalgebra to Set.

@[simp]

theorem LieSubalgebra.span_empty (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : LieSubalgebra.lieSpan R L ∅ = ⊥

@[simp]

theorem LieSubalgebra.span_univ (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] : LieSubalgebra.lieSpan R L Set.univ = ⊤

theorem LieSubalgebra.span_union (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (s : Set L) (t : Set L) : LieSubalgebra.lieSpan R L (s ∪ t) = LieSubalgebra.lieSpan R L s ⊔ LieSubalgebra.lieSpan R L t

theorem LieSubalgebra.span_iUnion (R : Type u) {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {ι : Sort u_1} (s : ι → Set L) : LieSubalgebra.lieSpan R L (⋃ (i : ι), s i) = ⨆ (i : ι), LieSubalgebra.lieSpan R L (s i)

noncomputable def LieEquiv.ofInjective {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Injective ↑f) : L₁ ≃ₗ⁅R⁆ { x // x ∈ LieHom.range f }

An injective Lie algebra morphism is an equivalence onto its range.

@[simp]

theorem LieEquiv.ofInjective_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (f : L₁ →ₗ⁅R⁆ L₂) (h : Function.Injective ↑f) (x : L₁) : ↑(↑(LieEquiv.ofInjective f h) x) = ↑f x

def LieEquiv.ofEq {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (L₁' : LieSubalgebra R L₁) (L₁'' : LieSubalgebra R L₁) (h : ↑L₁' = ↑L₁'') : { x // x ∈ L₁' } ≃ₗ⁅R⁆ { x // x ∈ L₁'' }

Lie subalgebras that are equal as sets are equivalent as Lie algebras.

@[simp]

theorem LieEquiv.ofEq_apply {R : Type u} {L₁ : Type v} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] (L : LieSubalgebra R L₁) (L' : LieSubalgebra R L₁) (h : ↑L = ↑L') (x : { x // x ∈ L }) : ↑(↑(LieEquiv.ofEq L L' h) x) = ↑x

def LieEquiv.lieSubalgebraMap {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁'' : LieSubalgebra R L₁) (e : L₁ ≃ₗ⁅R⁆ L₂) : { x // x ∈ L₁'' } ≃ₗ⁅R⁆ { x // x ∈ LieSubalgebra.map e.toLieHom L₁'' }

An equivalence of Lie algebras restricts to an equivalence from any Lie subalgebra onto its image.

@[simp]

theorem LieEquiv.lieSubalgebraMap_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁'' : LieSubalgebra R L₁) (e : L₁ ≃ₗ⁅R⁆ L₂) (x : { x // x ∈ L₁'' }) : ↑(↑(LieEquiv.lieSubalgebraMap L₁'' e) x) = ↑e ↑x

def LieEquiv.ofSubalgebras {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁' : LieSubalgebra R L₁) (L₂' : LieSubalgebra R L₂) (e : L₁ ≃ₗ⁅R⁆ L₂) (h : LieSubalgebra.map e.toLieHom L₁' = L₂') : { x // x ∈ L₁' } ≃ₗ⁅R⁆ { x // x ∈ L₂' }

An equivalence of Lie algebras restricts to an equivalence from any Lie subalgebra onto its image.

@[simp]

theorem LieEquiv.ofSubalgebras_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁' : LieSubalgebra R L₁) (L₂' : LieSubalgebra R L₂) (e : L₁ ≃ₗ⁅R⁆ L₂) (h : LieSubalgebra.map e.toLieHom L₁' = L₂') (x : { x // x ∈ L₁' }) : ↑(↑(LieEquiv.ofSubalgebras L₁' L₂' e h) x) = ↑e ↑x

@[simp]

theorem LieEquiv.ofSubalgebras_symm_apply {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R] [LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (L₁' : LieSubalgebra R L₁) (L₂' : LieSubalgebra R L₂) (e : L₁ ≃ₗ⁅R⁆ L₂) (h : LieSubalgebra.map e.toLieHom L₁' = L₂') (x : { x // x ∈ L₂' }) : ↑(↑(LieEquiv.symm (LieEquiv.ofSubalgebras L₁' L₂' e h)) x) = ↑(LieEquiv.symm e) ↑x