Free Lie algebras

Given a commutative ring R and a type X we construct the free Lie algebra on X with coefficients in R together with its universal property.

Main definitions

• FreeLieAlgebra
• FreeLieAlgebra.lift
• FreeLieAlgebra.of
• FreeLieAlgebra.universalEnvelopingEquivFreeAlgebra

Implementation details

Quotient of free non-unital, non-associative algebra

We follow N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1--3 and construct the free Lie algebra as a quotient of the free non-unital, non-associative algebra. Since we do not currently have definitions of ideals, lattices of ideals, and quotients for NonUnitalNonAssocSemiring, we construct our quotient using the low-level Quot function on an inductively-defined relation.

Alternative construction (needs PBW)

An alternative construction of the free Lie algebra on X is to start with the free unital associative algebra on X, regard it as a Lie algebra via the ring commutator, and take its smallest Lie subalgebra containing X. I.e.: LieSubalgebra.lieSpan R (FreeAlgebra R X) (Set.range (FreeAlgebra.ι R)).

However with this definition there does not seem to be an easy proof that the required universal property holds, and I don't know of a proof that avoids invoking the Poincaré–Birkhoff–Witt theorem. A related MathOverflow question is this one.

Tags

lie algebra, free algebra, non-unital, non-associative, universal property, forgetful functor, adjoint functor

inductive FreeLieAlgebra.Rel (R : Type u) (X : Type v) [CommRing R] : FreeNonUnitalNonAssocAlgebra R X → FreeNonUnitalNonAssocAlgebra R X → Prop

The quotient of lib R X by the equivalence relation generated by this relation will give us the free Lie algebra.

• lie_self {R : Type u} {X : Type v} [CommRing R] (a : FreeNonUnitalNonAssocAlgebra R X) : Rel R X (a * a) 0
• leibniz_lie {R : Type u} {X : Type v} [CommRing R] (a b c : FreeNonUnitalNonAssocAlgebra R X) : Rel R X (a * (b * c)) (a * b * c + b * (a * c))
• smul {R : Type u} {X : Type v} [CommRing R] (t : R) {a b : FreeNonUnitalNonAssocAlgebra R X} : Rel R X a b → Rel R X (t • a) (t • b)
• add_right {R : Type u} {X : Type v} [CommRing R] {a b : FreeNonUnitalNonAssocAlgebra R X} (c : FreeNonUnitalNonAssocAlgebra R X) : Rel R X a b → Rel R X (a + c) (b + c)
• mul_left {R : Type u} {X : Type v} [CommRing R] (a : FreeNonUnitalNonAssocAlgebra R X) {b c : FreeNonUnitalNonAssocAlgebra R X} : Rel R X b c → Rel R X (a * b) (a * c)
• mul_right {R : Type u} {X : Type v} [CommRing R] {a b : FreeNonUnitalNonAssocAlgebra R X} (c : FreeNonUnitalNonAssocAlgebra R X) : Rel R X a b → Rel R X (a * c) (b * c)

theorem FreeLieAlgebra.Rel.addLeft {R : Type u} {X : Type v} [CommRing R] (a : FreeNonUnitalNonAssocAlgebra R X) {b c : FreeNonUnitalNonAssocAlgebra R X} (h : Rel R X b c) : Rel R X (a + b) (a + c)

theorem FreeLieAlgebra.Rel.neg {R : Type u} {X : Type v} [CommRing R] {a b : FreeNonUnitalNonAssocAlgebra R X} (h : Rel R X a b) : Rel R X (-a) (-b)

theorem FreeLieAlgebra.Rel.subLeft {R : Type u} {X : Type v} [CommRing R] (a : FreeNonUnitalNonAssocAlgebra R X) {b c : FreeNonUnitalNonAssocAlgebra R X} (h : Rel R X b c) : Rel R X (a - b) (a - c)

theorem FreeLieAlgebra.Rel.subRight {R : Type u} {X : Type v} [CommRing R] {a b : FreeNonUnitalNonAssocAlgebra R X} (c : FreeNonUnitalNonAssocAlgebra R X) (h : Rel R X a b) : Rel R X (a - c) (b - c)

theorem FreeLieAlgebra.Rel.smulOfTower {R : Type u} {X : Type v} [CommRing R] {S : Type u_1} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (t : S) (a b : FreeNonUnitalNonAssocAlgebra R X) (h : Rel R X a b) : Rel R X (t • a) (t • b)

def FreeLieAlgebra (R : Type u) (X : Type v) [CommRing R] : Type (max u v)

If α is a type, and R is a commutative ring, then FreeLieAlgebra R α is the free Lie algebra over R generated by α. This is a Lie algebra over R equipped with a function FreeLieAlgebra.of R : α → FreeLieAlgebra R α which has the following universal property: if L is any Lie algebra over R, and f : α → L is any function, then this function is the composite of FreeLieAlgebra.of R and a unique Lie algebra homomorphism FreeLieAlgebra.lift R f : FreeLieAlgebra R α →ₗ⁅R⁆ L.

A typical element of FreeLieAlgebra R α is a R-linear combination of formal brackets of elements of α. For example if x and y are terms of type α and a and b are term of type R then (3 * a * a) • ⁅⁅x, y⁆, x⁆ - (2 * b - 1) • ⁅y, x⁆ is a "typical" element of FreeLieAlgebra R α.

instance instInhabitedFreeLieAlgebra (R : Type u) (X : Type v) [CommRing R] : Inhabited (FreeLieAlgebra R X)

instance FreeLieAlgebra.instSMulOfIsScalarTower (R : Type u) (X : Type v) [CommRing R] {S : Type u_1} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] : SMul S (FreeLieAlgebra R X)

instance FreeLieAlgebra.instIsCentralScalar (R : Type u) (X : Type v) [CommRing R] {S : Type u_1} [Monoid S] [DistribMulAction S R] [DistribMulAction Sᵐᵒᵖ R] [IsScalarTower S R R] [IsCentralScalar S R] : IsCentralScalar S (FreeLieAlgebra R X)

instance FreeLieAlgebra.instZero (R : Type u) (X : Type v) [CommRing R] : Zero (FreeLieAlgebra R X)

instance FreeLieAlgebra.instAdd (R : Type u) (X : Type v) [CommRing R] : Add (FreeLieAlgebra R X)

instance FreeLieAlgebra.instNeg (R : Type u) (X : Type v) [CommRing R] : Neg (FreeLieAlgebra R X)

instance FreeLieAlgebra.instSub (R : Type u) (X : Type v) [CommRing R] : Sub (FreeLieAlgebra R X)

instance FreeLieAlgebra.instAddGroup (R : Type u) (X : Type v) [CommRing R] : AddGroup (FreeLieAlgebra R X)

instance FreeLieAlgebra.instAddCommSemigroup (R : Type u) (X : Type v) [CommRing R] : AddCommSemigroup (FreeLieAlgebra R X)

instance FreeLieAlgebra.instAddCommGroup (R : Type u) (X : Type v) [CommRing R] : AddCommGroup (FreeLieAlgebra R X)

instance FreeLieAlgebra.instModuleOfIsScalarTower (R : Type u) (X : Type v) [CommRing R] {S : Type u_1} [Semiring S] [Module S R] [IsScalarTower S R R] : Module S (FreeLieAlgebra R X)

instance FreeLieAlgebra.instLieRing (R : Type u) (X : Type v) [CommRing R] : LieRing (FreeLieAlgebra R X)

Note that here we turn the Mul coming from the NonUnitalNonAssocSemiring structure on lib R X into a Bracket on FreeLieAlgebra.

instance FreeLieAlgebra.instLieAlgebra (R : Type u) (X : Type v) [CommRing R] : LieAlgebra R (FreeLieAlgebra R X)

def FreeLieAlgebra.of (R : Type u) {X : Type v} [CommRing R] : X → FreeLieAlgebra R X

The embedding of X into the free Lie algebra of X with coefficients in the commutative ring R.

def FreeLieAlgebra.liftAux (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → CommutatorRing L) : FreeNonUnitalNonAssocAlgebra R X →ₙₐ[R] CommutatorRing L

An auxiliary definition used to construct the equivalence lift below.

theorem FreeLieAlgebra.liftAux_map_smul (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (t : R) (a : FreeNonUnitalNonAssocAlgebra R X) : (liftAux R f) (t • a) = t • (liftAux R f) a

theorem FreeLieAlgebra.liftAux_map_add (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (a b : FreeNonUnitalNonAssocAlgebra R X) : (liftAux R f) (a + b) = (liftAux R f) a + (liftAux R f) b

theorem FreeLieAlgebra.liftAux_map_mul (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (a b : FreeNonUnitalNonAssocAlgebra R X) : (liftAux R f) (a * b) = ⁅(liftAux R f) a, (liftAux R f) b⁆

theorem FreeLieAlgebra.liftAux_spec (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (a b : FreeNonUnitalNonAssocAlgebra R X) (h : Rel R X a b) : (liftAux R f) a = (liftAux R f) b

def FreeLieAlgebra.mk (R : Type u) {X : Type v} [CommRing R] : FreeNonUnitalNonAssocAlgebra R X →ₙₐ[R] CommutatorRing (FreeLieAlgebra R X)

The quotient map as a NonUnitalAlgHom.

def FreeLieAlgebra.lift (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] : (X → L) ≃ (FreeLieAlgebra R X →ₗ⁅R⁆ L)

The functor X ↦ FreeLieAlgebra R X from the category of types to the category of Lie algebras over R is adjoint to the forgetful functor in the other direction.

@[simp]

theorem FreeLieAlgebra.lift_symm_apply (R : Type u) {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (F : FreeLieAlgebra R X →ₗ⁅R⁆ L) : (lift R).symm F = ⇑F ∘ of R

@[simp]

theorem FreeLieAlgebra.of_comp_lift {R : Type u} {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) : ⇑((lift R) f) ∘ of R = f

@[simp]

theorem FreeLieAlgebra.lift_unique {R : Type u} {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (g : FreeLieAlgebra R X →ₗ⁅R⁆ L) : ⇑g ∘ of R = f ↔ g = (lift R) f

@[simp]

theorem FreeLieAlgebra.lift_of_apply {R : Type u} {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (f : X → L) (x : X) : ((lift R) f) (of R x) = f x

@[simp]

theorem FreeLieAlgebra.lift_comp_of {R : Type u} {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] (F : FreeLieAlgebra R X →ₗ⁅R⁆ L) : (lift R) (⇑F ∘ of R) = F

theorem FreeLieAlgebra.hom_ext {R : Type u} {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] {F₁ F₂ : FreeLieAlgebra R X →ₗ⁅R⁆ L} (h : ∀ (x : X), F₁ (of R x) = F₂ (of R x)) : F₁ = F₂

theorem FreeLieAlgebra.hom_ext_iff {R : Type u} {X : Type v} [CommRing R] {L : Type w} [LieRing L] [LieAlgebra R L] {F₁ F₂ : FreeLieAlgebra R X →ₗ⁅R⁆ L} : F₁ = F₂ ↔ ∀ (x : X), F₁ (of R x) = F₂ (of R x)

def FreeLieAlgebra.universalEnvelopingEquivFreeAlgebra (R : Type u) (X : Type v) [CommRing R] : UniversalEnvelopingAlgebra R (FreeLieAlgebra R X) ≃ₐ[R] FreeAlgebra R X

The universal enveloping algebra of the free Lie algebra is just the free unital associative algebra.

@[simp]

theorem FreeLieAlgebra.universalEnvelopingEquivFreeAlgebra_apply (R : Type u) (X : Type v) [CommRing R] (a : UniversalEnvelopingAlgebra R (FreeLieAlgebra R X)) : (universalEnvelopingEquivFreeAlgebra R X) a = ((UniversalEnvelopingAlgebra.lift R) ((lift R) (FreeAlgebra.ι R))) a

@[simp]

theorem FreeLieAlgebra.universalEnvelopingEquivFreeAlgebra_symm_apply (R : Type u) (X : Type v) [CommRing R] (a : FreeAlgebra R X) : (universalEnvelopingEquivFreeAlgebra R X).symm a = ((FreeAlgebra.lift R) (⇑(UniversalEnvelopingAlgebra.ι R) ∘ of R)) a