# mathlibdocumentation

algebra.lie.universal_enveloping

# Universal enveloping algebra

Given a commutative ring R and a Lie algebra L over R, we construct the universal enveloping algebra of L, together with its universal property.

## Main definitions

• universal_enveloping_algebra: the universal enveloping algebra, endowed with an R-algebra structure.
• universal_enveloping_algebra.ι: the Lie algebra morphism from L to its universal enveloping algebra.
• universal_enveloping_algebra.lift: given an associative algebra A, together with a Lie algebra morphism f : L →ₗ⁅R⁆ A, lift R L f : universal_enveloping_algebra R L →ₐ[R] A is the unique morphism of algebras through which f factors.
• universal_enveloping_algebra.ι_comp_lift: states that the lift of a morphism is indeed part of a factorisation.
• universal_enveloping_algebra.lift_unique: states that lifts of morphisms are indeed unique.
• universal_enveloping_algebra.hom_ext: a restatement of lift_unique as an extensionality lemma.

## Tags

lie algebra, universal enveloping algebra, tensor algebra

inductive universal_enveloping_algebra.rel (R : Type u₁) (L : Type u₂) [comm_ring R] [lie_ring L] [ L] :
→ Prop
• lie_compat : ∀ (R : Type u₁) (L : Type u₂) [_inst_1 : [_inst_2 : lie_ring L] [_inst_3 : L] (x y : L), ( x,y + ( y) * x) (( x) * y)

The quotient by the ideal generated by this relation is the universal enveloping algebra.

Note that we have avoided using the more natural expression: | lie_compat (x y : L) : rel (ιₜ ⁅x, y⁆) ⁅ιₜ x, ιₜ y⁆ so that our construction needs only the semiring structure of the tensor algebra.

def universal_enveloping_algebra (R : Type u₁) (L : Type u₂) [comm_ring R] [lie_ring L] [ L] :
Type (max u₁ u₂)

The universal enveloping algebra of a Lie algebra.

Equations
@[instance]
def universal_enveloping_algebra.ring (R : Type u₁) (L : Type u₂) [comm_ring R] [lie_ring L] [ L] :

@[instance]
def universal_enveloping_algebra.inhabited (R : Type u₁) (L : Type u₂) [comm_ring R] [lie_ring L] [ L] :

@[instance]
def universal_enveloping_algebra.inst (R : Type u₁) (L : Type u₂) [comm_ring R] [lie_ring L] [ L] :

def universal_enveloping_algebra.mk_alg_hom (R : Type u₁) (L : Type u₂) [comm_ring R] [lie_ring L] [ L] :

The quotient map from the tensor algebra to the universal enveloping algebra as a morphism of associative algebras.

Equations
def universal_enveloping_algebra.ι (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] :

The natural Lie algebra morphism from a Lie algebra to its universal enveloping algebra.

Equations
def universal_enveloping_algebra.lift (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] {A : Type u₃} [ring A] [ A] :

The universal property of the universal enveloping algebra: Lie algebra morphisms into associative algebras lift to associative algebra morphisms from the universal enveloping algebra.

Equations
@[simp]
theorem universal_enveloping_algebra.lift_symm_apply (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] {A : Type u₃} [ring A] [ A] (F : →ₐ[R] A) :

@[simp]
theorem universal_enveloping_algebra.ι_comp_lift (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] {A : Type u₃} [ring A] [ A] (f : L →ₗ⁅R A) :

@[simp]
theorem universal_enveloping_algebra.lift_ι_apply (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] {A : Type u₃} [ring A] [ A] (f : L →ₗ⁅R A) (x : L) :
= f x

theorem universal_enveloping_algebra.lift_unique (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] {A : Type u₃} [ring A] [ A] (f : L →ₗ⁅R A) (g : →ₐ[R] A) :

@[ext]
theorem universal_enveloping_algebra.hom_ext (R : Type u₁) {L : Type u₂} [comm_ring R] [lie_ring L] [ L] {A : Type u₃} [ring A] [ A] {g₁ g₂ : →ₐ[R] A} (h : = ) :
g₁ = g₂