# Documentation

Mathlib.Algebra.Lie.Quotient

# 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.instHasQuotientLieSubmodule {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R 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} [] [] [] [Module R M] [] {N : LieSubmodule R L M} :
instance LieSubmodule.Quotient.module' {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] {N : LieSubmodule R L M} {S : Type u_1} [] [SMul S R] [Module S M] [] :
Module S (M N)
instance LieSubmodule.Quotient.module {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] {N : LieSubmodule R L M} :
Module R (M N)
instance LieSubmodule.Quotient.isCentralScalar {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] {N : LieSubmodule R L M} {S : Type u_1} [] [SMul S R] [Module S M] [] [SMul Sᵐᵒᵖ R] [] [] [] :
instance LieSubmodule.Quotient.inhabited {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] {N : LieSubmodule R L M} :
@[inline, reducible]
abbrev LieSubmodule.Quotient.mk {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] {N : LieSubmodule R L M} :
MM N

Map sending an element of M to the corresponding element of M/N, when N is a lie_submodule of the lie_module N.

Instances For
theorem LieSubmodule.Quotient.is_quotient_mk {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] {N : LieSubmodule R L M} (m : M) :
def LieSubmodule.Quotient.lieSubmoduleInvariant {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] {N : LieSubmodule R L M} :
L →ₗ[R] { x // x }

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.

Instances For
def LieSubmodule.Quotient.actionAsEndoMap {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) :

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.

Instances For
instance LieSubmodule.Quotient.actionAsEndoMapBracket {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule 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} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) :
instance LieSubmodule.Quotient.lieQuotientLieModule {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule 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} [] [] [] {I : LieIdeal R L} :
Bracket (L I) (L I)
@[simp]
theorem LieSubmodule.Quotient.mk_bracket {R : Type u} {L : Type v} [] [] [] {I : LieIdeal R L} (x : L) (y : L) :
instance LieSubmodule.Quotient.lieQuotientLieRing {R : Type u} {L : Type v} [] [] [] {I : LieIdeal R L} :
LieRing (L I)
instance LieSubmodule.Quotient.lieQuotientLieAlgebra {R : Type u} {L : Type v} [] [] [] {I : LieIdeal R L} :
LieAlgebra R (L I)
@[simp]
theorem LieSubmodule.Quotient.mk'_toFun {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) :
∀ (a : M),
def LieSubmodule.Quotient.mk' {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) :

LieSubmodule.Quotient.mk as a LieModuleHom.

Instances For
theorem LieSubmodule.Quotient.mk_eq_zero {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) {m : M} :
= 0 m N
@[simp]
theorem LieSubmodule.Quotient.mk_eq_zero' {R : Type u} {L : Type v} {M : Type w} [] [] [] [Module R M] [] (N : LieSubmodule R L M) {m : M} :
m N
@[simp]
theorem LieSubmodule.Quotient.mk'_ker {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) :
@[simp]
theorem LieSubmodule.Quotient.map_mk'_eq_bot_le {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) (N' : LieSubmodule R L M) :
N' N
theorem LieSubmodule.Quotient.lieModuleHom_ext {R : Type u} {L : Type v} {M : Type w} [] [] [] [] [Module R M] [] [LieModule R L M] (N : LieSubmodule R L M) ⦃f : M N →ₗ⁅R,L M ⦃g : M N →ₗ⁅R,L M :
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].

@[simp]
theorem LieHom.quotKerEquivRange_invFun {R : Type u_1} {L : Type u_2} {L' : Type u_3} [] [] [] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R L') :
∀ (a : { x // x }),
@[simp]
theorem LieHom.quotKerEquivRange_toFun {R : Type u_1} {L : Type u_2} {L' : Type u_3} [] [] [] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R L') (a : L ) :
↑() a = ↑() a
noncomputable def LieHom.quotKerEquivRange {R : Type u_1} {L : Type u_2} {L' : Type u_3} [] [] [] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R L') :
(L ) ≃ₗ⁅R { x // }

The first isomorphism theorem for morphisms of Lie algebras.

Instances For