algebra.lie.direct_sum - mathlib docs

@[instance]

def direct_sum.lie_ring_module {ι : Type v} {L : Type w₁} {M : ι → Type w} [lie_ring L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), lie_ring_module L (M i)] :

lie_ring_module L (⨁ (i : ι), M i)

Equations

direct_sum.lie_ring_module = {to_has_bracket := {bracket := λ (x : L) (m : ⨁ (i : ι), M i), dfinsupp.map_range (λ (i : ι) (m' : (λ (i : ι), M i) i), ⁅x,m'⁆) _ m}, add_lie := _, lie_add := _, leibniz_lie := _}

source

@[simp]

theorem direct_sum.lie_module_bracket_apply {ι : Type v} {L : Type w₁} {M : ι → Type w} [lie_ring L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), lie_ring_module L (M i)] (x : L) (m : ⨁ (i : ι), M i) (i : ι) :

⇑⁅x,m⁆ i = ⁅x,⇑m i⁆

source

@[instance]

def direct_sum.lie_module {R : Type u} {ι : Type v} [comm_ring R] {L : Type w₁} {M : ι → Type w} [lie_ring L] [lie_algebra R L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [Π (i : ι), lie_ring_module L (M i)] [Π (i : ι), lie_module R L (M i)] :

lie_module R L (⨁ (i : ι), M i)

Equations

direct_sum.lie_module = {smul_lie := _, lie_smul := _}

source

def direct_sum.lie_module_of (R : Type u) (ι : Type v) [comm_ring R] (L : Type w₁) (M : ι → Type w) [lie_ring L] [lie_algebra R L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [Π (i : ι), lie_ring_module L (M i)] [Π (i : ι), lie_module R L (M i)] [decidable_eq ι] (j : ι) :

M j →ₗ⁅R,L⁆ ⨁ (i : ι), M i

The inclusion of each component into a direct sum as a morphism of Lie modules.

Equations

direct_sum.lie_module_of R ι L M j = {to_fun := (direct_sum.lof R ι M j).to_fun, map_add' := _, map_smul' := _, map_lie' := _}

source

def direct_sum.lie_module_component (R : Type u) (ι : Type v) [comm_ring R] (L : Type w₁) (M : ι → Type w) [lie_ring L] [lie_algebra R L] [Π (i : ι), add_comm_group (M i)] [Π (i : ι), module R (M i)] [Π (i : ι), lie_ring_module L (M i)] [Π (i : ι), lie_module R L (M i)] (j : ι) :

(⨁ (i : ι), M i) →ₗ⁅R,L⁆ M j

The projection map onto one component, as a morphism of Lie modules.

Equations

direct_sum.lie_module_component R ι L M j = {to_fun := (direct_sum.component R ι M j).to_fun, map_add' := _, map_smul' := _, map_lie' := _}

source

@[instance]

def direct_sum.lie_ring {ι : Type v} (L : ι → Type w) [Π (i : ι), lie_ring (L i)] :

lie_ring (⨁ (i : ι), L i)

Equations

direct_sum.lie_ring L = {to_add_comm_group := {add := add_comm_group.add infer_instance, add_assoc := _, zero := add_comm_group.zero infer_instance, zero_add := _, add_zero := _, nsmul := add_comm_group.nsmul infer_instance, nsmul_zero' := _, nsmul_succ' := _, neg := add_comm_group.neg infer_instance, sub := add_comm_group.sub infer_instance, sub_eq_add_neg := _, gsmul := add_comm_group.gsmul infer_instance, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _}, to_has_bracket := {bracket := dfinsupp.zip_with (λ (i : ι) (x y : (λ (i : ι), L i) i), ⁅x,y⁆) _}, add_lie := _, lie_add := _, lie_self := _, leibniz_lie := _}

source

@[simp]

theorem direct_sum.bracket_apply {ι : Type v} (L : ι → Type w) [Π (i : ι), lie_ring (L i)] (x y : ⨁ (i : ι), L i) (i : ι) :

⇑⁅x,y⁆ i = ⁅⇑x i,⇑y i⁆

source

@[instance]

def direct_sum.lie_algebra {R : Type u} {ι : Type v} [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] :

lie_algebra R (⨁ (i : ι), L i)

Equations

direct_sum.lie_algebra L = {to_module := {to_distrib_mul_action := module.to_distrib_mul_action infer_instance, add_smul := _, zero_smul := _}, lie_smul := _}

source

@[simp]

theorem direct_sum.lie_algebra_of_apply (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] (j : ι) (ᾰ : L j) :

⇑(direct_sum.lie_algebra_of R ι L j) ᾰ = ⇑(direct_sum.of L j) ᾰ

source

def direct_sum.lie_algebra_of (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] (j : ι) :

L j →ₗ⁅R⁆ ⨁ (i : ι), L i

The inclusion of each component into the direct sum as morphism of Lie algebras.

Equations

direct_sum.lie_algebra_of R ι L j = {to_fun := ⇑(direct_sum.of L j), map_add' := _, map_smul' := _, map_lie' := _}

source

def direct_sum.lie_algebra_component (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] (j : ι) :

(⨁ (i : ι), L i) →ₗ⁅R⁆ L j

The projection map onto one component, as a morphism of Lie algebras.

Equations

direct_sum.lie_algebra_component R ι L j = {to_fun := ⇑(direct_sum.component R ι L j), map_add' := _, map_smul' := _, map_lie' := _}

source

@[simp]

theorem direct_sum.lie_algebra_component_apply (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] (j : ι) (ᾰ : ⨁ (i : ι), L i) :

⇑(direct_sum.lie_algebra_component R ι L j) ᾰ = ⇑(direct_sum.component R ι L j) ᾰ

source

@[ext]

theorem direct_sum.lie_algebra_ext (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] {x y : ⨁ (i : ι), L i} (h : ∀ (i : ι), ⇑(direct_sum.lie_algebra_component R ι L i) x = ⇑(direct_sum.lie_algebra_component R ι L i) y) :

x = y

source

theorem direct_sum.lie_of_of_ne (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] {i j : ι} (hij : j ≠ i) (x : L i) (y : L j) :

⁅⇑(direct_sum.of L i) x,⇑(direct_sum.of L j) y⁆ = 0

source

theorem direct_sum.lie_of_of_eq (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] {i j : ι} (hij : j = i) (x : L i) (y : L j) :

⁅⇑(direct_sum.of L i) x,⇑(direct_sum.of L j) y⁆ = ⇑(direct_sum.of L i) ⁅x,hij.rec_on y⁆

source

@[simp]

theorem direct_sum.lie_of (R : Type u) (ι : Type v) [comm_ring R] (L : ι → Type w) [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] {i j : ι} (x : L i) (y : L j) :

⁅⇑(direct_sum.of L i) x,⇑(direct_sum.of L j) y⁆ = dite (j = i) (λ (hij : j = i), ⇑(direct_sum.lie_algebra_of R ι L i) ⁅x,hij.rec_on y⁆) (λ (hij : ¬j = i), 0)

source

def direct_sum.to_lie_algebra {R : Type u} {ι : Type v} [comm_ring R] {L : ι → Type w} [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] (L' : Type w₁) [lie_ring L'] [lie_algebra R L'] (f : Π (i : ι), L i →ₗ⁅R⁆ L') (hf : ∀ (i j : ι), i ≠ j → ∀ (x : L i) (y : L j), ⁅⇑(f i) x,⇑(f j) y⁆ = 0) :

(⨁ (i : ι), L i) →ₗ⁅R⁆ L'

Given a family of Lie algebras L i, together with a family of morphisms of Lie algebras f i : L i →ₗ⁅R⁆ L' into a fixed Lie algebra L', we have a natural linear map: (⨁ i, L i) →ₗ[R] L'. If in addition ⁅f i x, f j y⁆ = 0 for any x ∈ L i and y ∈ L j (i ≠ j) then this map is a morphism of Lie algebras.

Equations

direct_sum.to_lie_algebra L' f hf = {to_fun := ⇑(direct_sum.to_module R ι L' (λ (i : ι), ↑(f i))), map_add' := _, map_smul' := _, map_lie' := _}

source

@[simp]

theorem direct_sum.to_lie_algebra_apply {R : Type u} {ι : Type v} [comm_ring R] {L : ι → Type w} [Π (i : ι), lie_ring (L i)] [Π (i : ι), lie_algebra R (L i)] [decidable_eq ι] (L' : Type w₁) [lie_ring L'] [lie_algebra R L'] (f : Π (i : ι), L i →ₗ⁅R⁆ L') (hf : ∀ (i j : ι), i ≠ j → ∀ (x : L i) (y : L j), ⁅⇑(f i) x,⇑(f j) y⁆ = 0) (ᾰ : ⨁ (i : ι), (λ (i : ι), L i) i) :

⇑(direct_sum.to_lie_algebra L' f hf) ᾰ = ⇑(direct_sum.to_module R ι L' (λ (i : ι), ↑(f i))) ᾰ

source

def direct_sum.lie_algebra_is_internal {R : Type u} {ι : Type v} [comm_ring R] {L : Type w} [lie_ring L] [lie_algebra R L] (I : ι → lie_ideal R L) [decidable_eq ι] :

Prop

Given a Lie algebra L and a family of ideals I i ⊆ L, informally this definition is the statement that L = ⨁ i, I i.

More formally, the inclusions give a natural map from the (external) direct sum to the enclosing Lie algebra: (⨁ i, I i) → L, and this definition is the proposition that this map is bijective.

Equations

direct_sum.lie_algebra_is_internal I = function.bijective ⇑(direct_sum.to_module R ι L (λ (i : ι), ↑((I i).incl)))

source

@[instance]

def direct_sum.lie_ring_of_ideals {R : Type u} {ι : Type v} [comm_ring R] {L : Type w} [lie_ring L] [lie_algebra R L] (I : ι → lie_ideal R L) :

lie_ring (⨁ (i : ι), ↥(I i))

The fact that this instance is necessary seems to be a bug in typeclass inference. See [this Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/ Typeclass.20resolution.20under.20binders/near/245151099).

Equations

direct_sum.lie_ring_of_ideals I = direct_sum.lie_ring (λ (i : ι), ↥(I i))

source

@[instance]

def direct_sum.lie_algebra_of_ideals {R : Type u} {ι : Type v} [comm_ring R] {L : Type w} [lie_ring L] [lie_algebra R L] (I : ι → lie_ideal R L) :

lie_algebra R (⨁ (i : ι), ↥(I i))

See direct_sum.lie_ring_of_ideals comment.

Equations

direct_sum.lie_algebra_of_ideals I = direct_sum.lie_algebra (λ (i : ι), ↥(I i))

mathlib documentation

Direct sums of Lie algebras and Lie modules #

Tags #