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normalizer : lie_submodule R L M | { carrier := { m | ∀ (x : L), ⁅x, m⁆ ∈ N },
add_mem' := λ m₁ m₂ hm₁ hm₂ x, by { rw lie_add, exact N.add_mem' (hm₁ x) (hm₂ x), },
zero_mem' := λ x, by simp,
smul_mem' := λ t m hm x, by { rw lie_smul, exact N.smul_mem' t (hm x), },
lie_mem := λ x m hm y, by { rw leibniz_lie, exact N.add_mem' (hm ⁅y, x⁆) (N.... | def | lie_submodule.normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"leibniz_lie",
"lie_add",
"lie_smul",
"lie_submodule"
] | The normalizer of a Lie submodule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_normalizer (m : M) :
m ∈ N.normalizer ↔ ∀ (x : L), ⁅x, m⁆ ∈ N | iff.rfl | lemma | lie_submodule.mem_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_normalizer : N ≤ N.normalizer | begin
intros m hm,
rw mem_normalizer,
exact λ x, N.lie_mem hm,
end | lemma | lie_submodule.le_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalizer_inf :
(N₁ ⊓ N₂).normalizer = N₁.normalizer ⊓ N₂.normalizer | by { ext, simp [← forall_and_distrib], } | lemma | lie_submodule.normalizer_inf | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"forall_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_normalizer :
monotone (normalizer : lie_submodule R L M → lie_submodule R L M) | begin
intros N₁ N₂ h m hm,
rw mem_normalizer at hm ⊢,
exact λ x, h (hm x),
end | lemma | lie_submodule.monotone_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"lie_submodule",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_normalizer (f : M' →ₗ⁅R,L⁆ M) :
N.normalizer.comap f = (N.comap f).normalizer | by { ext, simp, } | lemma | lie_submodule.comap_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_lie_le_iff_le_normalizer (N' : lie_submodule R L M) :
⁅(⊤ : lie_ideal R L), N⁆ ≤ N' ↔ N ≤ N'.normalizer | by { rw lie_le_iff, tauto, } | lemma | lie_submodule.top_lie_le_iff_le_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"lie_ideal",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_top_lie_normalizer :
galois_connection (λ N : lie_submodule R L M, ⁅(⊤ : lie_ideal R L), N⁆) normalizer | top_lie_le_iff_le_normalizer | lemma | lie_submodule.gc_top_lie_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"galois_connection",
"lie_ideal",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalizer_bot_eq_max_triv_submodule :
(⊥ : lie_submodule R L M).normalizer = lie_module.max_triv_submodule R L M | rfl | lemma | lie_submodule.normalizer_bot_eq_max_triv_submodule | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"lie_module.max_triv_submodule",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalizer : lie_subalgebra R L | { lie_mem' := λ y z hy hz x,
begin
rw [coe_bracket_of_module, mem_to_lie_submodule, leibniz_lie, ← lie_skew y, ← sub_eq_add_neg],
exact H.sub_mem (hz ⟨_, hy x⟩) (hy ⟨_, hz x⟩),
end,
.. H.to_lie_submodule.normalizer } | def | lie_subalgebra.normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"leibniz_lie",
"lie_skew",
"lie_subalgebra"
] | Regarding a Lie subalgebra `H ⊆ L` as a module over itself, its normalizer is in fact a Lie
subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_normalizer_iff' (x : L) : x ∈ H.normalizer ↔ ∀ (y : L), (y ∈ H) → ⁅y, x⁆ ∈ H | by { rw subtype.forall', refl, } | lemma | lie_subalgebra.mem_normalizer_iff' | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"subtype.forall'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_normalizer_iff (x : L) : x ∈ H.normalizer ↔ ∀ (y : L), (y ∈ H) → ⁅x, y⁆ ∈ H | begin
rw mem_normalizer_iff',
refine forall₂_congr (λ y hy, _),
rw [← lie_skew, neg_mem_iff],
end | lemma | lie_subalgebra.mem_normalizer_iff | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"forall₂_congr",
"lie_skew"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_normalizer : H ≤ H.normalizer | H.to_lie_submodule.le_normalizer | lemma | lie_subalgebra.le_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_normalizer_eq_normalizer :
(H.to_lie_submodule.normalizer : submodule R L) = H.normalizer | rfl | lemma | lie_subalgebra.coe_normalizer_eq_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_mem_sup_of_mem_normalizer {x y z : L} (hx : x ∈ H.normalizer)
(hy : y ∈ (R ∙ x) ⊔ ↑H) (hz : z ∈ (R ∙ x) ⊔ ↑H) : ⁅y, z⁆ ∈ (R ∙ x) ⊔ ↑H | begin
rw submodule.mem_sup at hy hz,
obtain ⟨u₁, hu₁, v, hv : v ∈ H, rfl⟩ := hy,
obtain ⟨u₂, hu₂, w, hw : w ∈ H, rfl⟩ := hz,
obtain ⟨t, rfl⟩ := submodule.mem_span_singleton.mp hu₁,
obtain ⟨s, rfl⟩ := submodule.mem_span_singleton.mp hu₂,
apply submodule.mem_sup_right,
simp only [lie_subalgebra.mem_coe_subm... | lemma | lie_subalgebra.lie_mem_sup_of_mem_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"add_lie",
"lie_add",
"lie_self",
"lie_smul",
"lie_subalgebra.mem_coe_submodule",
"smul_lie",
"smul_zero",
"submodule.mem_sup",
"submodule.mem_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_in_normalizer {x y : L} (hx : x ∈ H.normalizer) (hy : y ∈ H) : ⁅x,y⁆ ∈ H | begin
rw [← lie_skew, neg_mem_iff],
exact hx ⟨y, hy⟩,
end | lemma | lie_subalgebra.ideal_in_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"lie_skew"
] | A Lie subalgebra is an ideal of its normalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nested_lie_ideal_of_le_normalizer
{K : lie_subalgebra R L} (h₁ : H ≤ K) (h₂ : K ≤ H.normalizer) :
∃ (I : lie_ideal R K), (I : lie_subalgebra R K) = of_le h₁ | begin
rw exists_nested_lie_ideal_coe_eq_iff,
exact λ x y hx hy, ideal_in_normalizer (h₂ hx) hy,
end | lemma | lie_subalgebra.exists_nested_lie_ideal_of_le_normalizer | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"lie_ideal",
"lie_subalgebra"
] | A Lie subalgebra `H` is an ideal of any Lie subalgebra `K` containing `H` and contained in the
normalizer of `H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalizer_eq_self_iff :
H.normalizer = H ↔ (lie_module.max_triv_submodule R H $ L ⧸ H.to_lie_submodule) = ⊥ | begin
rw lie_submodule.eq_bot_iff,
refine ⟨λ h, _, λ h, le_antisymm (λ x hx, _) H.le_normalizer⟩,
{ rintros ⟨x⟩ hx,
suffices : x ∈ H, by simpa,
rw [← h, H.mem_normalizer_iff'],
intros y hy,
replace hx : ⁅_, lie_submodule.quotient.mk' _ x⁆ = 0 := hx ⟨y, hy⟩,
rwa [← lie_module_hom.map_lie, lie_s... | lemma | lie_subalgebra.normalizer_eq_self_iff | algebra.lie | src/algebra/lie/normalizer.lean | [
"algebra.lie.abelian",
"algebra.lie.ideal_operations",
"algebra.lie.quotient"
] | [
"lie_module.max_triv_submodule",
"lie_module_hom.map_lie",
"lie_submodule.eq_bot_iff",
"lie_submodule.quotient.mk'",
"lie_submodule.quotient.mk_eq_zero",
"submodule.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_def (x y : A) : ⁅x, y⁆ = x*y - y*x | rfl | lemma | ring.lie_def | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_iff_lie_eq {x y : A} : commute x y ↔ ⁅x, y⁆ = 0 | sub_eq_zero.symm | lemma | commute_iff_lie_eq | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.lie_eq {x y : A} (h : commute x y) : ⁅x, y⁆ = 0 | sub_eq_zero_of_eq h | lemma | commute.lie_eq | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_associative_ring : lie_ring A | { add_lie := by simp only [ring.lie_def, right_distrib, left_distrib,
sub_eq_add_neg, add_comm, add_left_comm, forall_const, eq_self_iff_true, neg_add_rev],
lie_add := by simp only [ring.lie_def, right_distrib, left_distrib,
sub_eq_add_neg, add_comm, add_left_comm, forall_const, eq_self_iff_true, ne... | instance | lie_ring.of_associative_ring | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"add_lie",
"forall_const",
"left_distrib",
"leibniz_lie",
"lie_add",
"lie_ring",
"lie_self",
"right_distrib",
"ring.lie_def"
] | An associative ring gives rise to a Lie ring by taking the bracket to be the ring commutator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_associative_ring_bracket (x y : A) : ⁅x, y⁆ = x*y - y*x | rfl | lemma | lie_ring.of_associative_ring_bracket | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_apply {α : Type*} (f g : α → A) (a : α) : ⁅f, g⁆ a = ⁅f a, g a⁆ | rfl | lemma | lie_ring.lie_apply | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ring_module.of_associative_module : lie_ring_module A M | { bracket := (•),
add_lie := add_smul,
lie_add := smul_add,
leibniz_lie :=
by simp [lie_ring.of_associative_ring_bracket, sub_smul, mul_smul, sub_add_cancel], } | def | lie_ring_module.of_associative_module | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"add_lie",
"add_smul",
"leibniz_lie",
"lie_add",
"lie_ring.of_associative_ring_bracket",
"lie_ring_module",
"smul_add",
"sub_smul"
] | We can regard a module over an associative ring `A` as a Lie ring module over `A` with Lie
bracket equal to its ring commutator.
Note that this cannot be a global instance because it would create a diamond when `M = A`,
specifically we can build two mathematically-different `has_bracket A A`s:
1. `@ring.has_bracket A... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_eq_smul (a : A) (m : M) : ⁅a, m⁆ = a • m | rfl | lemma | lie_eq_smul | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.of_associative_algebra : lie_algebra R A | { lie_smul := λ t x y,
by rw [lie_ring.of_associative_ring_bracket, lie_ring.of_associative_ring_bracket,
algebra.mul_smul_comm, algebra.smul_mul_assoc, smul_sub], } | instance | lie_algebra.of_associative_algebra | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"algebra.mul_smul_comm",
"algebra.smul_mul_assoc",
"lie_algebra",
"lie_ring.of_associative_ring_bracket",
"lie_smul",
"smul_sub"
] | An associative algebra gives rise to a Lie algebra by taking the bracket to be the ring
commutator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module.of_associative_module : lie_module R A M | { smul_lie := smul_assoc,
lie_smul := smul_algebra_smul_comm } | def | lie_module.of_associative_module | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_module",
"lie_smul",
"smul_algebra_smul_comm",
"smul_assoc",
"smul_lie"
] | A representation of an associative algebra `A` is also a representation of `A`, regarded as a
Lie algebra via the ring commutator.
See the comment at `lie_ring_module.of_associative_module` for why the possibility `M = A` means
this cannot be a global instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.End.lie_ring_module : lie_ring_module (module.End R M) M | lie_ring_module.of_associative_module | instance | module.End.lie_ring_module | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_ring_module",
"lie_ring_module.of_associative_module",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.End.lie_module : lie_module R (module.End R M) M | lie_module.of_associative_module | instance | module.End.lie_module | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_module",
"lie_module.of_associative_module",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_hom : A →ₗ⁅R⁆ B | { map_lie' := λ x y, show f ⁅x,y⁆ = ⁅f x,f y⁆,
by simp only [lie_ring.of_associative_ring_bracket, alg_hom.map_sub, alg_hom.map_mul],
..f.to_linear_map, } | def | alg_hom.to_lie_hom | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"alg_hom.map_mul",
"alg_hom.map_sub",
"lie_ring.of_associative_ring_bracket"
] | The map `of_associative_algebra` associating a Lie algebra to an associative algebra is
functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_lie_hom_coe : f.to_lie_hom = ↑f | rfl | lemma | alg_hom.to_lie_hom_coe | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_lie_hom : ((f : A →ₗ⁅R⁆ B) : A → B) = f | rfl | lemma | alg_hom.coe_to_lie_hom | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_hom_apply (x : A) : f.to_lie_hom x = f x | rfl | lemma | alg_hom.to_lie_hom_apply | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_hom_id : (alg_hom.id R A : A →ₗ⁅R⁆ A) = lie_hom.id | rfl | lemma | alg_hom.to_lie_hom_id | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"alg_hom.id",
"lie_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_hom_comp : (g.comp f : A →ₗ⁅R⁆ C) = (g : B →ₗ⁅R⁆ C).comp (f : A →ₗ⁅R⁆ B) | rfl | lemma | alg_hom.to_lie_hom_comp | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_hom_injective {f g : A →ₐ[R] B}
(h : (f : A →ₗ⁅R⁆ B) = (g : A →ₗ⁅R⁆ B)) : f = g | by { ext a, exact lie_hom.congr_fun h a, } | lemma | alg_hom.to_lie_hom_injective | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module.to_endomorphism : L →ₗ⁅R⁆ module.End R M | { to_fun := λ x,
{ to_fun := λ m, ⁅x, m⁆,
map_add' := lie_add x,
map_smul' := λ t, lie_smul t x, },
map_add' := λ x y, by { ext m, apply add_lie, },
map_smul' := λ t x, by { ext m, apply smul_lie, },
map_lie' := λ x y, by { ext m, apply lie_lie, }, } | def | lie_module.to_endomorphism | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"add_lie",
"lie_add",
"lie_lie",
"lie_smul",
"module.End",
"smul_lie"
] | A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.
See also `lie_module.to_module_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.ad : L →ₗ⁅R⁆ module.End R L | lie_module.to_endomorphism R L L | def | lie_algebra.ad | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_module.to_endomorphism",
"module.End"
] | The adjoint action of a Lie algebra on itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.ad_apply (x y : L) : lie_algebra.ad R L x y = ⁅x, y⁆ | rfl | lemma | lie_algebra.ad_apply | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_algebra.ad"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module.to_endomorphism_module_End :
lie_module.to_endomorphism R (module.End R M) M = lie_hom.id | by { ext g m, simp [lie_eq_smul], } | lemma | lie_module.to_endomorphism_module_End | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_eq_smul",
"lie_hom.id",
"lie_module.to_endomorphism",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra.to_endomorphism_eq (K : lie_subalgebra R L) {x : K} :
lie_module.to_endomorphism R K M x = lie_module.to_endomorphism R L M x | rfl | lemma | lie_subalgebra.to_endomorphism_eq | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_module.to_endomorphism",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra.to_endomorphism_mk (K : lie_subalgebra R L) {x : L} (hx : x ∈ K) :
lie_module.to_endomorphism R K M ⟨x, hx⟩ = lie_module.to_endomorphism R L M x | rfl | lemma | lie_subalgebra.to_endomorphism_mk | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_module.to_endomorphism",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_map_to_endomorphism_le :
(N : submodule R M).map (lie_module.to_endomorphism R L M x) ≤ N | begin
rintros n ⟨m, hm, rfl⟩,
exact N.lie_mem hm,
end | lemma | lie_submodule.coe_map_to_endomorphism_le | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_module.to_endomorphism",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_endomorphism_comp_subtype_mem (m : M) (hm : m ∈ (N : submodule R M)) :
(to_endomorphism R L M x).comp (N : submodule R M).subtype ⟨m, hm⟩ ∈ (N : submodule R M) | by simpa using N.lie_mem hm | lemma | lie_submodule.to_endomorphism_comp_subtype_mem | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_endomorphism_restrict_eq_to_endomorphism
(h := N.to_endomorphism_comp_subtype_mem x) :
(to_endomorphism R L M x).restrict h = to_endomorphism R L N x | by { ext, simp [linear_map.restrict_apply], } | lemma | lie_submodule.to_endomorphism_restrict_eq_to_endomorphism | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"linear_map.restrict_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.ad_eq_lmul_left_sub_lmul_right (A : Type v) [ring A] [algebra R A] :
(ad R A : A → module.End R A) = linear_map.mul_left R - linear_map.mul_right R | by { ext a b, simp [lie_ring.of_associative_ring_bracket], } | lemma | lie_algebra.ad_eq_lmul_left_sub_lmul_right | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"algebra",
"lie_ring.of_associative_ring_bracket",
"linear_map.mul_left",
"linear_map.mul_right",
"module.End",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra.ad_comp_incl_eq (K : lie_subalgebra R L) (x : K) :
(ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) | begin
ext y,
simp only [ad_apply, lie_hom.coe_to_linear_map, lie_subalgebra.coe_incl, linear_map.coe_comp,
lie_subalgebra.coe_bracket, function.comp_app],
end | lemma | lie_subalgebra.ad_comp_incl_eq | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_hom.coe_to_linear_map",
"lie_subalgebra",
"lie_subalgebra.coe_bracket",
"lie_subalgebra.coe_incl",
"linear_map.coe_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra_of_subalgebra (R : Type u) [comm_ring R] (A : Type v) [ring A] [algebra R A]
(A' : subalgebra R A) : lie_subalgebra R A | { lie_mem' := λ x y hx hy, by
{ change ⁅x, y⁆ ∈ A', change x ∈ A' at hx, change y ∈ A' at hy,
rw lie_ring.of_associative_ring_bracket,
have hxy := A'.mul_mem hx hy,
have hyx := A'.mul_mem hy hx,
exact submodule.sub_mem A'.to_submodule hxy hyx, },
..A'.to_submodule } | def | lie_subalgebra_of_subalgebra | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"algebra",
"comm_ring",
"lie_ring.of_associative_ring_bracket",
"lie_subalgebra",
"ring",
"subalgebra",
"submodule.sub_mem"
] | A subalgebra of an associative algebra is a Lie subalgebra of the associated Lie algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_conj : module.End R M₁ ≃ₗ⁅R⁆ module.End R M₂ | { map_lie' := λ f g, show e.conj ⁅f, g⁆ = ⁅e.conj f, e.conj g⁆,
by simp only [lie_ring.of_associative_ring_bracket, linear_map.mul_eq_comp, e.conj_comp,
linear_equiv.map_sub],
..e.conj } | def | linear_equiv.lie_conj | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_ring.of_associative_ring_bracket",
"linear_equiv.map_sub",
"linear_map.mul_eq_comp",
"module.End"
] | A linear equivalence of two modules induces a Lie algebra equivalence of their endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_conj_apply (f : module.End R M₁) : e.lie_conj f = e.conj f | rfl | lemma | linear_equiv.lie_conj_apply | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_conj_symm : e.lie_conj.symm = e.symm.lie_conj | rfl | lemma | linear_equiv.lie_conj_symm | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_equiv : A₁ ≃ₗ⁅R⁆ A₂ | { to_fun := e.to_fun,
map_lie' := λ x y, by simp [lie_ring.of_associative_ring_bracket],
..e.to_linear_equiv } | def | alg_equiv.to_lie_equiv | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [
"lie_ring.of_associative_ring_bracket"
] | An equivalence of associative algebras is an equivalence of associated Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_lie_equiv_apply (x : A₁) : e.to_lie_equiv x = e x | rfl | lemma | alg_equiv.to_lie_equiv_apply | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_equiv_symm_apply (x : A₂) : e.to_lie_equiv.symm x = e.symm x | rfl | lemma | alg_equiv.to_lie_equiv_symm_apply | algebra.lie | src/algebra/lie/of_associative.lean | [
"algebra.lie.basic",
"algebra.lie.subalgebra",
"algebra.lie.submodule",
"algebra.algebra.subalgebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group : add_comm_group (M ⧸ N) | submodule.quotient.add_comm_group _ | instance | lie_submodule.quotient.add_comm_group | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"add_comm_group",
"submodule.quotient.add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' {S : Type*} [semiring S] [has_smul S R] [module S M] [is_scalar_tower S R M] :
module S (M ⧸ N) | submodule.quotient.module' _ | instance | lie_submodule.quotient.module' | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"has_smul",
"is_scalar_tower",
"module",
"semiring",
"submodule.quotient.module'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module : module R (M ⧸ N) | submodule.quotient.module _ | instance | lie_submodule.quotient.module | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"module",
"submodule.quotient.module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_central_scalar {S : Type*} [semiring S]
[has_smul S R] [module S M] [is_scalar_tower S R M]
[has_smul Sᵐᵒᵖ R] [module Sᵐᵒᵖ M] [is_scalar_tower Sᵐᵒᵖ R M]
[is_central_scalar S M] : is_central_scalar S (M ⧸ N) | submodule.quotient.is_central_scalar _ | instance | lie_submodule.quotient.is_central_scalar | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"has_smul",
"is_central_scalar",
"is_scalar_tower",
"module",
"semiring",
"submodule.quotient.is_central_scalar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited : inhabited (M ⧸ N) | ⟨0⟩ | instance | lie_submodule.quotient.inhabited | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk : M → M ⧸ N | submodule.quotient.mk | abbreviation | lie_submodule.quotient.mk | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"submodule.quotient.mk"
] | Map sending an element of `M` to the corresponding element of `M/N`, when `N` is a
lie_submodule of the lie_module `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_quotient_mk (m : M) : quotient.mk' m = (mk m : M ⧸ N) | rfl | lemma | lie_submodule.quotient.is_quotient_mk | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_submodule_invariant : L →ₗ[R] submodule.compatible_maps N.to_submodule N.to_submodule | linear_map.cod_restrict _ (lie_module.to_endomorphism R L M) $ λ _ _, N.lie_mem | def | lie_submodule.quotient.lie_submodule_invariant | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_module.to_endomorphism",
"linear_map.cod_restrict",
"submodule.compatible_maps"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
action_as_endo_map : L →ₗ⁅R⁆ module.End R (M ⧸ N) | { map_lie' := λ x y, submodule.linear_map_qext _ $ linear_map.ext $ λ m,
congr_arg mk $ lie_lie _ _ _,
..linear_map.comp (submodule.mapq_linear (N : submodule R M) ↑N) lie_submodule_invariant } | def | lie_submodule.quotient.action_as_endo_map | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_lie",
"linear_map.comp",
"linear_map.ext",
"module.End",
"submodule",
"submodule.linear_map_qext",
"submodule.mapq_linear"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
action_as_endo_map_bracket : has_bracket L (M ⧸ N) | ⟨λ x n, action_as_endo_map N x n⟩ | instance | lie_submodule.quotient.action_as_endo_map_bracket | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"has_bracket"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_quotient_lie_ring_module : lie_ring_module L (M ⧸ N) | { bracket := has_bracket.bracket,
..lie_ring_module.comp_lie_hom _ (action_as_endo_map N) } | instance | lie_submodule.quotient.lie_quotient_lie_ring_module | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_ring_module",
"lie_ring_module.comp_lie_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_quotient_lie_module : lie_module R L (M ⧸ N) | lie_module.comp_lie_hom _ (action_as_endo_map N) | instance | lie_submodule.quotient.lie_quotient_lie_module | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_module",
"lie_module.comp_lie_hom"
] | The quotient of a Lie module by a Lie submodule, is a Lie module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_quotient_has_bracket : has_bracket (L ⧸ I) (L ⧸ I) | ⟨begin
intros x y,
apply quotient.lift_on₂' x y (λ x' y', mk ⁅x', y'⁆),
intros x₁ x₂ y₁ y₂ h₁ h₂,
apply (submodule.quotient.eq I.to_submodule).2,
rw submodule.quotient_rel_r_def at h₁ h₂,
have h : ⁅x₁, x₂⁆ - ⁅y₁, y₂⁆ = ⁅x₁, x₂ - y₂⁆ + ⁅x₁ - y₁, y₂⁆,
by simp [-lie_skew, sub_eq_add_neg, add_assoc],
rw h... | instance | lie_submodule.quotient.lie_quotient_has_bracket | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"has_bracket",
"lie_mem_left",
"lie_mem_right",
"lie_skew",
"quotient.lift_on₂'",
"submodule.add_mem",
"submodule.quotient.eq",
"submodule.quotient_rel_r_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bracket (x y : L) :
mk ⁅x, y⁆ = ⁅(mk x : L ⧸ I), (mk y : L ⧸ I)⁆ | rfl | lemma | lie_submodule.quotient.mk_bracket | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_quotient_lie_ring : lie_ring (L ⧸ I) | { add_lie := by { intros x' y' z', apply quotient.induction_on₃' x' y' z', intros x y z,
repeat { rw is_quotient_mk <|>
rw ←mk_bracket <|>
rw ←submodule.quotient.mk_add, },
apply congr_arg, apply add_lie, },
lie_add := by ... | instance | lie_submodule.quotient.lie_quotient_lie_ring | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"add_lie",
"leibniz_lie",
"lie_add",
"lie_ring",
"lie_self",
"quotient.induction_on'",
"quotient.induction_on₃'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_quotient_lie_algebra : lie_algebra R (L ⧸ I) | { lie_smul := by { intros t x' y', apply quotient.induction_on₂' x' y', intros x y,
repeat { rw is_quotient_mk <|>
rw ←mk_bracket <|>
rw ←submodule.quotient.mk_smul, },
apply congr_arg, apply lie_smul, } } | instance | lie_submodule.quotient.lie_quotient_lie_algebra | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_algebra",
"lie_smul",
"quotient.induction_on₂'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' : M →ₗ⁅R,L⁆ M ⧸ N | { to_fun := mk, map_lie' := λ r m, rfl, ..N.to_submodule.mkq} | def | lie_submodule.quotient.mk' | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"mk'"
] | `lie_submodule.quotient.mk` as a `lie_module_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_eq_zero {m : M} : mk' N m = 0 ↔ m ∈ N | submodule.quotient.mk_eq_zero N.to_submodule | lemma | lie_submodule.quotient.mk_eq_zero | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"mk'",
"submodule.quotient.mk_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_ker : (mk' N).ker = N | by { ext, simp, } | lemma | lie_submodule.quotient.mk'_ker | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk'_eq_bot_le : map (mk' N) N' = ⊥ ↔ N' ≤ N | by rw [← lie_module_hom.le_ker_iff_map, mk'_ker] | lemma | lie_submodule.quotient.map_mk'_eq_bot_le | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_module_hom.le_ker_iff_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module_hom_ext ⦃f g : M ⧸ N →ₗ⁅R,L⁆ M⦄ (h : f.comp (mk' N) = g.comp (mk' N)) :
f = g | lie_module_hom.ext $ λ x, quotient.induction_on' x $ lie_module_hom.congr_fun h | lemma | lie_submodule.quotient.lie_module_hom_ext | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_module_hom.congr_fun",
"lie_module_hom.ext",
"mk'",
"quotient.induction_on'"
] | Two `lie_module_hom`s from a quotient lie module are equal if their compositions with
`lie_submodule.quotient.mk'` are equal.
See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_ker_equiv_range : L ⧸ f.ker ≃ₗ⁅R⁆ f.range | { to_fun := (f : L →ₗ[R] L').quot_ker_equiv_range,
map_lie' :=
begin
rintros ⟨x⟩ ⟨y⟩,
rw [← set_like.coe_eq_coe, lie_subalgebra.coe_bracket],
simp only [submodule.quotient.quot_mk_eq_mk, linear_map.quot_ker_equiv_range_apply_mk,
← lie_submodule.quotient.mk_bracket, coe_to_linear_map, map_lie],
e... | def | lie_hom.quot_ker_equiv_range | algebra.lie | src/algebra/lie/quotient.lean | [
"algebra.lie.submodule",
"algebra.lie.of_associative",
"linear_algebra.isomorphisms"
] | [
"lie_subalgebra.coe_bracket",
"lie_submodule.quotient.mk_bracket",
"linear_map.quot_ker_equiv_range_apply_mk",
"set_like.coe_eq_coe",
"submodule.quotient.quot_mk_eq_mk"
] | The first isomorphism theorem for morphisms of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module.is_irreducible (R : Type u) (L : Type v) (M : Type w)
[comm_ring R] [lie_ring L] [lie_algebra R L] [add_comm_group M] [module R M]
[lie_ring_module L M] [lie_module R L M] : Prop | (irreducible : ∀ (N : lie_submodule R L M), N ≠ ⊥ → N = ⊤) | class | lie_module.is_irreducible | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"add_comm_group",
"comm_ring",
"irreducible",
"lie_algebra",
"lie_module",
"lie_ring",
"lie_ring_module",
"lie_submodule",
"module"
] | A Lie module is irreducible if it is zero or its only non-trivial Lie submodule is itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_simple extends lie_module.is_irreducible R L L : Prop | (non_abelian : ¬is_lie_abelian L) | class | lie_algebra.is_simple | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"is_lie_abelian",
"lie_module.is_irreducible"
] | A Lie algebra is simple if it is irreducible as a Lie module over itself via the adjoint
action, and it is non-Abelian. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_semisimple : Prop | (semisimple : radical R L = ⊥) | class | lie_algebra.is_semisimple | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [] | A semisimple Lie algebra is one with trivial radical.
Note that the label 'semisimple' is apparently not universally agreed
[upon](https://mathoverflow.net/questions/149391/on-radicals-of-a-lie-algebra#comment383669_149391)
for general coefficients. We are following [Seligman, page 15](seligman1967) and using the labe... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_semisimple_iff_no_solvable_ideals :
is_semisimple R L ↔ ∀ (I : lie_ideal R L), is_solvable R I → I = ⊥ | ⟨λ h, Sup_eq_bot.mp h.semisimple, λ h, ⟨Sup_eq_bot.mpr h⟩⟩ | lemma | lie_algebra.is_semisimple_iff_no_solvable_ideals | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"is_solvable",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_semisimple_iff_no_abelian_ideals :
is_semisimple R L ↔ ∀ (I : lie_ideal R L), is_lie_abelian I → I = ⊥ | begin
rw is_semisimple_iff_no_solvable_ideals,
split; intros h₁ I h₂,
{ haveI : is_lie_abelian I := h₂, apply h₁, exact lie_algebra.of_abelian_is_solvable R I, },
{ haveI : is_solvable R I := h₂, rw ← abelian_of_solvable_ideal_eq_bot_iff, apply h₁,
exact abelian_derived_abelian_of_ideal I, },
end | lemma | lie_algebra.is_semisimple_iff_no_abelian_ideals | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"is_lie_abelian",
"is_solvable",
"lie_algebra.of_abelian_is_solvable",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
center_eq_bot_of_semisimple [h : is_semisimple R L] : center R L = ⊥ | by { rw is_semisimple_iff_no_abelian_ideals at h, apply h, apply_instance, } | lemma | lie_algebra.center_eq_bot_of_semisimple | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_semisimple_of_is_simple [h : is_simple R L] : is_semisimple R L | begin
rw is_semisimple_iff_no_abelian_ideals,
intros I hI,
obtain @⟨⟨h₁⟩, h₂⟩ := id h,
by_contradiction contra,
rw [h₁ I contra, lie_abelian_iff_equiv_lie_abelian lie_ideal.top_equiv] at hI,
exact h₂ hI,
end | instance | lie_algebra.is_semisimple_of_is_simple | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"by_contradiction",
"lie_abelian_iff_equiv_lie_abelian",
"lie_ideal.top_equiv"
] | A simple Lie algebra is semisimple. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_of_semisimple_lie_abelian [is_semisimple R L] [h : is_lie_abelian L] :
subsingleton L | begin
rw [is_lie_abelian_iff_center_eq_top R L, center_eq_bot_of_semisimple] at h,
exact (lie_submodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h),
end | lemma | lie_algebra.subsingleton_of_semisimple_lie_abelian | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"is_lie_abelian",
"lie_submodule.subsingleton_iff",
"subsingleton_of_bot_eq_top"
] | A semisimple Abelian Lie algebra is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abelian_radical_of_semisimple [is_semisimple R L] : is_lie_abelian (radical R L) | by { rw is_semisimple.semisimple, exact is_lie_abelian_bot R L, } | lemma | lie_algebra.abelian_radical_of_semisimple | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"is_lie_abelian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abelian_radical_iff_solvable_is_abelian [is_noetherian R L] :
is_lie_abelian (radical R L) ↔ ∀ (I : lie_ideal R L), is_solvable R I → is_lie_abelian I | begin
split,
{ rintros h₁ I h₂,
rw lie_ideal.solvable_iff_le_radical at h₂,
exact (lie_ideal.hom_of_le_injective h₂).is_lie_abelian h₁, },
{ intros h, apply h, apply_instance, },
end | lemma | lie_algebra.abelian_radical_iff_solvable_is_abelian | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [
"is_lie_abelian",
"is_noetherian",
"is_solvable",
"lie_ideal",
"lie_ideal.hom_of_le_injective"
] | The two properties shown to be equivalent here are possible definitions for a Lie algebra
to be reductive.
Note that there is absolutely [no agreement](https://mathoverflow.net/questions/284713/) on what
the label 'reductive' should mean when the coefficients are not a field of characteristic zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ad_ker_eq_bot_of_semisimple [is_semisimple R L] : (ad R L).ker = ⊥ | by simp | lemma | lie_algebra.ad_ker_eq_bot_of_semisimple | algebra.lie | src/algebra/lie/semisimple.lean | [
"algebra.lie.solvable"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bilin_form.is_skew_adjoint_bracket (f g : module.End R M)
(hf : f ∈ B.skew_adjoint_submodule) (hg : g ∈ B.skew_adjoint_submodule) :
⁅f, g⁆ ∈ B.skew_adjoint_submodule | begin
rw mem_skew_adjoint_submodule at *,
have hfg : is_adjoint_pair B B (f * g) (g * f), { rw ←neg_mul_neg g f, exact hf.mul hg, },
have hgf : is_adjoint_pair B B (g * f) (f * g), { rw ←neg_mul_neg f g, exact hg.mul hf, },
change bilin_form.is_adjoint_pair B B (f * g - g * f) (-(f * g - g * f)), rw neg_sub,
... | lemma | bilin_form.is_skew_adjoint_bracket | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"bilin_form.is_adjoint_pair",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skew_adjoint_lie_subalgebra : lie_subalgebra R (module.End R M) | { lie_mem' := B.is_skew_adjoint_bracket, ..B.skew_adjoint_submodule } | def | skew_adjoint_lie_subalgebra | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"lie_subalgebra",
"module.End"
] | Given an `R`-module `M`, equipped with a bilinear form, the skew-adjoint endomorphisms form a
Lie subalgebra of the Lie algebra of endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_adjoint_lie_subalgebra_equiv :
skew_adjoint_lie_subalgebra (B.comp (↑e : N →ₗ[R] M) ↑e) ≃ₗ⁅R⁆ skew_adjoint_lie_subalgebra B | begin
apply lie_equiv.of_subalgebras _ _ e.lie_conj,
ext f,
simp only [lie_subalgebra.mem_coe, submodule.mem_map_equiv, lie_subalgebra.mem_map_submodule,
coe_coe],
exact (bilin_form.is_pair_self_adjoint_equiv (-B) B e f).symm,
end | def | skew_adjoint_lie_subalgebra_equiv | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"bilin_form.is_pair_self_adjoint_equiv",
"coe_coe",
"lie_equiv.of_subalgebras",
"lie_subalgebra.mem_coe",
"lie_subalgebra.mem_map_submodule",
"skew_adjoint_lie_subalgebra",
"submodule.mem_map_equiv"
] | An equivalence of modules with bilinear forms gives equivalence of Lie algebras of skew-adjoint
endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_adjoint_lie_subalgebra_equiv_apply
(f : skew_adjoint_lie_subalgebra (B.comp ↑e ↑e)) :
↑(skew_adjoint_lie_subalgebra_equiv B e f) = e.lie_conj f | by simp [skew_adjoint_lie_subalgebra_equiv] | lemma | skew_adjoint_lie_subalgebra_equiv_apply | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"skew_adjoint_lie_subalgebra",
"skew_adjoint_lie_subalgebra_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skew_adjoint_lie_subalgebra_equiv_symm_apply (f : skew_adjoint_lie_subalgebra B) :
↑((skew_adjoint_lie_subalgebra_equiv B e).symm f) = e.symm.lie_conj f | by simp [skew_adjoint_lie_subalgebra_equiv] | lemma | skew_adjoint_lie_subalgebra_equiv_symm_apply | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"skew_adjoint_lie_subalgebra",
"skew_adjoint_lie_subalgebra_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.lie_transpose (A B : matrix n n R) : ⁅A, B⁆ᵀ = ⁅Bᵀ, Aᵀ⁆ | show (A * B - B * A)ᵀ = (Bᵀ * Aᵀ - Aᵀ * Bᵀ), by simp | lemma | matrix.lie_transpose | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.is_skew_adjoint_bracket (A B : matrix n n R)
(hA : A ∈ skew_adjoint_matrices_submodule J) (hB : B ∈ skew_adjoint_matrices_submodule J) :
⁅A, B⁆ ∈ skew_adjoint_matrices_submodule J | begin
simp only [mem_skew_adjoint_matrices_submodule] at *,
change ⁅A, B⁆ᵀ ⬝ J = J ⬝ -⁅A, B⁆, change Aᵀ ⬝ J = J ⬝ -A at hA, change Bᵀ ⬝ J = J ⬝ -B at hB,
simp only [←matrix.mul_eq_mul] at *,
rw [matrix.lie_transpose, lie_ring.of_associative_ring_bracket,
lie_ring.of_associative_ring_bracket, sub_mul, mul_as... | lemma | matrix.is_skew_adjoint_bracket | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"lie_ring.of_associative_ring_bracket",
"matrix",
"matrix.lie_transpose",
"mem_skew_adjoint_matrices_submodule",
"mul_assoc",
"skew_adjoint_matrices_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skew_adjoint_matrices_lie_subalgebra : lie_subalgebra R (matrix n n R) | { lie_mem' := J.is_skew_adjoint_bracket, ..(skew_adjoint_matrices_submodule J) } | def | skew_adjoint_matrices_lie_subalgebra | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"lie_subalgebra",
"matrix",
"skew_adjoint_matrices_submodule"
] | The Lie subalgebra of skew-adjoint square matrices corresponding to a square matrix `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_skew_adjoint_matrices_lie_subalgebra (A : matrix n n R) :
A ∈ skew_adjoint_matrices_lie_subalgebra J ↔ A ∈ skew_adjoint_matrices_submodule J | iff.rfl | lemma | mem_skew_adjoint_matrices_lie_subalgebra | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"matrix",
"skew_adjoint_matrices_lie_subalgebra",
"skew_adjoint_matrices_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skew_adjoint_matrices_lie_subalgebra_equiv (P : matrix n n R) (h : invertible P) :
skew_adjoint_matrices_lie_subalgebra J ≃ₗ⁅R⁆ skew_adjoint_matrices_lie_subalgebra (Pᵀ ⬝ J ⬝ P) | lie_equiv.of_subalgebras _ _ (P.lie_conj h).symm
begin
ext A,
suffices : P.lie_conj h A ∈ skew_adjoint_matrices_submodule J ↔
A ∈ skew_adjoint_matrices_submodule (Pᵀ ⬝ J ⬝ P),
{ simp only [lie_subalgebra.mem_coe, submodule.mem_map_equiv, lie_subalgebra.mem_map_submodule,
coe_coe], exact this, },
simp ... | def | skew_adjoint_matrices_lie_subalgebra_equiv | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"coe_coe",
"invertible",
"is_unit_of_invertible",
"lie_equiv.of_subalgebras",
"lie_subalgebra.mem_coe",
"lie_subalgebra.mem_map_submodule",
"matrix",
"matrix.is_skew_adjoint",
"skew_adjoint_matrices_lie_subalgebra",
"skew_adjoint_matrices_submodule",
"submodule.mem_map_equiv"
] | An invertible matrix `P` gives a Lie algebra equivalence between those endomorphisms that are
skew-adjoint with respect to a square matrix `J` and those with respect to `PᵀJP`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_adjoint_matrices_lie_subalgebra_equiv_apply
(P : matrix n n R) (h : invertible P) (A : skew_adjoint_matrices_lie_subalgebra J) :
↑(skew_adjoint_matrices_lie_subalgebra_equiv J P h A) = P⁻¹ ⬝ ↑A ⬝ P | by simp [skew_adjoint_matrices_lie_subalgebra_equiv] | lemma | skew_adjoint_matrices_lie_subalgebra_equiv_apply | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"invertible",
"matrix",
"skew_adjoint_matrices_lie_subalgebra",
"skew_adjoint_matrices_lie_subalgebra_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
skew_adjoint_matrices_lie_subalgebra_equiv_transpose {m : Type w} [decidable_eq m] [fintype m]
(e : matrix n n R ≃ₐ[R] matrix m m R) (h : ∀ A, (e A)ᵀ = e (Aᵀ)) :
skew_adjoint_matrices_lie_subalgebra J ≃ₗ⁅R⁆ skew_adjoint_matrices_lie_subalgebra (e J) | lie_equiv.of_subalgebras _ _ e.to_lie_equiv
begin
ext A,
suffices : J.is_skew_adjoint (e.symm A) ↔ (e J).is_skew_adjoint A, by simpa [this],
simp [matrix.is_skew_adjoint, matrix.is_adjoint_pair, ← matrix.mul_eq_mul,
← h, ← function.injective.eq_iff e.injective],
end | def | skew_adjoint_matrices_lie_subalgebra_equiv_transpose | algebra.lie | src/algebra/lie/skew_adjoint.lean | [
"algebra.lie.matrix",
"linear_algebra.matrix.bilinear_form"
] | [
"fintype",
"function.injective.eq_iff",
"lie_equiv.of_subalgebras",
"matrix",
"matrix.is_adjoint_pair",
"matrix.is_skew_adjoint",
"matrix.mul_eq_mul",
"skew_adjoint_matrices_lie_subalgebra"
] | An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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