statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
coe_to_linear_map (f : M →ₗ⁅R,L⁆ N) : ((f : M →ₗ[R] N) : M → N) = f | rfl | lemma | lie_module_hom.coe_to_linear_map | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (f : M →ₗ⁅R,L⁆ N) (c : R) (x : M) : f (c • x) = c • f x | linear_map.map_smul (f : M →ₗ[R] N) c x | lemma | lie_module_hom.map_smul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (f : M →ₗ⁅R,L⁆ N) (x y : M) : f (x + y) = (f x) + (f y) | linear_map.map_add (f : M →ₗ[R] N) x y | lemma | lie_module_hom.map_add | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (f : M →ₗ⁅R,L⁆ N) (x y : M) : f (x - y) = (f x) - (f y) | linear_map.map_sub (f : M →ₗ[R] N) x y | lemma | lie_module_hom.map_sub | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (f : M →ₗ⁅R,L⁆ N) (x : M) : f (-x) = -(f x) | linear_map.map_neg (f : M →ₗ[R] N) x | lemma | lie_module_hom.map_neg | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lie (f : M →ₗ⁅R,L⁆ N) (x : L) (m : M) : f ⁅x, m⁆ = ⁅x, f m⁆ | lie_module_hom.map_lie' f | lemma | lie_module_hom.map_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lie₂ (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (x : L) (m : M) (n : N) :
⁅x, f m n⁆ = f ⁅x, m⁆ n + f m ⁅x, n⁆ | by simp only [sub_add_cancel, map_lie, lie_hom.lie_apply] | lemma | lie_module_hom.map_lie₂ | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_hom.lie_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero (f : M →ₗ⁅R,L⁆ N) : f 0 = 0 | linear_map.map_zero (f : M →ₗ[R] N) | lemma | lie_module_hom.map_zero | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : M →ₗ⁅R,L⁆ M | { map_lie' := λ x m, rfl,
.. (linear_map.id : M →ₗ[R] M) } | def | lie_module_hom.id | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.id"
] | The identity map is a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ((id : M →ₗ⁅R,L⁆ M) : M → M) = _root_.id | rfl | lemma | lie_module_hom.coe_id | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (x : M) : (id : M →ₗ⁅R,L⁆ M) x = x | rfl | lemma | lie_module_hom.id_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : M →ₗ⁅R,L⁆ N) : M → N) = 0 | rfl | lemma | lie_module_hom.coe_zero | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (m : M) : (0 : M →ₗ⁅R,L⁆ N) m = 0 | rfl | lemma | lie_module_hom.zero_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (M →ₗ⁅R,L⁆ N) (M → N) coe_fn | by { rintros ⟨⟨f, _⟩⟩ ⟨⟨g, _⟩⟩ ⟨h⟩, congr, } | lemma | lie_module_hom.coe_injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : M →ₗ⁅R,L⁆ N} (h : ∀ m, f m = g m) : f = g | coe_injective $ funext h | lemma | lie_module_hom.ext | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : M →ₗ⁅R,L⁆ N} : f = g ↔ ∀ m, f m = g m | ⟨by { rintro rfl m, refl, }, ext⟩ | lemma | lie_module_hom.ext_iff | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : M →ₗ⁅R,L⁆ N} (h : f = g) (x : M) : f x = g x | h ▸ rfl | lemma | lie_module_hom.congr_fun | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : M →ₗ⁅R,L⁆ N) (h) :
(⟨f, h⟩ : M →ₗ⁅R,L⁆ N) = f | by { ext, refl, } | lemma | lie_module_hom.mk_coe | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : M →ₗ[R] N) (h) :
((⟨f, h⟩ : M →ₗ⁅R,L⁆ N) : M → N) = f | by { ext, refl, } | lemma | lie_module_hom.coe_mk | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_mk (f : M →ₗ[R] N) (h) :
((⟨f, h⟩ : M →ₗ⁅R,L⁆ N) : M →ₗ[R] N) = f | by { ext, refl, } | lemma | lie_module_hom.coe_linear_mk | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) : M →ₗ⁅R,L⁆ P | { map_lie' := λ x m, by { change f (g ⁅x, m⁆) = ⁅x, f (g m)⁆, rw [map_lie, map_lie], },
..linear_map.comp f.to_linear_map g.to_linear_map } | def | lie_module_hom.comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.comp"
] | The composition of Lie module morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) (m : M) :
f.comp g m = f (g m) | rfl | lemma | lie_module_hom.comp_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) :
(f.comp g : M → P) = f ∘ g | rfl | lemma | lie_module_hom.coe_comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map_comp (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) :
(f.comp g : M →ₗ[R] P) = (f : N →ₗ[R] P).comp (g : M →ₗ[R] N) | rfl | lemma | lie_module_hom.coe_linear_map_comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse (f : M →ₗ⁅R,L⁆ N) (g : N → M)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) : N →ₗ⁅R,L⁆ M | { map_lie' := λ x n,
calc g ⁅x, n⁆ = g ⁅x, f (g n)⁆ : by rw h₂
... = g (f ⁅x, g n⁆) : by rw map_lie
... = ⁅x, g n⁆ : (h₁ _),
..linear_map.inverse f.to_linear_map g h₁ h₂ } | def | lie_module_hom.inverse | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.inverse"
] | The inverse of a bijective morphism of Lie modules is a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add (f g : M →ₗ⁅R,L⁆ N) : ⇑(f + g) = f + g | rfl | lemma | lie_module_hom.coe_add | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (f g : M →ₗ⁅R,L⁆ N) (m : M) : (f + g) m = f m + g m | rfl | lemma | lie_module_hom.add_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (f g : M →ₗ⁅R,L⁆ N) : ⇑(f - g) = f - g | rfl | lemma | lie_module_hom.coe_sub | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (f g : M →ₗ⁅R,L⁆ N) (m : M) : (f - g) m = f m - g m | rfl | lemma | lie_module_hom.sub_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (f : M →ₗ⁅R,L⁆ N) : ⇑(-f) = -f | rfl | lemma | lie_module_hom.coe_neg | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply (f : M →ₗ⁅R,L⁆ N) (m : M) : (-f) m = -(f m) | rfl | lemma | lie_module_hom.neg_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul : has_smul ℕ (M →ₗ⁅R,L⁆ N) | { smul := λ n f, { map_lie' := λ x m, by simp, ..(n • (f : M →ₗ[R] N)) } } | instance | lie_module_hom.has_nsmul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (n : ℕ) (f : M →ₗ⁅R,L⁆ N) : ⇑(n • f) = n • f | rfl | lemma | lie_module_hom.coe_nsmul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_apply (n : ℕ) (f : M →ₗ⁅R,L⁆ N) (m : M) : (n • f) m = n • f m | rfl | lemma | lie_module_hom.nsmul_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zsmul : has_smul ℤ (M →ₗ⁅R,L⁆ N) | { smul := λ z f, { map_lie' := λ x m, by simp, ..(z • (f : M →ₗ[R] N)) } } | instance | lie_module_hom.has_zsmul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zsmul (z : ℤ) (f : M →ₗ⁅R,L⁆ N) : ⇑(z • f) = z • f | rfl | lemma | lie_module_hom.coe_zsmul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_apply (z : ℤ) (f : M →ₗ⁅R,L⁆ N) (m : M) : (z • f) m = z • f m | rfl | lemma | lie_module_hom.zsmul_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (t : R) (f : M →ₗ⁅R,L⁆ N) : ⇑(t • f) = t • f | rfl | lemma | lie_module_hom.coe_smul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (t : R) (f : M →ₗ⁅R,L⁆ N) (m : M) : (t • f) m = t • (f m) | rfl | lemma | lie_module_hom.smul_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module_equiv extends M →ₗ⁅R,L⁆ N | (inv_fun : N → M)
(left_inv : function.left_inverse inv_fun to_fun)
(right_inv : function.right_inverse inv_fun to_fun) | structure | lie_module_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"inv_fun"
] | An equivalence of Lie algebra modules is a linear equivalence which is also a morphism of
Lie algebra modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_equiv (e : M ≃ₗ⁅R,L⁆ N) : M ≃ₗ[R] N | { ..e } | def | lie_module_equiv.to_linear_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | View an equivalence of Lie modules as a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_equiv (e : M ≃ₗ⁅R,L⁆ N) : M ≃ N | { ..e } | def | lie_module_equiv.to_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | View an equivalence of Lie modules as a type level equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_equiv : has_coe (M ≃ₗ⁅R,L⁆ N) (M ≃ N) | ⟨to_equiv⟩ | instance | lie_module_equiv.has_coe_to_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_lie_module_hom : has_coe (M ≃ₗ⁅R,L⁆ N) (M →ₗ⁅R,L⁆ N) | ⟨to_lie_module_hom⟩ | instance | lie_module_equiv.has_coe_to_lie_module_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_linear_equiv : has_coe (M ≃ₗ⁅R,L⁆ N) (M ≃ₗ[R] N) | ⟨to_linear_equiv⟩ | instance | lie_module_equiv.has_coe_to_linear_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (e : M ≃ₗ⁅R,L⁆ N) : function.injective e | e.to_equiv.injective | lemma | lie_module_equiv.injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : M →ₗ⁅R,L⁆ N) (inv_fun h₁ h₂) :
((⟨f, inv_fun, h₁, h₂⟩ : M ≃ₗ⁅R,L⁆ N) : M → N) = f | rfl | lemma | lie_module_equiv.coe_mk | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_lie_module_hom (e : M ≃ₗ⁅R,L⁆ N) :
((e : M →ₗ⁅R,L⁆ N) : M → N) = e | rfl | lemma | lie_module_equiv.coe_to_lie_module_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_equiv (e : M ≃ₗ⁅R,L⁆ N) : ((e : M ≃ₗ[R] N) : M → N) = e | rfl | lemma | lie_module_equiv.coe_to_linear_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_injective : function.injective (to_equiv : (M ≃ₗ⁅R,L⁆ N) → M ≃ N) | λ e₁ e₂ h, begin
rcases e₁ with ⟨⟨⟩⟩, rcases e₂ with ⟨⟨⟩⟩,
have inj := equiv.mk.inj h,
dsimp at inj,
apply lie_module_equiv.mk.inj_eq.mpr,
split,
{ congr,
ext,
rw inj.1 },
{ exact inj.2 },
end | lemma | lie_module_equiv.to_equiv_injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (e₁ e₂ : M ≃ₗ⁅R,L⁆ N) (h : ∀ m, e₁ m = e₂ m) : e₁ = e₂ | to_equiv_injective (equiv.ext h) | lemma | lie_module_equiv.ext | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (m : M) : (1 : (M ≃ₗ⁅R,L⁆ M)) m = m | rfl | lemma | lie_module_equiv.one_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : M ≃ₗ⁅R,L⁆ M | 1 | def | lie_module_equiv.refl | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | Lie module equivalences are reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_apply (m : M) : (refl : M ≃ₗ⁅R,L⁆ M) m = m | rfl | lemma | lie_module_equiv.refl_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : M ≃ₗ⁅R,L⁆ N) : N ≃ₗ⁅R,L⁆ M | { ..lie_module_hom.inverse e.to_lie_module_hom e.inv_fun e.left_inv e.right_inv,
..(e : M ≃ₗ[R] N).symm } | def | lie_module_equiv.symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_module_hom.inverse"
] | Lie module equivalences are syemmtric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (e : M ≃ₗ⁅R,L⁆ N) : ∀ x, e (e.symm x) = x | e.to_linear_equiv.apply_symm_apply | lemma | lie_module_equiv.apply_symm_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (e : M ≃ₗ⁅R,L⁆ N) : ∀ x, e.symm (e x) = x | e.to_linear_equiv.symm_apply_apply | lemma | lie_module_equiv.symm_apply_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (e : M ≃ₗ⁅R,L⁆ N) : e.symm.symm = e | by { ext, apply_fun e.symm using e.symm.injective, simp, } | lemma | lie_module_equiv.symm_symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) : M ≃ₗ⁅R,L⁆ P | { ..lie_module_hom.comp e₂.to_lie_module_hom e₁.to_lie_module_hom,
..linear_equiv.trans e₁.to_linear_equiv e₂.to_linear_equiv } | def | lie_module_equiv.trans | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_module_hom.comp",
"linear_equiv.trans"
] | Lie module equivalences are transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) (m : M) :
(e₁.trans e₂) m = e₂ (e₁ m) | rfl | lemma | lie_module_equiv.trans_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) :
(e₁.trans e₂).symm = e₂.symm.trans e₁.symm | rfl | lemma | lie_module_equiv.symm_trans | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_trans_symm (e : M ≃ₗ⁅R,L⁆ N) : e.trans e.symm = refl | ext _ _ e.symm_apply_apply | lemma | lie_module_equiv.self_trans_symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (e : M ≃ₗ⁅R,L⁆ N) : e.symm.trans e = refl | ext _ _ e.apply_symm_apply | lemma | lie_module_equiv.symm_trans_self | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generators
| H : B → generators
| E : B → generators
| F : B → generators | inductive | cartan_matrix.generators | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [] | The generators of the free Lie algebra from which we construct the Lie algebra of a Cartan
matrix as a quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
HH : B × B → free_lie_algebra R (generators B) | uncurry $ λ i j, ⁅H i, H j⁆ | def | cartan_matrix.relations.HH | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra"
] | The terms correpsonding to the `⁅H, H⁆`-relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
EF : B × B → free_lie_algebra R (generators B) | uncurry $ λ i j, if i = j then ⁅E i, F i⁆ - H i else ⁅E i, F j⁆ | def | cartan_matrix.relations.EF | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra"
] | The terms correpsonding to the `⁅E, F⁆`-relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
HE : B × B → free_lie_algebra R (generators B) | uncurry $ λ i j, ⁅H i, E j⁆ - (A i j) • E j | def | cartan_matrix.relations.HE | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra"
] | The terms correpsonding to the `⁅H, E⁆`-relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
HF : B × B → free_lie_algebra R (generators B) | uncurry $ λ i j, ⁅H i, F j⁆ + (A i j) • F j | def | cartan_matrix.relations.HF | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra"
] | The terms correpsonding to the `⁅H, F⁆`-relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ad_E : B × B → free_lie_algebra R (generators B) | uncurry $ λ i j, (ad (E i))^(-A i j).to_nat $ ⁅E i, E j⁆ | def | cartan_matrix.relations.ad_E | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra",
"to_nat"
] | The terms correpsonding to the `ad E`-relations.
Note that we use `int.to_nat` so that we can take the power and that we do not bother
restricting to the case `i ≠ j` since these relations are zero anyway. We also defensively
ensure this with `ad_E_of_eq_eq_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ad_F : B × B → free_lie_algebra R (generators B) | uncurry $ λ i j, (ad (F i))^(-A i j).to_nat $ ⁅F i, F j⁆ | def | cartan_matrix.relations.ad_F | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra",
"to_nat"
] | The terms correpsonding to the `ad F`-relations.
See also `ad_E` docstring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ad_E_of_eq_eq_zero (i : B) (h : A i i = 2) : ad_E R A ⟨i, i⟩ = 0 | have h' : (-2 : ℤ).to_nat = 0, { refl, },
by simp [ad_E, h, h'] | lemma | cartan_matrix.relations.ad_E_of_eq_eq_zero | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ad_F_of_eq_eq_zero (i : B) (h : A i i = 2) : ad_F R A ⟨i, i⟩ = 0 | have h' : (-2 : ℤ).to_nat = 0, { refl, },
by simp [ad_F, h, h'] | lemma | cartan_matrix.relations.ad_F_of_eq_eq_zero | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_set : set (free_lie_algebra R (generators B)) | (set.range $ HH R) ∪
(set.range $ EF R) ∪
(set.range $ HE R A) ∪
(set.range $ HF R A) ∪
(set.range $ ad_E R A) ∪
(set.range $ ad_F R A) | def | cartan_matrix.relations.to_set | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra",
"set.range"
] | The union of all the relations as a subset of the free Lie algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ideal : lie_ideal R (free_lie_algebra R (generators B)) | lie_submodule.lie_span R _ $ to_set R A | def | cartan_matrix.relations.to_ideal | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"free_lie_algebra",
"lie_ideal",
"lie_submodule.lie_span"
] | The ideal of the free Lie algebra generated by the relations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix.to_lie_algebra | free_lie_algebra R _ ⧸ cartan_matrix.relations.to_ideal R A | def | matrix.to_lie_algebra | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"cartan_matrix.relations.to_ideal",
"free_lie_algebra"
] | The Lie algebra corresponding to a Cartan matrix.
Note that it is defined for any matrix of integers. Its value for non-Cartan matrices should be
regarded as junk. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
E₆ : matrix (fin 6) (fin 6) ℤ | !![ 2, 0, -1, 0, 0, 0;
0, 2, 0, -1, 0, 0;
-1, 0, 2, -1, 0, 0;
0, -1, -1, 2, -1, 0;
0, 0, 0, -1, 2, -1;
... | def | cartan_matrix.E₆ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"matrix"
] | The Cartan matrix of type e₆. See [bourbaki1968] plate V, page 277.
The corresponding Dynkin diagram is:
```
o
|
o --- o --- o --- o --- o
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
E₇ : matrix (fin 7) (fin 7) ℤ | !![ 2, 0, -1, 0, 0, 0, 0;
0, 2, 0, -1, 0, 0, 0;
-1, 0, 2, -1, 0, 0, 0;
0, -1, -1, 2, -1, 0, 0;
0, 0, 0, -1, 2, -1, 0;
... | def | cartan_matrix.E₇ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"matrix"
] | The Cartan matrix of type e₇. See [bourbaki1968] plate VI, page 281.
The corresponding Dynkin diagram is:
```
o
|
o --- o --- o --- o --- o --- o
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
E₈ : matrix (fin 8) (fin 8) ℤ | !![ 2, 0, -1, 0, 0, 0, 0, 0;
0, 2, 0, -1, 0, 0, 0, 0;
-1, 0, 2, -1, 0, 0, 0, 0;
0, -1, -1, 2, -1, 0, 0, 0;
0, 0, 0, -1, 2, -1, 0, ... | def | cartan_matrix.E₈ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"matrix"
] | The Cartan matrix of type e₈. See [bourbaki1968] plate VII, page 285.
The corresponding Dynkin diagram is:
```
o
|
o --- o --- o --- o --- o --- o --- o
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
F₄ : matrix (fin 4) (fin 4) ℤ | !![ 2, -1, 0, 0;
-1, 2, -2, 0;
0, -1, 2, -1;
0, 0, -1, 2] | def | cartan_matrix.F₄ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"matrix"
] | The Cartan matrix of type f₄. See [bourbaki1968] plate VIII, page 288.
The corresponding Dynkin diagram is:
```
o --- o =>= o --- o
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
G₂ : matrix (fin 2) (fin 2) ℤ | !![ 2, -3;
-1, 2] | def | cartan_matrix.G₂ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [
"matrix"
] | The Cartan matrix of type g₂. See [bourbaki1968] plate IX, page 290.
The corresponding Dynkin diagram is:
```
o ≡>≡ o
```
Actually we are using the transpose of Bourbaki's matrix. This is to make this matrix consistent
with `cartan_matrix.F₄`, in the sense that all non-zero values below the diagonal are -1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e₆ | cartan_matrix.E₆.to_lie_algebra R | abbreviation | lie_algebra.e₆ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [] | The exceptional split Lie algebra of type e₆. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e₇ | cartan_matrix.E₇.to_lie_algebra R | abbreviation | lie_algebra.e₇ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [] | The exceptional split Lie algebra of type e₇. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
e₈ | cartan_matrix.E₈.to_lie_algebra R | abbreviation | lie_algebra.e₈ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [] | The exceptional split Lie algebra of type e₈. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
f₄ | cartan_matrix.F₄.to_lie_algebra R | abbreviation | lie_algebra.f₄ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [] | The exceptional split Lie algebra of type f₄. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
g₂ | cartan_matrix.G₂.to_lie_algebra R | abbreviation | lie_algebra.g₂ | algebra.lie | src/algebra/lie/cartan_matrix.lean | [
"algebra.lie.free",
"algebra.lie.quotient",
"data.matrix.notation"
] | [] | The exceptional split Lie algebra of type g₂. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_submodule.is_ucs_limit
{M : Type*} [add_comm_group M] [module R M] [lie_ring_module L M] [lie_module R L M]
(N : lie_submodule R L M) : Prop | ∃ k, ∀ l, k ≤ l → (⊥ : lie_submodule R L M).ucs l = N | def | lie_submodule.is_ucs_limit | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"add_comm_group",
"lie_module",
"lie_ring_module",
"lie_submodule",
"module"
] | Given a Lie module `M` of a Lie algebra `L`, `lie_submodule.is_ucs_limit` is the proposition
that a Lie submodule `N ⊆ M` is the limiting value for the upper central series.
This is a characteristic property of Cartan subalgebras with the roles of `L`, `M`, `N` played by
`H`, `L`, `H`, respectively. See `lie_subalgebr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cartan_subalgebra : Prop | (nilpotent : lie_algebra.is_nilpotent R H)
(self_normalizing : H.normalizer = H) | class | lie_subalgebra.is_cartan_subalgebra | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"lie_algebra.is_nilpotent"
] | A Cartan subalgebra is a nilpotent, self-normalizing subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normalizer_eq_self_of_is_cartan_subalgebra
(H : lie_subalgebra R L) [H.is_cartan_subalgebra] :
H.to_lie_submodule.normalizer = H.to_lie_submodule | by rw [← lie_submodule.coe_to_submodule_eq_iff, coe_normalizer_eq_normalizer,
is_cartan_subalgebra.self_normalizing, coe_to_lie_submodule] | lemma | lie_subalgebra.normalizer_eq_self_of_is_cartan_subalgebra | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"lie_subalgebra",
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_eq_self_of_is_cartan_subalgebra
(H : lie_subalgebra R L) [H.is_cartan_subalgebra] (k : ℕ) :
H.to_lie_submodule.ucs k = H.to_lie_submodule | begin
induction k with k ih,
{ simp, },
{ simp [ih], },
end | lemma | lie_subalgebra.ucs_eq_self_of_is_cartan_subalgebra | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"ih",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cartan_subalgebra_iff_is_ucs_limit :
H.is_cartan_subalgebra ↔ H.to_lie_submodule.is_ucs_limit | begin
split,
{ introsI h,
have h₁ : _root_.lie_algebra.is_nilpotent R H := by apply_instance,
obtain ⟨k, hk⟩ := H.to_lie_submodule.is_nilpotent_iff_exists_self_le_ucs.mp h₁,
replace hk : H.to_lie_submodule = lie_submodule.ucs k ⊥ :=
le_antisymm hk (lie_submodule.ucs_le_of_normalizer_eq_self
... | lemma | lie_subalgebra.is_cartan_subalgebra_iff_is_ucs_limit | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"lie_algebra.is_nilpotent",
"lie_subalgebra.coe_normalizer_eq_normalizer",
"lie_subalgebra.coe_to_lie_submodule",
"lie_subalgebra.coe_to_submodule_eq_iff",
"lie_subalgebra.ucs_eq_self_of_is_cartan_subalgebra",
"lie_submodule.lcs_le_iff",
"lie_submodule.ucs",
"lie_submodule.ucs_add",
"lie_submodule.u... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.normalizer_eq_top {R : Type u} {L : Type v}
[comm_ring R] [lie_ring L] [lie_algebra R L] (I : lie_ideal R L) :
(I : lie_subalgebra R L).normalizer = ⊤ | begin
ext x,
simpa only [lie_subalgebra.mem_normalizer_iff, lie_subalgebra.mem_top, iff_true]
using λ y hy, I.lie_mem hy
end | lemma | lie_ideal.normalizer_eq_top | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"comm_ring",
"lie_algebra",
"lie_ideal",
"lie_ring",
"lie_subalgebra",
"lie_subalgebra.mem_normalizer_iff",
"lie_subalgebra.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.top_is_cartan_subalgebra_of_nilpotent [lie_algebra.is_nilpotent R L] :
lie_subalgebra.is_cartan_subalgebra (⊤ : lie_subalgebra R L) | { nilpotent := infer_instance,
self_normalizing :=
by { rw [← top_coe_lie_subalgebra, normalizer_eq_top, top_coe_lie_subalgebra], }, } | instance | lie_algebra.top_is_cartan_subalgebra_of_nilpotent | algebra.lie | src/algebra/lie/cartan_subalgebra.lean | [
"algebra.lie.nilpotent",
"algebra.lie.normalizer"
] | [
"lie_algebra.is_nilpotent",
"lie_subalgebra",
"lie_subalgebra.is_cartan_subalgebra"
] | A nilpotent Lie algebra is its own Cartan subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_character | L →ₗ⁅R⁆ R | abbreviation | lie_algebra.lie_character | algebra.lie | src/algebra/lie/character.lean | [
"algebra.lie.abelian",
"algebra.lie.solvable",
"linear_algebra.dual"
] | [] | A character of a Lie algebra is a morphism to the scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_character_apply_lie (χ : lie_character R L) (x y : L) : χ ⁅x, y⁆ = 0 | by rw [lie_hom.map_lie, lie_ring.of_associative_ring_bracket, mul_comm, sub_self] | lemma | lie_algebra.lie_character_apply_lie | algebra.lie | src/algebra/lie/character.lean | [
"algebra.lie.abelian",
"algebra.lie.solvable",
"linear_algebra.dual"
] | [
"lie_hom.map_lie",
"lie_ring.of_associative_ring_bracket",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_character_apply_of_mem_derived
(χ : lie_character R L) {x : L} (h : x ∈ derived_series R L 1) : χ x = 0 | begin
rw [derived_series_def, derived_series_of_ideal_succ, derived_series_of_ideal_zero,
← lie_submodule.mem_coe_submodule, lie_submodule.lie_ideal_oper_eq_linear_span] at h,
apply submodule.span_induction h,
{ rintros y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩, apply lie_character_apply_lie, },
{ exact χ.map_zero, },
... | lemma | lie_algebra.lie_character_apply_of_mem_derived | algebra.lie | src/algebra/lie/character.lean | [
"algebra.lie.abelian",
"algebra.lie.solvable",
"linear_algebra.dual"
] | [
"derived_series",
"lie_hom.map_add",
"lie_hom.map_smul",
"lie_submodule.lie_ideal_oper_eq_linear_span",
"lie_submodule.mem_coe_submodule",
"smul_zero",
"submodule.span_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_character_equiv_linear_dual [is_lie_abelian L] :
lie_character R L ≃ module.dual R L | { to_fun := λ χ, (χ : L →ₗ[R] R),
inv_fun := λ ψ,
{ map_lie' := λ x y, by
rw [lie_module.is_trivial.trivial, lie_ring.of_associative_ring_bracket, mul_comm, sub_self,
linear_map.to_fun_eq_coe, linear_map.map_zero],
.. ψ, },
left_inv := λ χ, by { ext, refl, },
right_inv := λ ψ, by { ext, refl... | def | lie_algebra.lie_character_equiv_linear_dual | algebra.lie | src/algebra/lie/character.lean | [
"algebra.lie.abelian",
"algebra.lie.solvable",
"linear_algebra.dual"
] | [
"inv_fun",
"is_lie_abelian",
"lie_ring.of_associative_ring_bracket",
"linear_map.map_zero",
"linear_map.to_fun_eq_coe",
"module.dual",
"mul_comm"
] | For an Abelian Lie algebra, characters are just linear forms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix_trace_commutator_zero [fintype n] (X Y : matrix n n R) :
matrix.trace ⁅X, Y⁆ = 0 | calc _ = matrix.trace (X ⬝ Y) - matrix.trace (Y ⬝ X) : trace_sub _ _
... = matrix.trace (X ⬝ Y) - matrix.trace (X ⬝ Y) :
congr_arg (λ x, _ - x) (matrix.trace_mul_comm Y X)
... = 0 : sub_self _ | lemma | lie_algebra.matrix_trace_commutator_zero | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"matrix",
"matrix.trace",
"matrix.trace_mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sl [fintype n] : lie_subalgebra R (matrix n n R) | { lie_mem' := λ X Y _ _, linear_map.mem_ker.2 $ matrix_trace_commutator_zero _ _ _ _,
..linear_map.ker (matrix.trace_linear_map n R R) } | def | lie_algebra.special_linear.sl | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype",
"lie_subalgebra",
"linear_map.ker",
"matrix",
"matrix.trace_linear_map"
] | The special linear Lie algebra: square matrices of trace zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sl_bracket [fintype n] (A B : sl n R) : ⁅A, B⁆.val = A.val ⬝ B.val - B.val ⬝ A.val | rfl | lemma | lie_algebra.special_linear.sl_bracket | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Eb (h : j ≠ i) : sl n R | ⟨matrix.std_basis_matrix i j (1 : R),
show matrix.std_basis_matrix i j (1 : R) ∈ linear_map.ker (matrix.trace_linear_map n R R),
from matrix.std_basis_matrix.trace_zero i j (1 : R) h⟩ | def | lie_algebra.special_linear.Eb | algebra.lie | src/algebra/lie/classical.lean | [
"algebra.invertible",
"data.matrix.basis",
"data.matrix.dmatrix",
"algebra.lie.abelian",
"linear_algebra.matrix.trace",
"algebra.lie.skew_adjoint",
"linear_algebra.symplectic_group"
] | [
"linear_map.ker",
"matrix.std_basis_matrix",
"matrix.std_basis_matrix.trace_zero",
"matrix.trace_linear_map"
] | When j ≠ i, the elementary matrices are elements of sl n R, in fact they are part of a natural
basis of sl n R. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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