statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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map_bracket_eq : map f ⁅I, N⁆ = ⁅I, map f N⁆ | begin
rw [← coe_to_submodule_eq_iff, coe_submodule_map, lie_ideal_oper_eq_linear_span,
lie_ideal_oper_eq_linear_span, submodule.map_span],
congr,
ext m,
split,
{ rintros ⟨-, ⟨⟨x, ⟨n, hn⟩, rfl⟩, hm⟩⟩,
simp only [lie_module_hom.coe_to_linear_map, lie_module_hom.map_lie] at hm,
exact ⟨x, ⟨f n, (mem_m... | lemma | lie_submodule.map_bracket_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_module_hom.coe_to_linear_map",
"lie_module_hom.map_lie",
"mem_map",
"submodule.map_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_le : map f (comap f N₂) ≤ N₂ | (N₂ : set M₂).image_preimage_subset f | lemma | lie_submodule.map_comap_le | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_eq (hf : N₂ ≤ f.range) : map f (comap f N₂) = N₂ | begin
rw set_like.ext'_iff,
exact set.image_preimage_eq_of_subset hf,
end | lemma | lie_submodule.map_comap_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"set.image_preimage_eq_of_subset",
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_map : N ≤ comap f (map f N) | (N : set M).subset_preimage_image f | lemma | lie_submodule.le_comap_map | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map_eq (hf : f.ker = ⊥) : comap f (map f N) = N | begin
rw set_like.ext'_iff,
exact (N : set M).preimage_image_eq (f.ker_eq_bot.mp hf),
end | lemma | lie_submodule.comap_map_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bracket_eq (hf₁ : f.ker = ⊥) (hf₂ : N₂ ≤ f.range) :
comap f ⁅I, N₂⁆ = ⁅I, comap f N₂⁆ | begin
conv_lhs { rw ← map_comap_eq N₂ f hf₂, },
rw [← map_bracket_eq, comap_map_eq _ f hf₁],
end | lemma | lie_submodule.comap_bracket_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_incl : map N.incl (comap N.incl N') = N ⊓ N' | begin
rw ← coe_to_submodule_eq_iff,
exact (N : submodule R M).map_comap_subtype N',
end | lemma | lie_submodule.map_comap_incl | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bracket_le {I₁ I₂ : lie_ideal R L} : map f ⁅I₁, I₂⁆ ≤ ⁅map f I₁, map f I₂⁆ | begin
rw map_le_iff_le_comap, erw lie_submodule.lie_span_le,
intros x hx, obtain ⟨⟨y₁, hy₁⟩, ⟨y₂, hy₂⟩, hx⟩ := hx, rw ← hx,
let fy₁ : ↥(map f I₁) := ⟨f y₁, mem_map hy₁⟩,
let fy₂ : ↥(map f I₂) := ⟨f y₂, mem_map hy₂⟩,
change _ ∈ comap f ⁅map f I₁, map f I₂⁆,
simp only [submodule.coe_mk, mem_comap, lie_hom.map... | lemma | lie_ideal.map_bracket_le | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_hom.map_lie",
"lie_ideal",
"lie_submodule.lie_coe_mem_lie",
"lie_submodule.lie_span_le",
"mem_map",
"submodule.coe_mk"
] | Note that the inequality can be strict; e.g., the inclusion of an Abelian subalgebra of a
simple algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_bracket_eq {I₁ I₂ : lie_ideal R L} (h : function.surjective f) :
map f ⁅I₁, I₂⁆ = ⁅map f I₁, map f I₂⁆ | begin
suffices : ⁅map f I₁, map f I₂⁆ ≤ map f ⁅I₁, I₂⁆, { exact le_antisymm (map_bracket_le f) this, },
rw [← lie_submodule.coe_submodule_le_coe_submodule, coe_map_of_surjective h,
lie_submodule.lie_ideal_oper_eq_linear_span,
lie_submodule.lie_ideal_oper_eq_linear_span, linear_map.map_span],
apply submodu... | lemma | lie_ideal.map_bracket_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_ideal",
"lie_submodule.coe_submodule_le_coe_submodule",
"lie_submodule.lie_ideal_oper_eq_linear_span",
"submodule.span_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bracket_le {J₁ J₂ : lie_ideal R L'} : ⁅comap f J₁, comap f J₂⁆ ≤ comap f ⁅J₁, J₂⁆ | begin
rw ← map_le_iff_le_comap,
exact le_trans (map_bracket_le f) (lie_submodule.mono_lie _ _ _ _ map_comap_le map_comap_le),
end | lemma | lie_ideal.comap_bracket_le | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_ideal",
"lie_submodule.mono_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_incl {I₁ I₂ : lie_ideal R L} : map I₁.incl (comap I₁.incl I₂) = I₁ ⊓ I₂ | by { conv_rhs { rw ← I₁.incl_ideal_range, }, rw ← map_comap_eq, exact I₁.incl_is_ideal_morphism, } | lemma | lie_ideal.map_comap_incl | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bracket_eq {J₁ J₂ : lie_ideal R L'} (h : f.is_ideal_morphism) :
comap f ⁅f.ideal_range ⊓ J₁, f.ideal_range ⊓ J₂⁆ = ⁅comap f J₁, comap f J₂⁆ ⊔ f.ker | begin
rw [← lie_submodule.coe_to_submodule_eq_iff, comap_coe_submodule,
lie_submodule.sup_coe_to_submodule, f.ker_coe_submodule, ← submodule.comap_map_eq,
lie_submodule.lie_ideal_oper_eq_linear_span, lie_submodule.lie_ideal_oper_eq_linear_span,
linear_map.map_span],
congr, simp only [lie_hom.coe_to_line... | lemma | lie_ideal.comap_bracket_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_hom.coe_to_linear_map",
"lie_hom.map_lie",
"lie_ideal",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.lie_ideal_oper_eq_linear_span",
"lie_submodule.mem_inf",
"lie_submodule.sup_coe_to_submodule",
"submodule.coe_mk",
"submodule.comap_map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_bracket_eq {J₁ J₂ : lie_ideal R L'} (h : f.is_ideal_morphism) :
map f ⁅comap f J₁, comap f J₂⁆ = ⁅f.ideal_range ⊓ J₁, f.ideal_range ⊓ J₂⁆ | by { rw [← map_sup_ker_eq_map, ← comap_bracket_eq h, map_comap_eq h, inf_eq_right],
exact le_trans (lie_submodule.lie_le_left _ _) inf_le_left, } | lemma | lie_ideal.map_comap_bracket_eq | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"inf_eq_right",
"inf_le_left",
"lie_ideal",
"lie_submodule.lie_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bracket_incl {I₁ I₂ : lie_ideal R L} :
⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I ⊓ I₁, I ⊓ I₂⁆ | begin
conv_rhs { congr, skip, rw ← I.incl_ideal_range, }, rw comap_bracket_eq,
simp only [ker_incl, sup_bot_eq], exact I.incl_is_ideal_morphism,
end | lemma | lie_ideal.comap_bracket_incl | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"lie_ideal",
"sup_bot_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bracket_incl_of_le {I₁ I₂ : lie_ideal R L} (h₁ : I₁ ≤ I) (h₂ : I₂ ≤ I) :
⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I₁, I₂⁆ | by { rw comap_bracket_incl, rw ← inf_eq_right at h₁ h₂, rw [h₁, h₂], } | lemma | lie_ideal.comap_bracket_incl_of_le | algebra.lie | src/algebra/lie/ideal_operations.lean | [
"algebra.lie.submodule"
] | [
"inf_eq_right",
"lie_ideal"
] | This is a very useful result; it allows us to use the fact that inclusion distributes over the
Lie bracket operation on ideals, subject to the conditions shown. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_equiv_matrix' : module.End R (n → R) ≃ₗ⁅R⁆ matrix n n R | { map_lie' := λ T S,
begin
let f := @linear_map.to_matrix' R _ n n _ _,
change f (T.comp S - S.comp T) = (f T) * (f S) - (f S) * (f T),
have h : ∀ (T S : module.End R _), f (T.comp S) = (f T) ⬝ (f S) := linear_map.to_matrix'_comp,
rw [linear_equiv.map_sub, h, h, matrix.mul_eq_mul, matrix.mul_eq_mul],
... | def | lie_equiv_matrix' | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"linear_equiv.map_sub",
"linear_map.to_matrix'",
"linear_map.to_matrix'_comp",
"matrix",
"matrix.mul_eq_mul",
"module.End"
] | The natural equivalence between linear endomorphisms of finite free modules and square matrices
is compatible with the Lie algebra structures. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_equiv_matrix'_apply (f : module.End R (n → R)) :
lie_equiv_matrix' f = f.to_matrix' | rfl | lemma | lie_equiv_matrix'_apply | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"lie_equiv_matrix'",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_equiv_matrix'_symm_apply (A : matrix n n R) :
(@lie_equiv_matrix' R _ n _ _).symm A = A.to_lin' | rfl | lemma | lie_equiv_matrix'_symm_apply | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"lie_equiv_matrix'",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.lie_conj (P : matrix n n R) (h : invertible P) :
matrix n n R ≃ₗ⁅R⁆ matrix n n R | ((@lie_equiv_matrix' R _ n _ _).symm.trans (P.to_linear_equiv' h).lie_conj).trans lie_equiv_matrix' | def | matrix.lie_conj | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"invertible",
"lie_equiv_matrix'",
"matrix"
] | An invertible matrix induces a Lie algebra equivalence from the space of matrices to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix.lie_conj_apply (P A : matrix n n R) (h : invertible P) :
P.lie_conj h A = P ⬝ A ⬝ P⁻¹ | by simp [linear_equiv.conj_apply, matrix.lie_conj, linear_map.to_matrix'_comp,
linear_map.to_matrix'_to_lin'] | lemma | matrix.lie_conj_apply | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"invertible",
"linear_equiv.conj_apply",
"linear_map.to_matrix'_comp",
"linear_map.to_matrix'_to_lin'",
"matrix",
"matrix.lie_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.lie_conj_symm_apply (P A : matrix n n R) (h : invertible P) :
(P.lie_conj h).symm A = P⁻¹ ⬝ A ⬝ P | by simp [linear_equiv.symm_conj_apply, matrix.lie_conj, linear_map.to_matrix'_comp,
linear_map.to_matrix'_to_lin'] | lemma | matrix.lie_conj_symm_apply | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"invertible",
"linear_equiv.symm_conj_apply",
"linear_map.to_matrix'_comp",
"linear_map.to_matrix'_to_lin'",
"matrix",
"matrix.lie_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.reindex_lie_equiv : matrix n n R ≃ₗ⁅R⁆ matrix m m R | { to_fun := matrix.reindex e e,
map_lie' := λ M N, by simp only [lie_ring.of_associative_ring_bracket, matrix.reindex_apply,
matrix.submatrix_mul_equiv, matrix.mul_eq_mul, matrix.submatrix_sub, pi.sub_apply],
..(matrix.reindex_linear_equiv R R e e) } | def | matrix.reindex_lie_equiv | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"lie_ring.of_associative_ring_bracket",
"matrix",
"matrix.mul_eq_mul",
"matrix.reindex",
"matrix.reindex_apply",
"matrix.reindex_linear_equiv",
"matrix.submatrix_mul_equiv",
"matrix.submatrix_sub"
] | For square matrices, the natural map that reindexes a matrix's rows and columns with equivalent
types, `matrix.reindex`, is an equivalence of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix.reindex_lie_equiv_apply (M : matrix n n R) :
matrix.reindex_lie_equiv e M = matrix.reindex e e M | rfl | lemma | matrix.reindex_lie_equiv_apply | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"matrix",
"matrix.reindex",
"matrix.reindex_lie_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.reindex_lie_equiv_symm :
(matrix.reindex_lie_equiv e : _ ≃ₗ⁅R⁆ _).symm = matrix.reindex_lie_equiv e.symm | rfl | lemma | matrix.reindex_lie_equiv_symm | algebra.lie | src/algebra/lie/matrix.lean | [
"algebra.lie.of_associative",
"linear_algebra.matrix.reindex",
"linear_algebra.matrix.to_linear_equiv"
] | [
"matrix.reindex_lie_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs : lie_submodule R L M → lie_submodule R L M | (λ N, ⁅(⊤ : lie_ideal R L), N⁆)^[k] | def | lie_submodule.lcs | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_ideal",
"lie_submodule"
] | A generalisation of the lower central series. The zeroth term is a specified Lie submodule of
a Lie module. In the case when we specify the top ideal `⊤` of the Lie algebra, regarded as a Lie
module over itself, we get the usual lower central series of a Lie algebra.
It can be more convenient to work with this general... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcs_zero (N : lie_submodule R L M) : N.lcs 0 = N | rfl | lemma | lie_submodule.lcs_zero | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs_succ : N.lcs (k + 1) = ⁅(⊤ : lie_ideal R L), N.lcs k⁆ | function.iterate_succ_apply' (λ N', ⁅⊤, N'⁆) k N | lemma | lie_submodule.lcs_succ | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"function.iterate_succ_apply'",
"lie_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_central_series : lie_submodule R L M | (⊤ : lie_submodule R L M).lcs k | def | lie_module.lower_central_series | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_submodule",
"lower_central_series"
] | The lower central series of Lie submodules of a Lie module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_central_series_zero : lower_central_series R L M 0 = ⊤ | rfl | lemma | lie_module.lower_central_series_zero | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lower_central_series",
"lower_central_series_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_central_series_succ :
lower_central_series R L M (k + 1) = ⁅(⊤ : lie_ideal R L), lower_central_series R L M k⁆ | (⊤ : lie_submodule R L M).lcs_succ k | lemma | lie_module.lower_central_series_succ | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_ideal",
"lie_submodule",
"lower_central_series",
"lower_central_series_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs_le_self : N.lcs k ≤ N | begin
induction k with k ih,
{ simp, },
{ simp only [lcs_succ],
exact (lie_submodule.mono_lie_right _ _ ⊤ ih).trans (N.lie_le_right ⊤), },
end | lemma | lie_submodule.lcs_le_self | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_submodule.mono_lie_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_central_series_eq_lcs_comap :
lower_central_series R L N k = (N.lcs k).comap N.incl | begin
induction k with k ih,
{ simp, },
{ simp only [lcs_succ, lower_central_series_succ] at ⊢ ih,
have : N.lcs k ≤ N.incl.range,
{ rw N.range_incl,
apply lcs_le_self, },
rw [ih, lie_submodule.comap_bracket_eq _ _ N.incl N.ker_incl this], },
end | lemma | lie_submodule.lower_central_series_eq_lcs_comap | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_submodule.comap_bracket_eq",
"lower_central_series",
"lower_central_series_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_central_series_map_eq_lcs :
(lower_central_series R L N k).map N.incl = N.lcs k | begin
rw [lower_central_series_eq_lcs_comap, lie_submodule.map_comap_incl, inf_eq_right],
apply lcs_le_self,
end | lemma | lie_submodule.lower_central_series_map_eq_lcs | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"inf_eq_right",
"lie_submodule.map_comap_incl",
"lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_lower_central_series : antitone $ lower_central_series R L M | begin
intros l k,
induction k with k ih generalizing l;
intros h,
{ exact (le_zero_iff.mp h).symm ▸ le_rfl, },
{ rcases nat.of_le_succ h with hk | hk,
{ rw lower_central_series_succ,
exact (lie_submodule.mono_lie_right _ _ ⊤ (ih hk)).trans (lie_submodule.lie_le_right _ _), },
{ exact hk.symm ▸ l... | lemma | lie_module.antitone_lower_central_series | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"antitone",
"ih",
"le_rfl",
"lie_submodule.lie_le_right",
"lie_submodule.mono_lie_right",
"lower_central_series",
"lower_central_series_succ",
"nat.of_le_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivial_iff_lower_central_eq_bot : is_trivial L M ↔ lower_central_series R L M 1 = ⊥ | begin
split; intros h,
{ erw [eq_bot_iff, lie_submodule.lie_span_le], rintros m ⟨x, n, hn⟩, rw [← hn, h.trivial], simp,},
{ rw lie_submodule.eq_bot_iff at h, apply is_trivial.mk, intros x m, apply h,
apply lie_submodule.subset_lie_span, use [x, m], refl, },
end | lemma | lie_module.trivial_iff_lower_central_eq_bot | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"eq_bot_iff",
"lie_submodule.eq_bot_iff",
"lie_submodule.lie_span_le",
"lie_submodule.subset_lie_span",
"lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_to_endomorphism_mem_lower_central_series (x : L) (m : M) (k : ℕ) :
(to_endomorphism R L M x)^[k] m ∈ lower_central_series R L M k | begin
induction k with k ih,
{ simp only [function.iterate_zero], },
{ simp only [lower_central_series_succ, function.comp_app, function.iterate_succ',
to_endomorphism_apply_apply],
exact lie_submodule.lie_mem_lie _ _ (lie_submodule.mem_top x) ih, },
end | lemma | lie_module.iterate_to_endomorphism_mem_lower_central_series | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"function.iterate_succ'",
"function.iterate_zero",
"ih",
"lie_submodule.lie_mem_lie",
"lie_submodule.mem_top",
"lower_central_series",
"lower_central_series_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lower_central_series_le
{M₂ : Type w₁} [add_comm_group M₂] [module R M₂] [lie_ring_module L M₂] [lie_module R L M₂]
(k : ℕ) (f : M →ₗ⁅R,L⁆ M₂) :
lie_submodule.map f (lower_central_series R L M k) ≤ lower_central_series R L M₂ k | begin
induction k with k ih,
{ simp only [lie_module.lower_central_series_zero, le_top], },
{ simp only [lie_module.lower_central_series_succ, lie_submodule.map_bracket_eq],
exact lie_submodule.mono_lie_right _ _ ⊤ ih, },
end | lemma | lie_module.map_lower_central_series_le | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"add_comm_group",
"ih",
"le_top",
"lie_module",
"lie_module.lower_central_series_succ",
"lie_module.lower_central_series_zero",
"lie_ring_module",
"lie_submodule.map",
"lie_submodule.map_bracket_eq",
"lie_submodule.mono_lie_right",
"lower_central_series",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derived_series_le_lower_central_series (k : ℕ) :
derived_series R L k ≤ lower_central_series R L L k | begin
induction k with k h,
{ rw [derived_series_def, derived_series_of_ideal_zero, lower_central_series_zero],
exact le_rfl, },
{ have h' : derived_series R L k ≤ ⊤, { simp only [le_top], },
rw [derived_series_def, derived_series_of_ideal_succ, lower_central_series_succ],
exact lie_submodule.mono_lie... | lemma | lie_module.derived_series_le_lower_central_series | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"derived_series",
"le_rfl",
"le_top",
"lie_submodule.mono_lie",
"lower_central_series",
"lower_central_series_succ",
"lower_central_series_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent : Prop | (nilpotent : ∃ k, lower_central_series R L M k = ⊥) | class | lie_module.is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lower_central_series"
] | A Lie module is nilpotent if its lower central series reaches 0 (in a finite number of
steps). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_nilpotent_iff :
is_nilpotent R L M ↔ ∃ k, lower_central_series R L M k = ⊥ | ⟨λ h, h.nilpotent, λ h, ⟨h⟩⟩ | lemma | lie_module.is_nilpotent_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lower_central_series"
] | See also `lie_module.is_nilpotent_iff_exists_ucs_eq_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.lie_submodule.is_nilpotent_iff_exists_lcs_eq_bot (N : lie_submodule R L M) :
lie_module.is_nilpotent R L N ↔ ∃ k, N.lcs k = ⊥ | begin
rw is_nilpotent_iff,
refine exists_congr (λ k, _),
rw [N.lower_central_series_eq_lcs_comap k, lie_submodule.comap_incl_eq_bot,
inf_eq_right.mpr (N.lcs_le_self k)],
end | lemma | lie_submodule.is_nilpotent_iff_exists_lcs_eq_bot | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_module.is_nilpotent",
"lie_submodule",
"lie_submodule.comap_incl_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivial_is_nilpotent [is_trivial L M] : is_nilpotent R L M | ⟨by { use 1, change ⁅⊤, ⊤⁆ = ⊥, simp, }⟩ | instance | lie_module.trivial_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilpotent_endo_of_nilpotent_module [hM : is_nilpotent R L M] :
∃ (k : ℕ), ∀ (x : L), (to_endomorphism R L M x)^k = 0 | begin
unfreezingI { obtain ⟨k, hM⟩ := hM, },
use k,
intros x, ext m,
rw [linear_map.pow_apply, linear_map.zero_apply, ← @lie_submodule.mem_bot R L M, ← hM],
exact iterate_to_endomorphism_mem_lower_central_series R L M x m k,
end | lemma | lie_module.nilpotent_endo_of_nilpotent_module | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_submodule.mem_bot",
"linear_map.pow_apply",
"linear_map.zero_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_max_gen_zero_eigenspace_eq_top_of_nilpotent [is_nilpotent R L M] :
(⨅ (x : L), (to_endomorphism R L M x).maximal_generalized_eigenspace 0) = ⊤ | begin
ext m,
simp only [module.End.mem_maximal_generalized_eigenspace, submodule.mem_top, sub_zero, iff_true,
zero_smul, submodule.mem_infi],
intros x,
obtain ⟨k, hk⟩ := nilpotent_endo_of_nilpotent_module R L M,
use k, rw hk,
exact linear_map.zero_apply m,
end | lemma | lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"linear_map.zero_apply",
"module.End.mem_maximal_generalized_eigenspace",
"submodule.mem_infi",
"submodule.mem_top",
"zero_smul"
] | For a nilpotent Lie module, the weight space of the 0 weight is the whole module.
This result will be used downstream to show that weight spaces are Lie submodules, at which time
it will be possible to state it in the language of weight spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nilpotent_of_nilpotent_quotient {N : lie_submodule R L M}
(h₁ : N ≤ max_triv_submodule R L M) (h₂ : is_nilpotent R L (M ⧸ N)) : is_nilpotent R L M | begin
unfreezingI { obtain ⟨k, hk⟩ := h₂, },
use k+1,
simp only [lower_central_series_succ],
suffices : lower_central_series R L M k ≤ N,
{ replace this := lie_submodule.mono_lie_right _ _ ⊤ (le_trans this h₁),
rwa [ideal_oper_max_triv_submodule_eq_bot, le_bot_iff] at this, },
rw [← lie_submodule.quotie... | lemma | lie_module.nilpotent_of_nilpotent_quotient | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"le_bot_iff",
"lie_submodule",
"lie_submodule.mono_lie_right",
"lie_submodule.quotient.map_mk'_eq_bot_le",
"lie_submodule.quotient.mk'",
"lower_central_series",
"lower_central_series_succ"
] | If the quotient of a Lie module `M` by a Lie submodule on which the Lie algebra acts trivially
is nilpotent then `M` is nilpotent.
This is essentially the Lie module equivalent of the fact that a central
extension of nilpotent Lie algebras is nilpotent. See `lie_algebra.nilpotent_of_nilpotent_quotient`
below for the c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nilpotency_length : ℕ | Inf { k | lower_central_series R L M k = ⊥ } | def | lie_module.nilpotency_length | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lower_central_series"
] | Given a nilpotent Lie module `M` with lower central series `M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is
the natural number `k` (the number of inclusions).
For a non-nilpotent module, we use the junk value 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nilpotency_length_eq_zero_iff [is_nilpotent R L M] :
nilpotency_length R L M = 0 ↔ subsingleton M | begin
let s := { k | lower_central_series R L M k = ⊥ },
have hs : s.nonempty,
{ unfreezingI { obtain ⟨k, hk⟩ := (by apply_instance : is_nilpotent R L M), },
exact ⟨k, hk⟩, },
change Inf s = 0 ↔ _,
rw [← lie_submodule.subsingleton_iff R L M, ← subsingleton_iff_bot_eq_top,
← lower_central_series_zero... | lemma | lie_module.nilpotency_length_eq_zero_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_submodule",
"lie_submodule.subsingleton_iff",
"lower_central_series",
"lower_central_series_zero",
"nat.Inf_eq_zero",
"nat.Inf_mem",
"subsingleton_iff_bot_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nilpotency_length_eq_succ_iff (k : ℕ) :
nilpotency_length R L M = k + 1 ↔
lower_central_series R L M (k + 1) = ⊥ ∧ lower_central_series R L M k ≠ ⊥ | begin
let s := { k | lower_central_series R L M k = ⊥ },
change Inf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s,
have hs : ∀ k₁ k₂, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s,
{ rintros k₁ k₂ h₁₂ (h₁ : lower_central_series R L M k₁ = ⊥),
exact eq_bot_iff.mpr (h₁ ▸ antitone_lower_central_series R L M h₁₂), },
exact nat.Inf_upward_closed... | lemma | lie_module.nilpotency_length_eq_succ_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lower_central_series",
"nat.Inf_upward_closed_eq_succ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_central_series_last : lie_submodule R L M | match nilpotency_length R L M with
| 0 := ⊥
| k + 1 := lower_central_series R L M k
end | def | lie_module.lower_central_series_last | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_submodule",
"lower_central_series"
] | Given a non-trivial nilpotent Lie module `M` with lower central series
`M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is the `k-1`th term in the lower central series (the last
non-trivial term).
For a trivial or non-nilpotent module, this is the bottom submodule, `⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_central_series_last_le_max_triv :
lower_central_series_last R L M ≤ max_triv_submodule R L M | begin
rw lower_central_series_last,
cases h : nilpotency_length R L M with k,
{ exact bot_le, },
{ rw le_max_triv_iff_bracket_eq_bot,
rw [nilpotency_length_eq_succ_iff, lower_central_series_succ] at h,
exact h.1, },
end | lemma | lie_module.lower_central_series_last_le_max_triv | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"bot_le",
"lower_central_series_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_lower_central_series_last [nontrivial M] [is_nilpotent R L M] :
nontrivial (lower_central_series_last R L M) | begin
rw [lie_submodule.nontrivial_iff_ne_bot, lower_central_series_last],
cases h : nilpotency_length R L M,
{ rw [nilpotency_length_eq_zero_iff, ← not_nontrivial_iff_subsingleton] at h,
contradiction, },
{ rw nilpotency_length_eq_succ_iff at h,
exact h.2, },
end | lemma | lie_module.nontrivial_lower_central_series_last | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_submodule.nontrivial_iff_ne_bot",
"nontrivial",
"not_nontrivial_iff_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial_max_triv_of_is_nilpotent [nontrivial M] [is_nilpotent R L M] :
nontrivial (max_triv_submodule R L M) | set.nontrivial_mono
(lower_central_series_last_le_max_triv R L M)
(nontrivial_lower_central_series_last R L M) | lemma | lie_module.nontrivial_max_triv_of_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"nontrivial",
"set.nontrivial_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lcs_range_to_endomorphism_eq (k : ℕ) :
(lower_central_series R (to_endomorphism R L M).range M k : submodule R M) =
lower_central_series R L M k | begin
induction k with k ih,
{ simp, },
{ simp only [lower_central_series_succ, lie_submodule.lie_ideal_oper_eq_linear_span',
← (lower_central_series R (to_endomorphism R L M).range M k).mem_coe_submodule, ih],
congr,
ext m,
split,
{ rintros ⟨⟨-, ⟨y, rfl⟩⟩, -, n, hn, rfl⟩,
exact ⟨y, li... | lemma | lie_module.coe_lcs_range_to_endomorphism_eq | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_hom.mem_range_self",
"lie_submodule.lie_ideal_oper_eq_linear_span'",
"lie_submodule.mem_top",
"lower_central_series",
"lower_central_series_succ",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_range_to_endomorphism_iff :
is_nilpotent R (to_endomorphism R L M).range M ↔ is_nilpotent R L M | begin
split;
rintros ⟨k, hk⟩;
use k;
rw ← lie_submodule.coe_to_submodule_eq_iff at ⊢ hk;
simpa using hk,
end | lemma | lie_module.is_nilpotent_range_to_endomorphism_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs (k : ℕ) : lie_submodule R L M → lie_submodule R L M | normalizer^[k] | def | lie_submodule.ucs | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_submodule"
] | The upper (aka ascending) central series.
See also `lie_submodule.lcs`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ucs_zero : N.ucs 0 = N | rfl | lemma | lie_submodule.ucs_zero | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_succ (k : ℕ) :
N.ucs (k + 1) = (N.ucs k).normalizer | function.iterate_succ_apply' normalizer k N | lemma | lie_submodule.ucs_succ | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"function.iterate_succ_apply'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_add (k l : ℕ) :
N.ucs (k + l) = (N.ucs l).ucs k | function.iterate_add_apply normalizer k l N | lemma | lie_submodule.ucs_add | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"function.iterate_add_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_mono (k : ℕ) (h : N₁ ≤ N₂) :
N₁.ucs k ≤ N₂.ucs k | begin
induction k with k ih, { simpa, },
simp only [ucs_succ],
mono,
end | lemma | lie_submodule.ucs_mono | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
N₁.ucs k = N₁ | by { induction k with k ih, { simp, }, { rwa [ucs_succ, ih], }, } | lemma | lie_submodule.ucs_eq_self_of_normalizer_eq_self | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : lie_submodule R L M).ucs k ≤ N₁ | by { rw ← ucs_eq_self_of_normalizer_eq_self h k, mono, simp, } | lemma | lie_submodule.ucs_le_of_normalizer_eq_self | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_submodule"
] | If a Lie module `M` contains a self-normalizing Lie submodule `N`, then all terms of the upper
central series of `M` are contained in `N`.
An important instance of this situation arises from a Cartan subalgebra `H ⊆ L` with the roles of
`L`, `M`, `N` played by `H`, `L`, `H`, respectively. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcs_add_le_iff (l k : ℕ) :
N₁.lcs (l + k) ≤ N₂ ↔ N₁.lcs l ≤ N₂.ucs k | begin
revert l,
induction k with k ih, { simp, },
intros l,
rw [(by abel : l + (k + 1) = l + 1 + k), ih, ucs_succ, lcs_succ, top_lie_le_iff_le_normalizer],
end | lemma | lie_submodule.lcs_add_le_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs_le_iff (k : ℕ) :
N₁.lcs k ≤ N₂ ↔ N₁ ≤ N₂.ucs k | by { convert lcs_add_le_iff 0 k, rw zero_add, } | lemma | lie_submodule.lcs_le_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_lcs_ucs (k : ℕ):
galois_connection (λ (N : lie_submodule R L M), N.lcs k) (λ (N : lie_submodule R L M), N.ucs k) | λ N₁ N₂, lcs_le_iff k | lemma | lie_submodule.gc_lcs_ucs | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"galois_connection",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ lie_module.lower_central_series R L M k ≤ N | by { rw [eq_top_iff, ← lcs_le_iff], refl, } | lemma | lie_submodule.ucs_eq_top_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"eq_top_iff",
"lie_module.lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.lie_module.is_nilpotent_iff_exists_ucs_eq_top :
lie_module.is_nilpotent R L M ↔ ∃ k, (⊥ : lie_submodule R L M).ucs k = ⊤ | by { rw lie_module.is_nilpotent_iff, exact exists_congr (λ k, by simp [ucs_eq_top_iff]), } | lemma | lie_module.is_nilpotent_iff_exists_ucs_eq_top | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_module.is_nilpotent",
"lie_module.is_nilpotent_iff",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ucs_comap_incl (k : ℕ) :
((⊥ : lie_submodule R L M).ucs k).comap N.incl = (⊥ : lie_submodule R L N).ucs k | by { induction k with k ih, { exact N.ker_incl, }, { simp [← ih], }, } | lemma | lie_submodule.ucs_comap_incl | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_nilpotent_iff_exists_self_le_ucs :
lie_module.is_nilpotent R L N ↔ ∃ k, N ≤ (⊥ : lie_submodule R L M).ucs k | by simp_rw [lie_module.is_nilpotent_iff_exists_ucs_eq_top, ← ucs_comap_incl, comap_incl_eq_top] | lemma | lie_submodule.is_nilpotent_iff_exists_self_le_ucs | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_module.is_nilpotent",
"lie_module.is_nilpotent_iff_exists_ucs_eq_top",
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.lie_module_lcs_map_eq (k : ℕ) :
(lower_central_series R L M k : submodule R M).map g = lower_central_series R L₂ M₂ k | begin
induction k with k ih,
{ simp [linear_map.range_eq_top, hg], },
{ suffices : g '' {m | ∃ (x : L) n, n ∈ lower_central_series R L M k ∧ ⁅x, n⁆ = m} =
{m | ∃ (x : L₂) n, n ∈ lower_central_series R L M k ∧ ⁅x, g n⁆ = m},
{ simp only [← lie_submodule.mem_coe_submodule] at this,
simp [← ... | lemma | function.surjective.lie_module_lcs_map_eq | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_submodule.lie_ideal_oper_eq_linear_span'",
"lie_submodule.mem_coe_submodule",
"linear_map.range_eq_top",
"lower_central_series",
"submodule",
"submodule.map_span",
"submodule.span_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.lie_module_is_nilpotent [is_nilpotent R L M] : is_nilpotent R L₂ M₂ | begin
obtain ⟨k, hk⟩ := id (by apply_instance : is_nilpotent R L M),
use k,
rw ← lie_submodule.coe_to_submodule_eq_iff at ⊢ hk,
simp [← hf.lie_module_lcs_map_eq hg hfg k, hk],
end | lemma | function.surjective.lie_module_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.lie_module_is_nilpotent_iff (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ[R] M₂)
(hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆) :
is_nilpotent R L M ↔ is_nilpotent R L₂ M₂ | begin
split;
introsI h,
{ have hg : surjective (g : M →ₗ[R] M₂) := g.surjective,
exact f.surjective.lie_module_is_nilpotent hg hfg, },
{ have hg : surjective (g.symm : M₂ →ₗ[R] M) := g.symm.surjective,
refine f.symm.surjective.lie_module_is_nilpotent hg (λ x m, _),
rw [linear_equiv.coe_coe, lie_equi... | lemma | equiv.lie_module_is_nilpotent_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_equiv.coe_to_lie_hom",
"linear_equiv.coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module.is_nilpotent_of_top_iff :
is_nilpotent R (⊤ : lie_subalgebra R L) M ↔ is_nilpotent R L M | equiv.lie_module_is_nilpotent_iff lie_subalgebra.top_equiv (1 : M ≃ₗ[R] M) (λ x m, rfl) | lemma | lie_module.is_nilpotent_of_top_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"equiv.lie_module_is_nilpotent_iff",
"is_nilpotent",
"lie_subalgebra",
"lie_subalgebra.top_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_solvable_of_is_nilpotent (R : Type u) (L : Type v)
[comm_ring R] [lie_ring L] [lie_algebra R L] [hL : lie_module.is_nilpotent R L L] :
lie_algebra.is_solvable R L | begin
obtain ⟨k, h⟩ : ∃ k, lie_module.lower_central_series R L L k = ⊥ := hL.nilpotent,
use k, rw ← le_bot_iff at h ⊢,
exact le_trans (lie_module.derived_series_le_lower_central_series R L k) h,
end | instance | lie_algebra.is_solvable_of_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"comm_ring",
"le_bot_iff",
"lie_algebra",
"lie_algebra.is_solvable",
"lie_module.derived_series_le_lower_central_series",
"lie_module.is_nilpotent",
"lie_module.lower_central_series",
"lie_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_nilpotent (R : Type u) (L : Type v)
[comm_ring R] [lie_ring L] [lie_algebra R L] : Prop | lie_module.is_nilpotent R L L | abbreviation | lie_algebra.is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"comm_ring",
"lie_algebra",
"lie_module.is_nilpotent",
"lie_ring"
] | We say a Lie algebra is nilpotent when it is nilpotent as a Lie module over itself via the
adjoint representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.nilpotent_ad_of_nilpotent_algebra [is_nilpotent R L] :
∃ (k : ℕ), ∀ (x : L), (ad R L x)^k = 0 | lie_module.nilpotent_endo_of_nilpotent_module R L L | lemma | lie_algebra.nilpotent_ad_of_nilpotent_algebra | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_module.nilpotent_endo_of_nilpotent_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent [is_nilpotent R L] :
(⨅ (x : L), (ad R L x).maximal_generalized_eigenspace 0) = ⊤ | lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L | lemma | lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent"
] | See also `lie_algebra.zero_root_space_eq_top_of_nilpotent`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lower_central_series_ideal_quot_eq {I : lie_ideal R L} (k : ℕ) :
(lower_central_series R L (L ⧸ I) k : submodule R (L ⧸ I)) =
lower_central_series R (L ⧸ I) (L ⧸ I) k | begin
induction k with k ih,
{ simp only [lie_submodule.top_coe_submodule, lie_module.lower_central_series_zero], },
{ simp only [lie_module.lower_central_series_succ, lie_submodule.lie_ideal_oper_eq_linear_span],
congr,
ext x,
split,
{ rintros ⟨⟨y, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩,
erw [← lie... | lemma | coe_lower_central_series_ideal_quot_eq | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_ideal",
"lie_module.lower_central_series_succ",
"lie_module.lower_central_series_zero",
"lie_submodule.lie_ideal_oper_eq_linear_span",
"lie_submodule.mem_coe_submodule",
"lie_submodule.mem_top",
"lie_submodule.top_coe_submodule",
"lower_central_series",
"submodule"
] | Given an ideal `I` of a Lie algebra `L`, the lower central series of `L ⧸ I` is the same
whether we regard `L ⧸ I` as an `L` module or an `L ⧸ I` module.
TODO: This result obviously generalises but the generalisation requires the missing definition of
morphisms between Lie modules over different Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module.coe_lower_central_series_ideal_le {I : lie_ideal R L} (k : ℕ) :
(lower_central_series R I I k : submodule R I) ≤ lower_central_series R L I k | begin
induction k with k ih,
{ simp, },
{ simp only [lie_module.lower_central_series_succ, lie_submodule.lie_ideal_oper_eq_linear_span],
apply submodule.span_mono,
rintros x ⟨⟨y, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩,
exact ⟨⟨y.val, lie_submodule.mem_top _⟩, ⟨z, ih hz⟩, rfl⟩, },
end | lemma | lie_module.coe_lower_central_series_ideal_le | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_ideal",
"lie_module.lower_central_series_succ",
"lie_submodule.lie_ideal_oper_eq_linear_span",
"lie_submodule.mem_top",
"lower_central_series",
"submodule",
"submodule.span_mono"
] | Note that the below inequality can be strict. For example the ideal of strictly-upper-triangular
2x2 matrices inside the Lie algebra of upper-triangular 2x2 matrices with `k = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.nilpotent_of_nilpotent_quotient {I : lie_ideal R L}
(h₁ : I ≤ center R L) (h₂ : is_nilpotent R (L ⧸ I)) : is_nilpotent R L | begin
suffices : lie_module.is_nilpotent R L (L ⧸ I),
{ exact lie_module.nilpotent_of_nilpotent_quotient R L L h₁ this, },
unfreezingI { obtain ⟨k, hk⟩ := h₂, },
use k,
simp [← lie_submodule.coe_to_submodule_eq_iff, coe_lower_central_series_ideal_quot_eq, hk],
end | lemma | lie_algebra.nilpotent_of_nilpotent_quotient | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"coe_lower_central_series_ideal_quot_eq",
"is_nilpotent",
"lie_ideal",
"lie_module.is_nilpotent",
"lie_module.nilpotent_of_nilpotent_quotient",
"lie_submodule.coe_to_submodule_eq_iff"
] | A central extension of nilpotent Lie algebras is nilpotent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra.non_trivial_center_of_is_nilpotent [nontrivial L] [is_nilpotent R L] :
nontrivial $ center R L | lie_module.nontrivial_max_triv_of_is_nilpotent R L L | lemma | lie_algebra.non_trivial_center_of_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_module.nontrivial_max_triv_of_is_nilpotent",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.map_lower_central_series_le (k : ℕ) {f : L →ₗ⁅R⁆ L'} :
lie_ideal.map f (lower_central_series R L L k) ≤ lower_central_series R L' L' k | begin
induction k with k ih,
{ simp only [lie_module.lower_central_series_zero, le_top], },
{ simp only [lie_module.lower_central_series_succ],
exact le_trans (lie_ideal.map_bracket_le f) (lie_submodule.mono_lie _ _ _ _ le_top ih), },
end | lemma | lie_ideal.map_lower_central_series_le | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"le_top",
"lie_ideal.map",
"lie_ideal.map_bracket_le",
"lie_module.lower_central_series_succ",
"lie_module.lower_central_series_zero",
"lie_submodule.mono_lie",
"lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.lower_central_series_map_eq (k : ℕ) {f : L →ₗ⁅R⁆ L'}
(h : function.surjective f) :
lie_ideal.map f (lower_central_series R L L k) = lower_central_series R L' L' k | begin
have h' : (⊤ : lie_ideal R L).map f = ⊤,
{ rw ←f.ideal_range_eq_map,
exact f.ideal_range_eq_top_of_surjective h, },
induction k with k ih,
{ simp only [lie_module.lower_central_series_zero], exact h', },
{ simp only [lie_module.lower_central_series_succ, lie_ideal.map_bracket_eq f h, ih, h'], },
end | lemma | lie_ideal.lower_central_series_map_eq | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"ih",
"lie_ideal",
"lie_ideal.map",
"lie_ideal.map_bracket_eq",
"lie_module.lower_central_series_succ",
"lie_module.lower_central_series_zero",
"lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.lie_algebra_is_nilpotent [h₁ : is_nilpotent R L'] {f : L →ₗ⁅R⁆ L'}
(h₂ : function.injective f) : is_nilpotent R L | { nilpotent :=
begin
obtain ⟨k, hk⟩ := id h₁,
use k,
apply lie_ideal.bot_of_map_eq_bot h₂, rw [eq_bot_iff, ← hk],
apply lie_ideal.map_lower_central_series_le,
end, } | lemma | function.injective.lie_algebra_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"eq_bot_iff",
"is_nilpotent",
"lie_ideal.bot_of_map_eq_bot",
"lie_ideal.map_lower_central_series_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.lie_algebra_is_nilpotent [h₁ : is_nilpotent R L] {f : L →ₗ⁅R⁆ L'}
(h₂ : function.surjective f) : is_nilpotent R L' | { nilpotent :=
begin
obtain ⟨k, hk⟩ := id h₁,
use k,
rw [← lie_ideal.lower_central_series_map_eq k h₂, hk],
simp only [lie_ideal.map_eq_bot_iff, bot_le],
end, } | lemma | function.surjective.lie_algebra_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"bot_le",
"is_nilpotent",
"lie_ideal.lower_central_series_map_eq",
"lie_ideal.map_eq_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_equiv.nilpotent_iff_equiv_nilpotent (e : L ≃ₗ⁅R⁆ L') :
is_nilpotent R L ↔ is_nilpotent R L' | begin
split; introsI h,
{ exact e.symm.injective.lie_algebra_is_nilpotent, },
{ exact e.injective.lie_algebra_is_nilpotent, },
end | lemma | lie_equiv.nilpotent_iff_equiv_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_hom.is_nilpotent_range [is_nilpotent R L] (f : L →ₗ⁅R⁆ L') :
is_nilpotent R f.range | f.surjective_range_restrict.lie_algebra_is_nilpotent | lemma | lie_hom.is_nilpotent_range | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_nilpotent_range_ad_iff :
is_nilpotent R (ad R L).range ↔ is_nilpotent R L | begin
refine ⟨λ h, _, _⟩,
{ have : (ad R L).ker = center R L, { simp, },
exact lie_algebra.nilpotent_of_nilpotent_quotient (le_of_eq this)
((ad R L).quot_ker_equiv_range.nilpotent_iff_equiv_nilpotent.mpr h), },
{ introsI h,
exact (ad R L).is_nilpotent_range, },
end | lemma | lie_algebra.is_nilpotent_range_ad_iff | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_algebra.nilpotent_of_nilpotent_quotient"
] | Note that this result is not quite a special case of
`lie_module.is_nilpotent_range_to_endomorphism_iff` which concerns nilpotency of the
`(ad R L).range`-module `L`, whereas this result concerns nilpotency of the `(ad R L).range`-module
`(ad R L).range`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcs : lie_submodule R L M | (λ N, ⁅I, N⁆)^[k] ⊤ | def | lie_ideal.lcs | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_submodule"
] | Given a Lie module `M` over a Lie algebra `L` together with an ideal `I` of `L`, this is the
lower central series of `M` as an `I`-module. The advantage of using this definition instead of
`lie_module.lower_central_series R I M` is that its terms are Lie submodules of `M` as an
`L`-module, rather than just as an `I`-mo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lcs_zero : I.lcs M 0 = ⊤ | rfl | lemma | lie_ideal.lcs_zero | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs_succ : I.lcs M (k + 1) = ⁅I, I.lcs M k⁆ | function.iterate_succ_apply' (λ N, ⁅I, N⁆) k ⊤ | lemma | lie_ideal.lcs_succ | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"function.iterate_succ_apply'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcs_top : (⊤ : lie_ideal R L).lcs M k = lower_central_series R L M k | rfl | lemma | lie_ideal.lcs_top | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"lie_ideal",
"lower_central_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lcs_eq : (I.lcs M k : submodule R M) = lower_central_series R I M k | begin
induction k with k ih,
{ simp, },
{ simp_rw [lower_central_series_succ, lcs_succ, lie_submodule.lie_ideal_oper_eq_linear_span',
← (I.lcs M k).mem_coe_submodule, ih, lie_submodule.mem_coe_submodule,
lie_submodule.mem_top, exists_true_left, (I : lie_subalgebra R L).coe_bracket_of_module],
cong... | lemma | lie_ideal.coe_lcs_eq | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"exists_true_left",
"ih",
"lie_subalgebra",
"lie_submodule.lie_ideal_oper_eq_linear_span'",
"lie_submodule.mem_coe_submodule",
"lie_submodule.mem_top",
"lower_central_series",
"lower_central_series_succ",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.ad_nilpotent_of_nilpotent {a : A} (h : is_nilpotent a) :
is_nilpotent (lie_algebra.ad R A a) | begin
rw lie_algebra.ad_eq_lmul_left_sub_lmul_right,
have hl : is_nilpotent (linear_map.mul_left R a),
{ rwa linear_map.is_nilpotent_mul_left_iff, },
have hr : is_nilpotent (linear_map.mul_right R a),
{ rwa linear_map.is_nilpotent_mul_right_iff, },
have := @linear_map.commute_mul_left_right R A _ _ _ _ _ a ... | lemma | lie_algebra.ad_nilpotent_of_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_algebra.ad",
"lie_algebra.ad_eq_lmul_left_sub_lmul_right",
"linear_map.commute_mul_left_right",
"linear_map.is_nilpotent_mul_left_iff",
"linear_map.is_nilpotent_mul_right_iff",
"linear_map.mul_left",
"linear_map.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra.is_nilpotent_ad_of_is_nilpotent_ad {L : Type v} [lie_ring L] [lie_algebra R L]
(K : lie_subalgebra R L) {x : K} (h : is_nilpotent (lie_algebra.ad R L ↑x)) :
is_nilpotent (lie_algebra.ad R K x) | begin
obtain ⟨n, hn⟩ := h,
use n,
exact linear_map.submodule_pow_eq_zero_of_pow_eq_zero (K.ad_comp_incl_eq x) hn,
end | lemma | lie_subalgebra.is_nilpotent_ad_of_is_nilpotent_ad | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_algebra",
"lie_algebra.ad",
"lie_ring",
"lie_subalgebra",
"linear_map.submodule_pow_eq_zero_of_pow_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra.is_nilpotent_ad_of_is_nilpotent {L : lie_subalgebra R A} {x : L}
(h : is_nilpotent (x : A)) : is_nilpotent (lie_algebra.ad R L x) | L.is_nilpotent_ad_of_is_nilpotent_ad $ lie_algebra.ad_nilpotent_of_nilpotent R h | lemma | lie_algebra.is_nilpotent_ad_of_is_nilpotent | algebra.lie | src/algebra/lie/nilpotent.lean | [
"algebra.lie.solvable",
"algebra.lie.quotient",
"algebra.lie.normalizer",
"linear_algebra.eigenspace.basic",
"ring_theory.nilpotent"
] | [
"is_nilpotent",
"lie_algebra.ad",
"lie_algebra.ad_nilpotent_of_nilpotent",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutator_ring (L : Type v) : Type v | L | def | commutator_ring | algebra.lie | src/algebra/lie/non_unital_non_assoc_algebra.lean | [
"algebra.hom.non_unital_alg",
"algebra.lie.basic"
] | [] | Type synonym for turning a `lie_ring` into a `non_unital_non_assoc_semiring`.
A `lie_ring` can be regarded as a `non_unital_non_assoc_semiring` by turning its
`has_bracket` (denoted `⁅, ⁆`) into a `has_mul` (denoted `*`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_scalar_tower : is_scalar_tower R (commutator_ring L) (commutator_ring L) | ⟨smul_lie⟩ | instance | lie_algebra.is_scalar_tower | algebra.lie | src/algebra/lie/non_unital_non_assoc_algebra.lean | [
"algebra.hom.non_unital_alg",
"algebra.lie.basic"
] | [
"commutator_ring",
"is_scalar_tower"
] | Regarding the `lie_ring` of a `lie_algebra` as a `non_unital_non_assoc_semiring`, we can
reinterpret the `smul_lie` law as an `is_scalar_tower`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_comm_class : smul_comm_class R (commutator_ring L) (commutator_ring L) | ⟨λ t x y, (lie_smul t x y).symm⟩ | instance | lie_algebra.smul_comm_class | algebra.lie | src/algebra/lie/non_unital_non_assoc_algebra.lean | [
"algebra.hom.non_unital_alg",
"algebra.lie.basic"
] | [
"commutator_ring",
"lie_smul",
"smul_comm_class"
] | Regarding the `lie_ring` of a `lie_algebra` as a `non_unital_non_assoc_semiring`, we can
reinterpret the `lie_smul` law as an `smul_comm_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_non_unital_alg_hom (f : L →ₗ⁅R⁆ L₂) : commutator_ring L →ₙₐ[R] commutator_ring L₂ | { to_fun := f,
map_zero' := f.map_zero,
map_mul' := f.map_lie,
..f } | def | lie_hom.to_non_unital_alg_hom | algebra.lie | src/algebra/lie/non_unital_non_assoc_algebra.lean | [
"algebra.hom.non_unital_alg",
"algebra.lie.basic"
] | [
"commutator_ring"
] | Regarding the `lie_ring` of a `lie_algebra` as a `non_unital_non_assoc_semiring`, we can
regard a `lie_hom` as a `non_unital_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_non_unital_alg_hom_injective :
function.injective (to_non_unital_alg_hom : _ → (commutator_ring L →ₙₐ[R] commutator_ring L₂)) | λ f g h, ext $ non_unital_alg_hom.congr_fun h | lemma | lie_hom.to_non_unital_alg_hom_injective | algebra.lie | src/algebra/lie/non_unital_non_assoc_algebra.lean | [
"algebra.hom.non_unital_alg",
"algebra.lie.basic"
] | [
"commutator_ring",
"non_unital_alg_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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