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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
range : lie_submodule R L N | (lie_submodule.map f ⊤).copy (set.range f) set.image_univ.symm | def | lie_module_hom.range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule",
"lie_submodule.map",
"set.range"
] | The range of a morphism of Lie modules `f : M → N` is a Lie submodule of `N`.
See Note [range copy pattern]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_range : (f.range : set N) = set.range f | rfl | lemma | lie_module_hom.coe_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_submodule_range : (f.range : submodule R N) = (f : M →ₗ[R] N).range | rfl | lemma | lie_module_hom.coe_submodule_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range (n : N) : n ∈ f.range ↔ ∃ m, f m = n | iff.rfl | lemma | lie_module_hom.mem_range | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_top : lie_submodule.map f ⊤ = f.range | by { ext, simp [lie_submodule.mem_map], } | lemma | lie_module_hom.map_top | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.map",
"lie_submodule.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_incl : N.incl.ker = ⊥ | by simp [← lie_submodule.coe_to_submodule_eq_iff] | lemma | lie_submodule.ker_incl | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_incl : N.incl.range = N | by simp [← lie_submodule.coe_to_submodule_eq_iff] | lemma | lie_submodule.range_incl | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_incl_self : comap N.incl N = ⊤ | by simp [← lie_submodule.coe_to_submodule_eq_iff] | lemma | lie_submodule.comap_incl_self | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_submodule.coe_to_submodule_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_subalgebra.top_equiv : (⊤ : lie_subalgebra R L) ≃ₗ⁅R⁆ L | { inv_fun := λ x, ⟨x, set.mem_univ x⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, rfl,
..(⊤ : lie_subalgebra R L).incl, } | def | lie_subalgebra.top_equiv | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"inv_fun",
"lie_subalgebra",
"set.mem_univ"
] | The natural equivalence between the 'top' Lie subalgebra and the enclosing Lie algebra.
This is the Lie subalgebra version of `submodule.top_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_subalgebra.top_equiv_apply (x : (⊤ : lie_subalgebra R L)) :
lie_subalgebra.top_equiv x = x | rfl | lemma | lie_subalgebra.top_equiv_apply | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_subalgebra",
"lie_subalgebra.top_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal.top_equiv : (⊤ : lie_ideal R L) ≃ₗ⁅R⁆ L | lie_subalgebra.top_equiv | def | lie_ideal.top_equiv | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_subalgebra.top_equiv"
] | The natural equivalence between the 'top' Lie ideal and the enclosing Lie algebra.
This is the Lie ideal version of `submodule.top_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ideal.top_equiv_apply (x : (⊤ : lie_ideal R L)) :
lie_ideal.top_equiv x = x | rfl | lemma | lie_ideal.top_equiv_apply | algebra.lie | src/algebra/lie/submodule.lean | [
"algebra.lie.subalgebra",
"ring_theory.noetherian"
] | [
"lie_ideal",
"lie_ideal.top_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_bracket_aux (x : L) : module.End R (M ⊗[R] N) | (to_endomorphism R L M x).rtensor N + (to_endomorphism R L N x).ltensor M | def | tensor_product.lie_module.has_bracket_aux | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"module.End"
] | It is useful to define the bracket via this auxiliary function so that we have a type-theoretic
expression of the fact that `L` acts by linear endomorphisms. It simplifies the proofs in
`lie_ring_module` below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ring_module : lie_ring_module L (M ⊗[R] N) | { bracket := λ x, has_bracket_aux x,
add_lie := λ x y t, by
{ simp only [has_bracket_aux, linear_map.ltensor_add, linear_map.rtensor_add, lie_hom.map_add,
linear_map.add_apply], abel, },
lie_add := λ x, linear_map.map_add _,
leibniz_lie := λ x y t, by
{ suffices : (has_bracket_aux x).c... | instance | tensor_product.lie_module.lie_ring_module | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"add_lie",
"leibniz_lie",
"lie_add",
"lie_hom.map_add",
"lie_hom.map_lie",
"lie_ring.of_associative_ring_bracket",
"lie_ring_module",
"linear_map.add_apply",
"linear_map.coe_comp",
"linear_map.comp_apply",
"linear_map.compr₂_apply",
"linear_map.ltensor_add",
"linear_map.ltensor_sub",
"line... | The tensor product of two Lie modules is a Lie ring module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module : lie_module R L (M ⊗[R] N) | { smul_lie := λ c x t, by
{ change has_bracket_aux (c • x) _ = c • has_bracket_aux _ _,
simp only [has_bracket_aux, smul_add, linear_map.rtensor_smul, linear_map.smul_apply,
linear_map.ltensor_smul, lie_hom.map_smul, linear_map.add_apply], },
lie_smul := λ c x, linear_map.map_smul _ c, } | instance | tensor_product.lie_module.lie_module | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_hom.map_smul",
"lie_module",
"lie_smul",
"linear_map.add_apply",
"linear_map.ltensor_smul",
"linear_map.map_smul",
"linear_map.rtensor_smul",
"linear_map.smul_apply",
"smul_add",
"smul_lie"
] | The tensor product of two Lie modules is a Lie module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_tmul_right (x : L) (m : M) (n : N) :
⁅x, m ⊗ₜ[R] n⁆ = ⁅x, m⁆ ⊗ₜ n + m ⊗ₜ ⁅x, n⁆ | show has_bracket_aux x (m ⊗ₜ[R] n) = _,
by simp only [has_bracket_aux, linear_map.rtensor_tmul, to_endomorphism_apply_apply,
linear_map.add_apply, linear_map.ltensor_tmul] | lemma | tensor_product.lie_module.lie_tmul_right | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"linear_map.add_apply",
"linear_map.ltensor_tmul",
"linear_map.rtensor_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : (M →ₗ[R] N →ₗ[R] P) ≃ₗ⁅R,L⁆ (M ⊗[R] N →ₗ[R] P) | { map_lie' := λ x f, by
{ ext m n, simp only [mk_apply, linear_map.compr₂_apply, lie_tmul_right, linear_map.sub_apply,
lift.equiv_apply, linear_equiv.to_fun_eq_coe, lie_hom.lie_apply, linear_map.map_add],
abel, },
..tensor_product.lift.equiv R M N P } | def | tensor_product.lie_module.lift | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_hom.lie_apply",
"lift",
"linear_equiv.to_fun_eq_coe",
"linear_map.compr₂_apply",
"linear_map.map_add",
"linear_map.sub_apply",
"tensor_product.lift.equiv"
] | The universal property for tensor product of modules of a Lie algebra: the `R`-linear
tensor-hom adjunction is equivariant with respect to the `L` action. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_apply (f : M →ₗ[R] N →ₗ[R] P) (m : M) (n : N) :
lift R L M N P f (m ⊗ₜ n) = f m n | rfl | lemma | tensor_product.lie_module.lift_apply | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lie : (M →ₗ⁅R,L⁆ N →ₗ[R] P) ≃ₗ[R] (M ⊗[R] N →ₗ⁅R,L⁆ P) | (max_triv_linear_map_equiv_lie_module_hom.symm ≪≫ₗ
↑(max_triv_equiv (lift R L M N P))) ≪≫ₗ
max_triv_linear_map_equiv_lie_module_hom | def | tensor_product.lie_module.lift_lie | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lift"
] | A weaker form of the universal property for tensor product of modules of a Lie algebra.
Note that maps `f` of type `M →ₗ⁅R,L⁆ N →ₗ[R] P` are exactly those `R`-bilinear maps satisfying
`⁅x, f m n⁆ = f ⁅x, m⁆ n + f m ⁅x, n⁆` for all `x, m, n` (see e.g, `lie_module_hom.map_lie₂`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lift_lie_eq_lift_coe (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) :
⇑(lift_lie R L M N P f) = lift R L M N P f | begin
suffices : (lift_lie R L M N P f : M ⊗[R] N →ₗ[R] P) = lift R L M N P f,
{ rw [← this, lie_module_hom.coe_to_linear_map], },
ext m n,
simp only [lift_lie, linear_equiv.trans_apply, lie_module_equiv.coe_to_linear_equiv,
coe_linear_map_max_triv_linear_map_equiv_lie_module_hom, coe_max_triv_equiv_apply,
... | lemma | tensor_product.lie_module.coe_lift_lie_eq_lift_coe | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_module_equiv.coe_to_linear_equiv",
"lie_module_hom.coe_to_linear_map",
"lift",
"linear_equiv.trans_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lie_apply (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (m : M) (n : N) :
lift_lie R L M N P f (m ⊗ₜ n) = f m n | by simp only [coe_lift_lie_eq_lift_coe, lie_module_hom.coe_to_linear_map, lift_apply] | lemma | tensor_product.lie_module.lift_lie_apply | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_module_hom.coe_to_linear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : M →ₗ⁅R,L⁆ P) (g : N →ₗ⁅R,L⁆ Q) : M ⊗[R] N →ₗ⁅R,L⁆ P ⊗[R] Q | { map_lie' := λ x t, by
{ simp only [linear_map.to_fun_eq_coe],
apply t.induction_on,
{ simp only [linear_map.map_zero, lie_zero], },
{ intros m n, simp only [lie_module_hom.coe_to_linear_map, lie_tmul_right,
lie_module_hom.map_lie, map_tmul, linear_map.map_add], },
{ intros t₁ t₂ ... | def | tensor_product.lie_module.map | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_add",
"lie_module_hom.coe_to_linear_map",
"lie_module_hom.map_lie",
"lie_zero",
"linear_map.map_add",
"linear_map.map_zero",
"linear_map.to_fun_eq_coe"
] | A pair of Lie module morphisms `f : M → P` and `g : N → Q`, induce a Lie module morphism:
`M ⊗ N → P ⊗ Q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_linear_map_map (f : M →ₗ⁅R,L⁆ P) (g : N →ₗ⁅R,L⁆ Q) :
(map f g : M ⊗[R] N →ₗ[R] P ⊗[R] Q) = tensor_product.map (f : M →ₗ[R] P) (g : N →ₗ[R] Q) | rfl | lemma | tensor_product.lie_module.coe_linear_map_map | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"tensor_product.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_tmul (f : M →ₗ⁅R,L⁆ P) (g : N →ₗ⁅R,L⁆ Q) (m : M) (n : N) :
map f g (m ⊗ₜ n) = (f m) ⊗ₜ (g n) | map_tmul f g m n | lemma | tensor_product.lie_module.map_tmul | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_incl (M' : lie_submodule R L M) (N' : lie_submodule R L N) :
M' ⊗[R] N' →ₗ⁅R,L⁆ M ⊗[R] N | map M'.incl N'.incl | def | tensor_product.lie_module.map_incl | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_submodule"
] | Given Lie submodules `M' ⊆ M` and `N' ⊆ N`, this is the natural map: `M' ⊗ N' → M ⊗ N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_incl_def (M' : lie_submodule R L M) (N' : lie_submodule R L N) :
map_incl M' N' = map M'.incl N'.incl | rfl | lemma | tensor_product.lie_module.map_incl_def | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_module_hom : L ⊗[R] M →ₗ⁅R,L⁆ M | tensor_product.lie_module.lift_lie R L L M M
{ map_lie' := λ x m, by { ext n, simp [lie_ring.of_associative_ring_bracket], },
..(to_endomorphism R L M : L →ₗ[R] M →ₗ[R] M), } | def | lie_module.to_module_hom | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_ring.of_associative_ring_bracket",
"tensor_product.lie_module.lift_lie"
] | The action of the Lie algebra on one of its modules, regarded as a morphism of Lie modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_hom_apply (x : L) (m : M) :
to_module_hom R L M (x ⊗ₜ m) = ⁅x, m⁆ | by simp only [to_module_hom, tensor_product.lie_module.lift_lie_apply, to_endomorphism_apply_apply,
lie_hom.coe_to_linear_map, lie_module_hom.coe_mk, linear_map.coe_mk, linear_map.to_fun_eq_coe] | lemma | lie_module.to_module_hom_apply | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_hom.coe_to_linear_map",
"lie_module_hom.coe_mk",
"linear_map.coe_mk",
"linear_map.to_fun_eq_coe",
"tensor_product.lie_module.lift_lie_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ideal_oper_eq_tensor_map_range :
⁅I, N⁆ = ((to_module_hom R L M).comp (map_incl I N : ↥I ⊗ ↥N →ₗ⁅R,L⁆ L ⊗ M)).range | begin
rw [← coe_to_submodule_eq_iff, lie_ideal_oper_eq_linear_span, lie_module_hom.coe_submodule_range,
lie_module_hom.coe_linear_map_comp, linear_map.range_comp, map_incl_def, coe_linear_map_map,
tensor_product.map_range_eq_span_tmul, submodule.map_span],
congr, ext m, split,
{ rintros ⟨⟨x, hx⟩, ⟨n, hn⟩,... | lemma | lie_submodule.lie_ideal_oper_eq_tensor_map_range | algebra.lie | src/algebra/lie/tensor_product.lean | [
"algebra.lie.abelian"
] | [
"lie_module_hom.coe_linear_map_comp",
"lie_module_hom.coe_submodule_range",
"linear_map.range_comp",
"submodule.map_span",
"tensor_product.map_range_eq_span_tmul"
] | A useful alternative characterisation of Lie ideal operations on Lie submodules.
Given a Lie ideal `I ⊆ L` and a Lie submodule `N ⊆ M`, by tensoring the inclusion maps and then
applying the action of `L` on `M`, we obtain morphism of Lie modules `f : I ⊗ N → L ⊗ M → M`.
This lemma states that `⁅I, N⁆ = range f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel : tensor_algebra R L → tensor_algebra R L → Prop
| lie_compat (x y : L) : rel (ιₜ ⁅x, y⁆ + (ιₜ y) * (ιₜ x)) ((ιₜ x) * (ιₜ y)) | inductive | universal_enveloping_algebra.rel | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"rel",
"tensor_algebra"
] | The quotient by the ideal generated by this relation is the universal enveloping algebra.
Note that we have avoided using the more natural expression:
| lie_compat (x y : L) : rel (ιₜ ⁅x, y⁆) ⁅ιₜ x, ιₜ y⁆
so that our construction needs only the semiring structure of the tensor algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
universal_enveloping_algebra | ring_quot (universal_enveloping_algebra.rel R L) | def | universal_enveloping_algebra | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"ring_quot",
"universal_enveloping_algebra.rel"
] | The universal enveloping algebra of a Lie algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_alg_hom : tensor_algebra R L →ₐ[R] universal_enveloping_algebra R L | ring_quot.mk_alg_hom R (rel R L) | def | universal_enveloping_algebra.mk_alg_hom | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"mk_alg_hom",
"rel",
"tensor_algebra",
"universal_enveloping_algebra"
] | The quotient map from the tensor algebra to the universal enveloping algebra as a morphism of
associative algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι : L →ₗ⁅R⁆ universal_enveloping_algebra R L | { map_lie' := λ x y, by
{ suffices : mk_alg_hom R L (ιₜ ⁅x, y⁆ + (ιₜ y) * (ιₜ x)) = mk_alg_hom R L ((ιₜ x) * (ιₜ y)),
{ rw alg_hom.map_mul at this, simp [lie_ring.of_associative_ring_bracket, ← this], },
exact ring_quot.mk_alg_hom_rel _ (rel.lie_compat x y), },
..(mk_alg_hom R L).to_linear_map.comp ιₜ } | def | universal_enveloping_algebra.ι | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"alg_hom.map_mul",
"lie_ring.of_associative_ring_bracket",
"mk_alg_hom",
"universal_enveloping_algebra"
] | The natural Lie algebra morphism from a Lie algebra to its universal enveloping algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (L →ₗ⁅R⁆ A) ≃ (universal_enveloping_algebra R L →ₐ[R] A) | { to_fun := λ f,
ring_quot.lift_alg_hom R ⟨tensor_algebra.lift R (f : L →ₗ[R] A),
begin
intros a b h, induction h with x y,
simp only [lie_ring.of_associative_ring_bracket,
map_add, tensor_algebra.lift_ι_apply, lie_hom.coe_to_linear_map, lie_hom.map_lie,
map_mul, sub_add_cancel],
... | def | universal_enveloping_algebra.lift | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"alg_hom.coe_to_lie_hom",
"alg_hom.comp_to_linear_map",
"alg_hom.to_linear_map_apply",
"inv_fun",
"lie_hom.coe_comp",
"lie_hom.coe_linear_map_comp",
"lie_hom.coe_mk",
"lie_hom.coe_to_linear_map",
"lie_hom.map_lie",
"lie_ring.of_associative_ring_bracket",
"lift",
"linear_map.coe_comp",
"linea... | The universal property of the universal enveloping algebra: Lie algebra morphisms into
associative algebras lift to associative algebra morphisms from the universal enveloping algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_symm_apply (F : universal_enveloping_algebra R L →ₐ[R] A) :
(lift R).symm F = (F : universal_enveloping_algebra R L →ₗ⁅R⁆ A).comp (ι R) | rfl | lemma | universal_enveloping_algebra.lift_symm_apply | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"lift",
"universal_enveloping_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_comp_lift : (lift R f) ∘ (ι R) = f | funext $ lie_hom.ext_iff.mp $ (lift R).symm_apply_apply f | lemma | universal_enveloping_algebra.ι_comp_lift | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_ι_apply (x : L) : lift R f (ι R x) = f x | by rw [←function.comp_apply (lift R f) (ι R) x, ι_comp_lift] | lemma | universal_enveloping_algebra.lift_ι_apply | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (g : universal_enveloping_algebra R L →ₐ[R] A) :
g ∘ (ι R) = f ↔ g = lift R f | begin
refine iff.trans _ (lift R).symm_apply_eq,
split; {intro h, ext, simp [←h] },
end | lemma | universal_enveloping_algebra.lift_unique | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"lift",
"lift_unique",
"universal_enveloping_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {g₁ g₂ : universal_enveloping_algebra R L →ₐ[R] A}
(h : (g₁ : universal_enveloping_algebra R L →ₗ⁅R⁆ A).comp (ι R) =
(g₂ : universal_enveloping_algebra R L →ₗ⁅R⁆ A).comp (ι R)) :
g₁ = g₂ | have h' : (lift R).symm g₁ = (lift R).symm g₂, { ext, simp [h], },
(lift R).symm.injective h' | lemma | universal_enveloping_algebra.hom_ext | algebra.lie | src/algebra/lie/universal_enveloping.lean | [
"algebra.lie.of_associative",
"algebra.ring_quot",
"linear_algebra.tensor_algebra.basic"
] | [
"hom_ext",
"lift",
"universal_enveloping_algebra"
] | See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pre_weight_space (χ : L → R) : submodule R M | ⨅ (x : L), (to_endomorphism R L M x).maximal_generalized_eigenspace (χ x) | def | lie_module.pre_weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"submodule"
] | Given a Lie module `M` over a Lie algebra `L`, the pre-weight space of `M` with respect to a
map `χ : L → R` is the simultaneous generalized eigenspace of the action of all `x : L` on `M`,
with eigenvalues `χ x`.
See also `lie_module.weight_space`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_pre_weight_space (χ : L → R) (m : M) :
m ∈ pre_weight_space M χ ↔ ∀ x, ∃ (k : ℕ), ((to_endomorphism R L M x - (χ x) • 1)^k) m = 0 | by simp [pre_weight_space, -linear_map.pow_apply] | lemma | lie_module.mem_pre_weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"linear_map.pow_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pre_weight_space_zero_le_ker_of_is_noetherian [is_noetherian R M] (x : L) :
∃ (k : ℕ), pre_weight_space M (0 : L → R) ≤ ((to_endomorphism R L M x)^k).ker | begin
use (to_endomorphism R L M x).maximal_generalized_eigenspace_index 0,
simp only [← module.End.generalized_eigenspace_zero, pre_weight_space, pi.zero_apply, infi_le,
← (to_endomorphism R L M x).maximal_generalized_eigenspace_eq],
end | lemma | lie_module.exists_pre_weight_space_zero_le_ker_of_is_noetherian | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"infi_le",
"is_noetherian",
"module.End.generalized_eigenspace_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weight_vector_multiplication (M₁ : Type w₁) (M₂ : Type w₂) (M₃ : Type w₃)
[add_comm_group M₁] [module R M₁] [lie_ring_module L M₁] [lie_module R L M₁]
[add_comm_group M₂] [module R M₂] [lie_ring_module L M₂] [lie_module R L M₂]
[add_comm_group M₃] [module R M₃] [lie_ring_module L M₃] [lie_module R L M₃]
(g : M₁ ... | begin
/- Unpack the statement of the goal. -/
intros m₃,
simp only [lie_module_hom.coe_to_linear_map, pi.add_apply, function.comp_app,
mem_pre_weight_space, linear_map.coe_comp, tensor_product.map_incl, exists_imp_distrib,
linear_map.mem_range],
rintros t rfl x,
/- Set up some notation. -/
let F : ... | lemma | lie_module.weight_vector_multiplication | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"add_comm_group",
"add_smul",
"commute",
"curry_apply",
"exists_imp_distrib",
"lie_module",
"lie_module_hom.coe_to_linear_map",
"lie_module_hom.map_add",
"lie_module_hom.map_smul",
"lie_module_hom.map_sub",
"lie_module_hom.map_zero",
"lie_ring_module",
"linear_map.add_apply",
"linear_map.c... | See also `bourbaki1975b` Chapter VII §1.1, Proposition 2 (ii). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_mem_pre_weight_space_of_mem_pre_weight_space {χ₁ χ₂ : L → R} {x : L} {m : M}
(hx : x ∈ pre_weight_space L χ₁) (hm : m ∈ pre_weight_space M χ₂) :
⁅x, m⁆ ∈ pre_weight_space M (χ₁ + χ₂) | begin
apply lie_module.weight_vector_multiplication L L M M (to_module_hom R L M) χ₁ χ₂,
simp only [lie_module_hom.coe_to_linear_map, function.comp_app, linear_map.coe_comp,
tensor_product.map_incl, linear_map.mem_range],
use [⟨x, hx⟩ ⊗ₜ ⟨m, hm⟩],
simp only [submodule.subtype_apply, to_module_hom_apply, ten... | lemma | lie_module.lie_mem_pre_weight_space_of_mem_pre_weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_module.weight_vector_multiplication",
"lie_module_hom.coe_to_linear_map",
"linear_map.coe_comp",
"linear_map.mem_range",
"submodule.subtype_apply",
"tensor_product.map_incl",
"tensor_product.map_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weight_space [lie_algebra.is_nilpotent R L] (χ : L → R) : lie_submodule R L M | { lie_mem := λ x m hm,
begin
rw ← zero_add χ,
refine lie_mem_pre_weight_space_of_mem_pre_weight_space _ hm,
suffices : pre_weight_space L (0 : L → R) = ⊤, { simp only [this, submodule.mem_top], },
exact lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent R L,
end,
.. pre_weight_space M χ ... | def | lie_module.weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent",
"lie_algebra.is_nilpotent",
"lie_submodule",
"submodule.mem_top"
] | If a Lie algebra is nilpotent, then pre-weight spaces are Lie submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_weight_space [lie_algebra.is_nilpotent R L] (χ : L → R) (m : M) :
m ∈ weight_space M χ ↔ m ∈ pre_weight_space M χ | iff.rfl | lemma | lie_module.mem_weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_algebra.is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_weight_space_eq_top_of_nilpotent'
[lie_algebra.is_nilpotent R L] [is_nilpotent R L M] :
weight_space M (0 : L → R) = ⊤ | begin
rw [← lie_submodule.coe_to_submodule_eq_iff, lie_submodule.top_coe_submodule],
exact infi_max_gen_zero_eigenspace_eq_top_of_nilpotent R L M,
end | lemma | lie_module.zero_weight_space_eq_top_of_nilpotent' | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_nilpotent",
"lie_algebra.is_nilpotent",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.top_coe_submodule"
] | See also the more useful form `lie_module.zero_weight_space_eq_top_of_nilpotent`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_weight_space_of_top [lie_algebra.is_nilpotent R L] (χ : L → R) :
(weight_space M (χ ∘ (⊤ : lie_subalgebra R L).incl) : submodule R M) = weight_space M χ | begin
ext m,
simp only [weight_space, lie_submodule.coe_to_submodule_mk, lie_subalgebra.coe_bracket_of_module,
function.comp_app, mem_pre_weight_space],
split; intros h x,
{ obtain ⟨k, hk⟩ := h ⟨x, set.mem_univ x⟩, use k, exact hk, },
{ obtain ⟨k, hk⟩ := h x, use k, exact hk, },
end | lemma | lie_module.coe_weight_space_of_top | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_algebra.is_nilpotent",
"lie_subalgebra",
"lie_subalgebra.coe_bracket_of_module",
"lie_submodule.coe_to_submodule_mk",
"set.mem_univ",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_weight_space_eq_top_of_nilpotent
[lie_algebra.is_nilpotent R L] [is_nilpotent R L M] :
weight_space M (0 : (⊤ : lie_subalgebra R L) → R) = ⊤ | begin
/- We use `coe_weight_space_of_top` as a trick to circumvent the fact that we don't (yet) know
`is_nilpotent R (⊤ : lie_subalgebra R L) M` is equivalent to `is_nilpotent R L M`. -/
have h₀ : (0 : L → R) ∘ (⊤ : lie_subalgebra R L).incl = 0, { ext, refl, },
rw [← lie_submodule.coe_to_submodule_eq_iff, lie... | lemma | lie_module.zero_weight_space_eq_top_of_nilpotent | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_nilpotent",
"lie_algebra.is_nilpotent",
"lie_subalgebra",
"lie_submodule.coe_to_submodule_eq_iff",
"lie_submodule.top_coe_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_weight (χ : lie_character R H) : Prop | weight_space M χ ≠ ⊥ | def | lie_module.is_weight | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [] | Given a Lie module `M` of a Lie algebra `L`, a weight of `M` with respect to a nilpotent
subalgebra `H ⊆ L` is a Lie character whose corresponding weight space is non-empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_weight_zero_of_nilpotent
[nontrivial M] [lie_algebra.is_nilpotent R L] [is_nilpotent R L M] :
is_weight (⊤ : lie_subalgebra R L) M 0 | by { rw [is_weight, lie_hom.coe_zero, zero_weight_space_eq_top_of_nilpotent], exact top_ne_bot, } | lemma | lie_module.is_weight_zero_of_nilpotent | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_nilpotent",
"lie_algebra.is_nilpotent",
"lie_hom.coe_zero",
"lie_subalgebra",
"nontrivial",
"top_ne_bot"
] | For a non-trivial nilpotent Lie module over a nilpotent Lie algebra, the zero character is a
weight with respect to the `⊤` Lie subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_nilpotent_to_endomorphism_weight_space_zero
[lie_algebra.is_nilpotent R L] [is_noetherian R M] (x : L) :
_root_.is_nilpotent $ to_endomorphism R L (weight_space M (0 : L → R)) x | begin
obtain ⟨k, hk⟩ := exists_pre_weight_space_zero_le_ker_of_is_noetherian R M x,
use k,
ext ⟨m, hm⟩,
rw [linear_map.zero_apply, lie_submodule.coe_zero, submodule.coe_eq_zero,
← lie_submodule.to_endomorphism_restrict_eq_to_endomorphism, linear_map.pow_restrict,
← set_like.coe_eq_coe, linear_map.restri... | lemma | lie_module.is_nilpotent_to_endomorphism_weight_space_zero | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_noetherian",
"lie_algebra.is_nilpotent",
"lie_submodule.coe_zero",
"lie_submodule.to_endomorphism_restrict_eq_to_endomorphism",
"linear_map.pow_restrict",
"linear_map.restrict_apply",
"linear_map.zero_apply",
"set_like.coe_eq_coe",
"submodule.coe_eq_zero",
"submodule.coe_mk",
"submodule.coe_... | A (nilpotent) Lie algebra acts nilpotently on the zero weight space of a Noetherian Lie
module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
root_space (χ : H → R) : lie_submodule R H L | weight_space L χ | abbreviation | lie_algebra.root_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_submodule"
] | Given a nilpotent Lie subalgebra `H ⊆ L`, the root space of a map `χ : H → R` is the weight
space of `L` regarded as a module of `H` via the adjoint action. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_root_space_eq_top_of_nilpotent [h : is_nilpotent R L] :
root_space (⊤ : lie_subalgebra R L) 0 = ⊤ | zero_weight_space_eq_top_of_nilpotent L | lemma | lie_algebra.zero_root_space_eq_top_of_nilpotent | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_nilpotent",
"lie_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_root | is_weight H L | abbreviation | lie_algebra.is_root | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [] | A root of a Lie algebra `L` with respect to a nilpotent subalgebra `H ⊆ L` is a weight of `L`,
regarded as a module of `H` via the adjoint action. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
root_space_comap_eq_weight_space (χ : H → R) :
(root_space H χ).comap H.incl' = weight_space H χ | begin
ext x,
let f : H → module.End R L := λ y, to_endomorphism R H L y - (χ y) • 1,
let g : H → module.End R H := λ y, to_endomorphism R H H y - (χ y) • 1,
suffices : (∀ (y : H), ∃ (k : ℕ), ((f y)^k).comp (H.incl : H →ₗ[R] L) x = 0) ↔
∀ (y : H), ∃ (k : ℕ), (H.incl : H →ₗ[R] L).comp ((g y)^k) x = ... | lemma | lie_algebra.root_space_comap_eq_weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_hom.coe_to_linear_map",
"lie_subalgebra.coe_bracket",
"lie_subalgebra.coe_bracket_of_module",
"lie_subalgebra.coe_incl",
"lie_subalgebra.coe_incl'",
"lie_submodule.mem_comap",
"linear_map.coe_comp",
"linear_map.commute_pow_left_of_commute",
"linear_map.one_apply",
"linear_map.smul_apply",
"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_mem_weight_space_of_mem_weight_space {χ₁ χ₂ : H → R} {x : L} {m : M}
(hx : x ∈ root_space H χ₁) (hm : m ∈ weight_space M χ₂) : ⁅x, m⁆ ∈ weight_space M (χ₁ + χ₂) | begin
apply lie_module.weight_vector_multiplication
H L M M ((to_module_hom R L M).restrict_lie H) χ₁ χ₂,
simp only [lie_module_hom.coe_to_linear_map, function.comp_app, linear_map.coe_comp,
tensor_product.map_incl, linear_map.mem_range],
use [⟨x, hx⟩ ⊗ₜ ⟨m, hm⟩],
simp only [submodule.subtype_apply, to_... | lemma | lie_algebra.lie_mem_weight_space_of_mem_weight_space | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_module.weight_vector_multiplication",
"lie_module_hom.coe_restrict_lie",
"lie_module_hom.coe_to_linear_map",
"linear_map.coe_comp",
"linear_map.mem_range",
"submodule.coe_mk",
"submodule.subtype_apply",
"tensor_product.map_incl",
"tensor_product.map_tmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
root_space_weight_space_product_aux {χ₁ χ₂ χ₃ : H → R} (hχ : χ₁ + χ₂ = χ₃) :
(root_space H χ₁) →ₗ[R] (weight_space M χ₂) →ₗ[R] (weight_space M χ₃) | { to_fun := λ x,
{ to_fun :=
λ m, ⟨⁅(x : L), (m : M)⁆,
hχ ▸ (lie_mem_weight_space_of_mem_weight_space x.property m.property) ⟩,
map_add' := λ m n, by { simp only [lie_submodule.coe_add, lie_add], refl, },
map_smul' := λ t m, by { conv_lhs { congr, rw [lie_submodule.coe_smul, lie_smul]... | def | lie_algebra.root_space_weight_space_product_aux | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"add_lie",
"lie_add",
"lie_smul",
"lie_submodule.coe_add",
"lie_submodule.coe_smul",
"linear_map.add_apply",
"linear_map.coe_mk",
"linear_map.smul_apply",
"ring_hom.id_apply",
"smul_lie",
"subtype.coe_mk"
] | Auxiliary definition for `root_space_weight_space_product`,
which is close to the deterministic timeout limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
root_space_weight_space_product (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃) :
(root_space H χ₁) ⊗[R] (weight_space M χ₂) →ₗ⁅R,H⁆ weight_space M χ₃ | lift_lie R H (root_space H χ₁) (weight_space M χ₂) (weight_space M χ₃)
{ to_linear_map := root_space_weight_space_product_aux R L H M hχ,
map_lie' := λ x y, by ext m; rw [root_space_weight_space_product_aux,
lie_hom.lie_apply, lie_submodule.coe_sub, linear_map.coe_mk,
linear_map.coe_mk, subtype.coe_mk, subtyp... | def | lie_algebra.root_space_weight_space_product | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_hom.lie_apply",
"lie_lie",
"lie_subalgebra.coe_bracket_of_module",
"lie_submodule.coe_bracket",
"lie_submodule.coe_sub",
"linear_map.coe_mk",
"subtype.coe_mk"
] | Given a nilpotent Lie subalgebra `H ⊆ L` together with `χ₁ χ₂ : H → R`, there is a natural
`R`-bilinear product of root vectors and weight vectors, compatible with the actions of `H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_root_space_weight_space_product_tmul
(χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃) (x : root_space H χ₁) (m : weight_space M χ₂) :
(root_space_weight_space_product R L H M χ₁ χ₂ χ₃ hχ (x ⊗ₜ m) : M) = ⁅(x : L), (m : M)⁆ | by simp only [root_space_weight_space_product, root_space_weight_space_product_aux,
lift_apply, lie_module_hom.coe_to_linear_map,
coe_lift_lie_eq_lift_coe, submodule.coe_mk, linear_map.coe_mk, lie_module_hom.coe_mk] | lemma | lie_algebra.coe_root_space_weight_space_product_tmul | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_module_hom.coe_mk",
"lie_module_hom.coe_to_linear_map",
"linear_map.coe_mk",
"submodule.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
root_space_product (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃) :
(root_space H χ₁) ⊗[R] (root_space H χ₂) →ₗ⁅R,H⁆ root_space H χ₃ | root_space_weight_space_product R L H L χ₁ χ₂ χ₃ hχ | def | lie_algebra.root_space_product | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [] | Given a nilpotent Lie subalgebra `H ⊆ L` together with `χ₁ χ₂ : H → R`, there is a natural
`R`-bilinear product of root vectors, compatible with the actions of `H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
root_space_product_def :
root_space_product R L H = root_space_weight_space_product R L H L | rfl | lemma | lie_algebra.root_space_product_def | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
root_space_product_tmul
(χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃) (x : root_space H χ₁) (y : root_space H χ₂) :
(root_space_product R L H χ₁ χ₂ χ₃ hχ (x ⊗ₜ y) : L) = ⁅(x : L), (y : L)⁆ | by simp only [root_space_product_def, coe_root_space_weight_space_product_tmul] | lemma | lie_algebra.root_space_product_tmul | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_root_subalgebra : lie_subalgebra R L | { lie_mem' := λ x y hx hy, by
{ let xy : (root_space H 0) ⊗[R] (root_space H 0) := ⟨x, hx⟩ ⊗ₜ ⟨y, hy⟩,
suffices : (root_space_product R L H 0 0 0 (add_zero 0) xy : L) ∈ root_space H 0,
{ rwa [root_space_product_tmul, subtype.coe_mk, subtype.coe_mk] at this, },
exact (root_space_product R L H 0 0 0 (add_ze... | def | lie_algebra.zero_root_subalgebra | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_subalgebra",
"submodule",
"subtype.coe_mk"
] | Given a nilpotent Lie subalgebra `H ⊆ L`, the root space of the zero map `0 : H → R` is a Lie
subalgebra of `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero_root_subalgebra :
(zero_root_subalgebra R L H : submodule R L) = root_space H 0 | rfl | lemma | lie_algebra.coe_zero_root_subalgebra | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_zero_root_subalgebra (x : L) :
x ∈ zero_root_subalgebra R L H ↔ ∀ (y : H), ∃ (k : ℕ), ((to_endomorphism R H L y)^k) x = 0 | by simp only [zero_root_subalgebra, mem_weight_space, mem_pre_weight_space, pi.zero_apply, sub_zero,
set_like.mem_coe, zero_smul, lie_submodule.mem_coe_submodule, submodule.mem_carrier,
lie_subalgebra.mem_mk_iff] | lemma | lie_algebra.mem_zero_root_subalgebra | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_subalgebra.mem_mk_iff",
"lie_submodule.mem_coe_submodule",
"set_like.mem_coe",
"submodule.mem_carrier",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lie_submodule_le_root_space_zero : H.to_lie_submodule ≤ root_space H 0 | begin
intros x hx,
simp only [lie_subalgebra.mem_to_lie_submodule] at hx,
simp only [mem_weight_space, mem_pre_weight_space, pi.zero_apply, sub_zero, zero_smul],
intros y,
unfreezingI { obtain ⟨k, hk⟩ := (infer_instance : is_nilpotent R H) },
use k,
let f : module.End R H := to_endomorphism R H H y,
let... | lemma | lie_algebra.to_lie_submodule_le_root_space_zero | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_nilpotent",
"lie_subalgebra.coe_bracket",
"lie_subalgebra.coe_bracket_of_module",
"lie_subalgebra.mem_to_lie_submodule",
"lie_submodule.mem_bot",
"linear_map.coe_comp",
"linear_map.commute_pow_left_of_commute",
"linear_map.pow_apply",
"module.End",
"submodule",
"submodule.coe_eq_zero",
"su... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_zero_root_subalgebra : H ≤ zero_root_subalgebra R L H | begin
rw [← lie_subalgebra.coe_submodule_le_coe_submodule, ← H.coe_to_lie_submodule,
coe_zero_root_subalgebra, lie_submodule.coe_submodule_le_coe_submodule],
exact to_lie_submodule_le_root_space_zero R L H,
end | lemma | lie_algebra.le_zero_root_subalgebra | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_subalgebra.coe_submodule_le_coe_submodule",
"lie_submodule.coe_submodule_le_coe_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_root_subalgebra_normalizer_eq_self :
(zero_root_subalgebra R L H).normalizer = zero_root_subalgebra R L H | begin
refine le_antisymm _ (lie_subalgebra.le_normalizer _),
intros x hx,
rw lie_subalgebra.mem_normalizer_iff at hx,
rw mem_zero_root_subalgebra,
rintros ⟨y, hy⟩,
specialize hx y (le_zero_root_subalgebra R L H hy),
rw mem_zero_root_subalgebra at hx,
obtain ⟨k, hk⟩ := hx ⟨y, hy⟩,
rw [← lie_skew, linea... | lemma | lie_algebra.zero_root_subalgebra_normalizer_eq_self | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_skew",
"lie_subalgebra.coe_bracket_of_module",
"lie_subalgebra.le_normalizer",
"lie_subalgebra.mem_normalizer_iff",
"linear_map.coe_comp",
"linear_map.iterate_succ",
"linear_map.map_neg",
"submodule.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cartan_of_zero_root_subalgebra_eq (h : zero_root_subalgebra R L H = H) :
H.is_cartan_subalgebra | { nilpotent := infer_instance,
self_normalizing := by { rw ← h, exact zero_root_subalgebra_normalizer_eq_self R L H, } } | lemma | lie_algebra.is_cartan_of_zero_root_subalgebra_eq | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [] | If the zero root subalgebra of a nilpotent Lie subalgebra `H` is just `H` then `H` is a Cartan
subalgebra.
When `L` is Noetherian, it follows from Engel's theorem that the converse holds. See
`lie_algebra.zero_root_subalgebra_eq_iff_is_cartan` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_root_subalgebra_eq_of_is_cartan (H : lie_subalgebra R L)
[H.is_cartan_subalgebra] [is_noetherian R L] :
zero_root_subalgebra R L H = H | begin
refine le_antisymm _ (le_zero_root_subalgebra R L H),
suffices : root_space H 0 ≤ H.to_lie_submodule, { exact λ x hx, this hx, },
obtain ⟨k, hk⟩ := (root_space H 0).is_nilpotent_iff_exists_self_le_ucs.mp (by apply_instance),
exact hk.trans (lie_submodule.ucs_le_of_normalizer_eq_self (by simp) k),
end | lemma | lie_algebra.zero_root_subalgebra_eq_of_is_cartan | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_noetherian",
"lie_subalgebra",
"lie_submodule.ucs_le_of_normalizer_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_root_subalgebra_eq_iff_is_cartan [is_noetherian R L] :
zero_root_subalgebra R L H = H ↔ H.is_cartan_subalgebra | ⟨is_cartan_of_zero_root_subalgebra_eq R L H, by { introsI, simp, }⟩ | lemma | lie_algebra.zero_root_subalgebra_eq_iff_is_cartan | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"is_noetherian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
weight_space' (χ : H → R) : lie_submodule R (zero_root_subalgebra R L H) M | { lie_mem := λ x m hm, by
{ have hx : (x : L) ∈ root_space H 0,
{ rw [← lie_submodule.mem_coe_submodule, ← coe_zero_root_subalgebra], exact x.property, },
rw ← zero_add χ,
exact lie_mem_weight_space_of_mem_weight_space hx hm, },
.. (weight_space M χ : submodule R M) } | def | lie_module.weight_space' | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"lie_submodule",
"lie_submodule.mem_coe_submodule",
"submodule"
] | A priori, weight spaces are Lie submodules over the Lie subalgebra `H` used to define them.
However they are naturally Lie submodules over the (in general larger) Lie subalgebra
`zero_root_subalgebra R L H`. Even though it is often the case that
`zero_root_subalgebra R L H = H`, it is likely to be useful to have the fl... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_weight_space' (χ : H → R) :
(weight_space' M χ : submodule R M) = weight_space M χ | rfl | lemma | lie_module.coe_weight_space' | algebra.lie | src/algebra/lie/weights.lean | [
"algebra.lie.nilpotent",
"algebra.lie.tensor_product",
"algebra.lie.character",
"algebra.lie.engel",
"algebra.lie.cartan_subalgebra",
"linear_algebra.eigenspace.basic",
"ring_theory.tensor_product"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_linear_map : (M →ₗ[A] N) →ₗ[k] (M →ₗ[k] N) | { to_fun := linear_map.restrict_scalars k,
map_add' := by tidy,
map_smul' := by tidy, } | def | linear_map.restrict_scalars_linear_map | algebra.module | src/algebra/module/algebra.lean | [
"algebra.module.basic",
"algebra.algebra.basic"
] | [
"linear_map.restrict_scalars"
] | Restriction of scalars for linear maps between modules over a `k`-algebra is itself `k`-linear. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module (R : Type u) (M : Type v) [semiring R]
[add_comm_monoid M] extends distrib_mul_action R M | (add_smul : ∀(r s : R) (x : M), (r + s) • x = r • x + s • x)
(zero_smul : ∀x : M, (0 : R) • x = 0) | class | module | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_comm_monoid",
"add_smul",
"distrib_mul_action",
"semiring",
"zero_smul"
] | A module is a generalization of vector spaces to a scalar semiring.
It consists of a scalar semiring `R` and an additive monoid of "vectors" `M`,
connected by a "scalar multiplication" operation `r • x : M`
(where `r : R` and `x : M`) with some natural associativity and
distributivity axioms similar to those on... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.to_mul_action_with_zero :
mul_action_with_zero R M | { smul_zero := smul_zero,
zero_smul := module.zero_smul,
..(infer_instance : mul_action R M) } | instance | module.to_mul_action_with_zero | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"mul_action",
"mul_action_with_zero",
"smul_zero",
"zero_smul"
] | A module over a semiring automatically inherits a `mul_action_with_zero` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_monoid.nat_module : module ℕ M | { one_smul := one_nsmul,
mul_smul := λ m n a, mul_nsmul a m n,
smul_add := λ n a b, nsmul_add a b n,
smul_zero := nsmul_zero,
zero_smul := zero_nsmul,
add_smul := λ r s x, add_nsmul x r s } | instance | add_comm_monoid.nat_module | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_smul",
"module",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.End.nat_cast_def (n : ℕ) :
(↑n : add_monoid.End M) = distrib_mul_action.to_add_monoid_End ℕ M n | rfl | lemma | add_monoid.End.nat_cast_def | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_monoid.End",
"distrib_mul_action.to_add_monoid_End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_smul : (r + s) • x = r • x + s • x | module.add_smul r s x | theorem | add_smul | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex.combo_self {a b : R} (h : a + b = 1) (x : M) : a • x + b • x = x | by rw [←add_smul, h, one_smul] | lemma | convex.combo_self | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_smul : (2 : R) • x = x + x | by rw [bit0, add_smul, one_smul] | theorem | two_smul | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_smul",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_smul' : (2 : R) • x = bit0 x | two_smul R x | theorem | two_smul' | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_two_smul_add_inv_of_two_smul [invertible (2 : R)] (x : M) :
(⅟2 : R) • x + (⅟2 : R) • x = x | convex.combo_self inv_of_two_add_inv_of_two _ | lemma | inv_of_two_smul_add_inv_of_two_smul | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"convex.combo_self",
"inv_of_two_add_inv_of_two",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.module [add_comm_monoid M₂] [has_smul R M₂] (f : M₂ →+ M)
(hf : injective f) (smul : ∀ (c : R) x, f (c • x) = c • f x) :
module R M₂ | { smul := (•),
add_smul := λ c₁ c₂ x, hf $ by simp only [smul, f.map_add, add_smul],
zero_smul := λ x, hf $ by simp only [smul, zero_smul, f.map_zero],
.. hf.distrib_mul_action f smul } | def | function.injective.module | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_comm_monoid",
"add_smul",
"has_smul",
"module",
"zero_smul"
] | Pullback a `module` structure along an injective additive monoid homomorphism.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.module [add_comm_monoid M₂] [has_smul R M₂] (f : M →+ M₂)
(hf : surjective f) (smul : ∀ (c : R) x, f (c • x) = c • f x) :
module R M₂ | { smul := (•),
add_smul := λ c₁ c₂ x, by { rcases hf x with ⟨x, rfl⟩,
simp only [add_smul, ← smul, ← f.map_add] },
zero_smul := λ x, by { rcases hf x with ⟨x, rfl⟩, simp only [← f.map_zero, ← smul, zero_smul] },
.. hf.distrib_mul_action f smul } | def | function.surjective.module | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_comm_monoid",
"add_smul",
"has_smul",
"module",
"zero_smul"
] | Pushforward a `module` structure along a surjective additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.module_left {R S M : Type*} [semiring R] [add_comm_monoid M]
[module R M] [semiring S] [has_smul S M]
(f : R →+* S) (hf : function.surjective f) (hsmul : ∀ c (x : M), f c • x = c • x) :
module S M | { smul := (•),
zero_smul := λ x, by rw [← f.map_zero, hsmul, zero_smul],
add_smul := hf.forall₂.mpr (λ a b x, by simp only [← f.map_add, hsmul, add_smul]),
.. hf.distrib_mul_action_left f.to_monoid_hom hsmul } | def | function.surjective.module_left | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_comm_monoid",
"add_smul",
"has_smul",
"module",
"semiring",
"zero_smul"
] | Push forward the action of `R` on `M` along a compatible surjective map `f : R →+* S`.
See also `function.surjective.mul_action_left` and `function.surjective.distrib_mul_action_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.comp_hom [semiring S] (f : S →+* R) :
module S M | { smul := has_smul.comp.smul f,
add_smul := λ r s x, by simp [add_smul],
.. mul_action_with_zero.comp_hom M f.to_monoid_with_zero_hom,
.. distrib_mul_action.comp_hom M (f : S →* R) } | def | module.comp_hom | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_smul",
"distrib_mul_action.comp_hom",
"has_smul.comp.smul",
"module",
"mul_action_with_zero.comp_hom",
"semiring"
] | Compose a `module` with a `ring_hom`, with action `f s • m`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.to_add_monoid_End : R →+* add_monoid.End M | { map_zero' := add_monoid_hom.ext $ λ r, by simp,
map_add' := λ x y, add_monoid_hom.ext $ λ r, by simp [add_smul],
..distrib_mul_action.to_add_monoid_End R M } | def | module.to_add_monoid_End | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_monoid.End",
"add_smul",
"distrib_mul_action.to_add_monoid_End"
] | `(•)` as an `add_monoid_hom`.
This is a stronger version of `distrib_mul_action.to_add_monoid_End` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_add_hom : R →+ M →+ M | (module.to_add_monoid_End R M).to_add_monoid_hom | def | smul_add_hom | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"module.to_add_monoid_End"
] | A convenience alias for `module.to_add_monoid_End` as an `add_monoid_hom`, usually to allow the
use of `add_monoid_hom.flip`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_add_hom_apply (r : R) (x : M) :
smul_add_hom R M r x = r • x | rfl | lemma | smul_add_hom_apply | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"smul_add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.eq_zero_of_zero_eq_one (zero_eq_one : (0 : R) = 1) : x = 0 | by rw [←one_smul R x, ←zero_eq_one, zero_smul] | lemma | module.eq_zero_of_zero_eq_one | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_add_one_sub_smul {R : Type*} [ring R] [module R M]
{r : R} {m : M} : r • m + (1 - r) • m = m | by rw [← add_smul, add_sub_cancel'_right, one_smul] | lemma | smul_add_one_sub_smul | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_smul",
"module",
"one_smul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.add_comm_monoid_to_add_comm_group [ring R] [add_comm_monoid M] [module R M] :
add_comm_group M | { neg := λ a, (-1 : R) • a,
add_left_neg := λ a, show (-1 : R) • a + a = 0, by
{ nth_rewrite 1 ← one_smul _ a,
rw [← add_smul, add_left_neg, zero_smul] },
..(infer_instance : add_comm_monoid M), } | def | module.add_comm_monoid_to_add_comm_group | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_comm_group",
"add_comm_monoid",
"add_smul",
"module",
"one_smul",
"ring",
"zero_smul"
] | An `add_comm_monoid` that is a `module` over a `ring` carries a natural `add_comm_group`
structure.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_group.int_module : module ℤ M | { one_smul := one_zsmul,
mul_smul := λ m n a, mul_zsmul a m n,
smul_add := λ n a b, zsmul_add a b n,
smul_zero := zsmul_zero,
zero_smul := zero_zsmul,
add_smul := λ r s x, add_zsmul x r s } | instance | add_comm_group.int_module | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_smul",
"module",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.End.int_cast_def (z : ℤ) :
(↑z : add_monoid.End M) = distrib_mul_action.to_add_monoid_End ℤ M z | rfl | lemma | add_monoid.End.int_cast_def | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_monoid.End",
"distrib_mul_action.to_add_monoid_End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.core extends has_smul R M | (smul_add : ∀(r : R) (x y : M), r • (x + y) = r • x + r • y)
(add_smul : ∀(r s : R) (x : M), (r + s) • x = r • x + s • x)
(mul_smul : ∀(r s : R) (x : M), (r * s) • x = r • s • x)
(one_smul : ∀x : M, (1 : R) • x = x) | structure | module.core | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_smul",
"has_smul",
"one_smul",
"smul_add"
] | A structure containing most informations as in a module, except the fields `zero_smul`
and `smul_zero`. As these fields can be deduced from the other ones when `M` is an `add_comm_group`,
this provides a way to construct a module structure by checking less properties, in
`module.of_core`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module.of_core (H : module.core R M) : module R M | by letI := H.to_has_smul; exact
{ zero_smul := λ x, (add_monoid_hom.mk' (λ r : R, r • x) (λ r s, H.add_smul r s x)).map_zero,
smul_zero := λ r, (add_monoid_hom.mk' ((•) r) (H.smul_add r)).map_zero,
..H } | def | module.of_core | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"module",
"module.core",
"smul_zero",
"zero_smul"
] | Define `module` without proving `zero_smul` and `smul_zero` by using an auxiliary
structure `module.core`, when the underlying space is an `add_comm_group`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.combo_eq_smul_sub_add [module R M] {x y : M} {a b : R} (h : a + b = 1) :
a • x + b • y = b • (y - x) + x | calc
a • x + b • y = (b • y - b • x) + (a • x + b • x) : by abel
... = b • (y - x) + x : by rw [smul_sub, convex.combo_self h] | lemma | convex.combo_eq_smul_sub_add | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"convex.combo_self",
"module",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.ext' {R : Type*} [semiring R] {M : Type*} [add_comm_monoid M] (P Q : module R M)
(w : ∀ (r : R) (m : M), by { haveI := P, exact r • m } = by { haveI := Q, exact r • m }) :
P = Q | begin
ext,
exact w _ _
end | lemma | module.ext' | algebra.module | src/algebra/module/basic.lean | [
"algebra.smul_with_zero",
"group_theory.group_action.group",
"tactic.abel"
] | [
"add_comm_monoid",
"module",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.