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is_lie_abelian_iff_center_eq_top : is_lie_abelian L ↔ center R L = ⊤ | lie_module.is_trivial_iff_max_triv_eq_top R L L | lemma | lie_algebra.is_lie_abelian_iff_center_eq_top | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"is_lie_abelian",
"lie_module.is_trivial_iff_max_triv_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_submodule.trivial_lie_oper_zero [lie_module.is_trivial L M] : ⁅I, N⁆ = ⊥ | begin
suffices : ⁅I, N⁆ ≤ ⊥, from le_bot_iff.mp this,
rw [lie_ideal_oper_eq_span, lie_submodule.lie_span_le],
rintros m ⟨x, n, h⟩, rw trivial_lie_zero at h, simp [← h],
end | lemma | lie_submodule.trivial_lie_oper_zero | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"lie_module.is_trivial",
"lie_submodule.lie_span_le",
"trivial_lie_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_submodule.lie_abelian_iff_lie_self_eq_bot : is_lie_abelian I ↔ ⁅I, I⁆ = ⊥ | begin
simp only [_root_.eq_bot_iff, lie_ideal_oper_eq_span, lie_submodule.lie_span_le,
lie_submodule.bot_coe, set.subset_singleton_iff, set.mem_set_of_eq, exists_imp_distrib],
refine ⟨λ h z x y hz, hz.symm.trans (((I : lie_subalgebra R L).coe_bracket x y).symm.trans
((coe_zero_iff_zero _ _).mpr (by apply h.... | lemma | lie_submodule.lie_abelian_iff_lie_self_eq_bot | algebra.lie | src/algebra/lie/abelian.lean | [
"algebra.lie.of_associative",
"algebra.lie.ideal_operations"
] | [
"exists_imp_distrib",
"is_lie_abelian",
"lie_subalgebra",
"lie_submodule.bot_coe",
"lie_submodule.lie_span_le",
"set.subset_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket' : (A ⊗[R] L) →ₗ[R] (A ⊗[R] L) →ₗ[R] A ⊗[R] L | tensor_product.curry $
(tensor_product.map (linear_map.mul' R _) (lie_module.to_module_hom R L L : L ⊗[R] L →ₗ[R] L))
∘ₗ ↑(tensor_product.tensor_tensor_tensor_comm R A L A L) | def | lie_algebra.extend_scalars.bracket' | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [
"lie_module.to_module_hom",
"linear_map.mul'",
"tensor_product.curry",
"tensor_product.map",
"tensor_product.tensor_tensor_tensor_comm"
] | The Lie bracket on the extension of a Lie algebra `L` over `R` by an algebra `A` over `R`.
In fact this bracket is fully `A`-bilinear but without a significant upgrade to our mixed-scalar
support in the tensor product library, it is far easier to bootstrap like this, starting with the
definition below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bracket'_tmul (s t : A) (x y : L) :
bracket' R A L (s ⊗ₜ[R] x) (t ⊗ₜ[R] y) = (s*t) ⊗ₜ ⁅x, y⁆ | by simp [bracket'] | lemma | lie_algebra.extend_scalars.bracket'_tmul | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket_def (x y : A ⊗[R] L) : ⁅x, y⁆ = bracket' R A L x y | rfl | lemma | lie_algebra.extend_scalars.bracket_def | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s*t) ⊗ₜ ⁅x, y⁆ | by rw [bracket_def, bracket'_tmul] | lemma | lie_algebra.extend_scalars.bracket_tmul | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 | begin
simp only [bracket_def],
apply x.induction_on,
{ simp only [linear_map.map_zero, eq_self_iff_true, linear_map.zero_apply], },
{ intros a l,
simp only [bracket'_tmul, tensor_product.tmul_zero, eq_self_iff_true, lie_self], },
{ intros z₁ z₂ h₁ h₂,
suffices : bracket' R A L z₁ z₂ + bracket' R A L z... | lemma | lie_algebra.extend_scalars.bracket_lie_self | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [
"lie_self",
"lie_skew",
"linear_map.add_apply",
"linear_map.map_add",
"linear_map.map_zero",
"linear_map.zero_apply",
"mul_comm",
"tensor_product.tmul_neg",
"tensor_product.tmul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket_leibniz_lie (x y z : A ⊗[R] L) : ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆ | begin
simp only [bracket_def],
apply x.induction_on,
{ simp only [linear_map.map_zero, add_zero, eq_self_iff_true, linear_map.zero_apply], },
{ intros a₁ l₁,
apply y.induction_on,
{ simp only [linear_map.map_zero, add_zero, eq_self_iff_true, linear_map.zero_apply], },
{ intros a₂ l₂,
apply z.i... | lemma | lie_algebra.extend_scalars.bracket_leibniz_lie | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [
"leibniz_lie",
"linear_map.add_apply",
"linear_map.map_add",
"linear_map.map_zero",
"linear_map.zero_apply",
"mul_assoc",
"mul_left_comm",
"tensor_product.tmul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ = a • ⁅x, y⁆ | begin
apply x.induction_on,
{ simp only [zero_lie, smul_zero], },
{ intros a₁ l₁, apply y.induction_on,
{ simp only [lie_zero, smul_zero], },
{ intros a₂ l₂,
simp only [bracket_def, bracket', tensor_product.smul_tmul', mul_left_comm a₁ a a₂,
tensor_product.curry_apply, linear_map.mul'_apply,... | lemma | lie_algebra.extend_scalars.bracket_lie_smul | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [
"add_lie",
"algebra.id.smul_eq_mul",
"lie_add",
"lie_zero",
"linear_equiv.coe_coe",
"linear_map.coe_comp",
"linear_map.mul'_apply",
"mul_left_comm",
"smul_add",
"smul_zero",
"tensor_product.curry_apply",
"tensor_product.map_tmul",
"tensor_product.smul_tmul'",
"tensor_product.tensor_tensor_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra : lie_algebra A (A ⊗[R] L) | { lie_smul := bracket_lie_smul R A L, } | instance | lie_algebra.extend_scalars.lie_algebra | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [
"lie_algebra",
"lie_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra [comm_ring R] [algebra R A] : lie_algebra R (restrict_scalars R A L) | { lie_smul := λ t x y, (lie_smul (algebra_map R A t)
(restrict_scalars.add_equiv R A L x) (restrict_scalars.add_equiv R A L y) : _) } | instance | lie_algebra.restrict_scalars.lie_algebra | algebra.lie | src/algebra/lie/base_change.lean | [
"algebra.algebra.restrict_scalars",
"algebra.lie.tensor_product"
] | [
"algebra",
"algebra_map",
"comm_ring",
"lie_algebra",
"lie_smul",
"restrict_scalars",
"restrict_scalars.add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ring (L : Type v) extends add_comm_group L, has_bracket L L | (add_lie : ∀ (x y z : L), ⁅x + y, z⁆ = ⁅x, z⁆ + ⁅y, z⁆)
(lie_add : ∀ (x y z : L), ⁅x, y + z⁆ = ⁅x, y⁆ + ⁅x, z⁆)
(lie_self : ∀ (x : L), ⁅x, x⁆ = 0)
(leibniz_lie : ∀ (x y z : L), ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆) | class | lie_ring | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_comm_group",
"add_lie",
"has_bracket",
"leibniz_lie",
"lie_add",
"lie_self"
] | A Lie ring is an additive group with compatible product, known as the bracket, satisfying the
Jacobi identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_algebra (R : Type u) (L : Type v) [comm_ring R] [lie_ring L]
extends module R L | (lie_smul : ∀ (t : R) (x y : L), ⁅x, t • y⁆ = t • ⁅x, y⁆) | class | lie_algebra | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"comm_ring",
"lie_ring",
"lie_smul",
"module"
] | A Lie algebra is a module with compatible product, known as the bracket, satisfying the Jacobi
identity. Forgetting the scalar multiplication, every Lie algebra is a Lie ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ring_module (L : Type v) (M : Type w)
[lie_ring L] [add_comm_group M] extends has_bracket L M | (add_lie : ∀ (x y : L) (m : M), ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆)
(lie_add : ∀ (x : L) (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆)
(leibniz_lie : ∀ (x y : L) (m : M), ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆) | class | lie_ring_module | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_comm_group",
"add_lie",
"has_bracket",
"leibniz_lie",
"lie_add",
"lie_ring"
] | A Lie ring module is an additive group, together with an additive action of a
Lie ring on this group, such that the Lie bracket acts as the commutator of endomorphisms.
(For representations of Lie *algebras* see `lie_module`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module (R : Type u) (L : Type v) (M : Type w)
[comm_ring R] [lie_ring L] [lie_algebra R L] [add_comm_group M] [module R M]
[lie_ring_module L M] | (smul_lie : ∀ (t : R) (x : L) (m : M), ⁅t • x, m⁆ = t • ⁅x, m⁆)
(lie_smul : ∀ (t : R) (x : L) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) | class | lie_module | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_comm_group",
"comm_ring",
"lie_algebra",
"lie_ring",
"lie_ring_module",
"lie_smul",
"module",
"smul_lie"
] | A Lie module is a module over a commutative ring, together with a linear action of a Lie
algebra on this module, such that the Lie bracket acts as the commutator of endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_lie : ⁅x + y, m⁆ = ⁅x, m⁆ + ⁅y, m⁆ | lie_ring_module.add_lie x y m | lemma | add_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_add : ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆ | lie_ring_module.lie_add x m n | lemma | lie_add | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_lie : ⁅t • x, m⁆ = t • ⁅x, m⁆ | lie_module.smul_lie t x m | lemma | smul_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_smul : ⁅x, t • m⁆ = t • ⁅x, m⁆ | lie_module.lie_smul t x m | lemma | lie_smul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leibniz_lie : ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆ | lie_ring_module.leibniz_lie x y m | lemma | leibniz_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_zero : ⁅x, 0⁆ = (0 : M) | (add_monoid_hom.mk' _ (lie_add x)).map_zero | lemma | lie_zero | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_lie : ⁅(0 : L), m⁆ = 0 | (add_monoid_hom.mk' (λ (x : L), ⁅x, m⁆) (λ x y, add_lie x y m)).map_zero | lemma | zero_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_self : ⁅x, x⁆ = 0 | lie_ring.lie_self x | lemma | lie_self | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ring_self_module : lie_ring_module L L | { ..(infer_instance : lie_ring L) } | instance | lie_ring_self_module | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_ring",
"lie_ring_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_skew : -⁅y, x⁆ = ⁅x, y⁆ | have h : ⁅x + y, x⁆ + ⁅x + y, y⁆ = 0, { rw ← lie_add, apply lie_self, },
by simpa [neg_eq_iff_add_eq_zero] using h | lemma | lie_skew | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_add",
"lie_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_algebra_self_module : lie_module R L L | { smul_lie := λ t x m, by rw [←lie_skew, ←lie_skew x m, lie_algebra.lie_smul, smul_neg],
lie_smul := by apply lie_algebra.lie_smul, } | instance | lie_algebra_self_module | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_module",
"lie_smul",
"smul_lie",
"smul_neg"
] | Every Lie algebra is a module over itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_lie : ⁅-x, m⁆ = -⁅x, m⁆ | by { rw [←sub_eq_zero, sub_neg_eq_add, ←add_lie], simp, } | lemma | neg_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_neg : ⁅x, -m⁆ = -⁅x, m⁆ | by { rw [←sub_eq_zero, sub_neg_eq_add, ←lie_add], simp, } | lemma | lie_neg | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_lie : ⁅x - y, m⁆ = ⁅x, m⁆ - ⁅y, m⁆ | by simp [sub_eq_add_neg] | lemma | sub_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_sub : ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆ | by simp [sub_eq_add_neg] | lemma | lie_sub | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_lie (n : ℕ) : ⁅n • x, m⁆ = n • ⁅x, m⁆ | add_monoid_hom.map_nsmul ⟨λ (x : L), ⁅x, m⁆, zero_lie m, λ _ _, add_lie _ _ _⟩ _ _ | lemma | nsmul_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_lie",
"zero_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_nsmul (n : ℕ) : ⁅x, n • m⁆ = n • ⁅x, m⁆ | add_monoid_hom.map_nsmul ⟨λ (m : M), ⁅x, m⁆, lie_zero x, λ _ _, lie_add _ _ _⟩ _ _ | lemma | lie_nsmul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_add",
"lie_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_lie (a : ℤ) : ⁅a • x, m⁆ = a • ⁅x, m⁆ | add_monoid_hom.map_zsmul ⟨λ (x : L), ⁅x, m⁆, zero_lie m, λ _ _, add_lie _ _ _⟩ _ _ | lemma | zsmul_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_lie",
"zero_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_zsmul (a : ℤ) : ⁅x, a • m⁆ = a • ⁅x, m⁆ | add_monoid_hom.map_zsmul ⟨λ (m : M), ⁅x, m⁆, lie_zero x, λ _ _, lie_add _ _ _⟩ _ _ | lemma | lie_zsmul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_add",
"lie_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_lie : ⁅⁅x, y⁆, m⁆ = ⁅x, ⁅y, m⁆⁆ - ⁅y, ⁅x, m⁆⁆ | by rw [leibniz_lie, add_sub_cancel] | lemma | lie_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"leibniz_lie"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_jacobi : ⁅x, ⁅y, z⁆⁆ + ⁅y, ⁅z, x⁆⁆ + ⁅z, ⁅x, y⁆⁆ = 0 | by { rw [← neg_neg ⁅x, y⁆, lie_neg z, lie_skew y x, ← lie_skew, lie_lie], abel, } | lemma | lie_jacobi | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_lie",
"lie_neg",
"lie_skew"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_ring.int_lie_algebra : lie_algebra ℤ L | { lie_smul := λ n x y, lie_zsmul x y n, } | instance | lie_ring.int_lie_algebra | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_algebra",
"lie_smul",
"lie_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_hom.lie_apply (f : M →ₗ[R] N) (x : L) (m : M) :
⁅x, f⁆ m = ⁅x, f m⁆ - f ⁅x, m⁆ | rfl | lemma | lie_hom.lie_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_hom (R : Type u) (L : Type v) (L' : Type w)
[comm_ring R] [lie_ring L] [lie_algebra R L] [lie_ring L'] [lie_algebra R L']
extends L →ₗ[R] L' | (map_lie' : ∀ {x y : L}, to_fun ⁅x, y⁆ = ⁅to_fun x, to_fun y⁆) | structure | lie_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"comm_ring",
"lie_algebra",
"lie_ring"
] | A morphism of Lie algebras is a linear map respecting the bracket operations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂ | h
initialize_simps_projections lie_hom (to_linear_map_to_fun → apply) | def | lie_hom.simps.apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_hom"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_linear_map (f : L₁ →ₗ⁅R⁆ L₂) : ((f : L₁ →ₗ[R] L₂) : L₁ → L₂) = f | rfl | lemma | lie_hom.coe_to_linear_map | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : L₁ →ₗ⁅R⁆ L₂) : f.to_fun = ⇑f | rfl | lemma | lie_hom.to_fun_eq_coe | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (f : L₁ →ₗ⁅R⁆ L₂) (c : R) (x : L₁) : f (c • x) = c • f x | linear_map.map_smul (f : L₁ →ₗ[R] L₂) c x | lemma | lie_hom.map_smul | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f (x + y) = (f x) + (f y) | linear_map.map_add (f : L₁ →ₗ[R] L₂) x y | lemma | lie_hom.map_add | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f (x - y) = (f x) - (f y) | linear_map.map_sub (f : L₁ →ₗ[R] L₂) x y | lemma | lie_hom.map_sub | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (f : L₁ →ₗ⁅R⁆ L₂) (x : L₁) : f (-x) = -(f x) | linear_map.map_neg (f : L₁ →ₗ[R] L₂) x | lemma | lie_hom.map_neg | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.map_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lie (f : L₁ →ₗ⁅R⁆ L₂) (x y : L₁) : f ⁅x, y⁆ = ⁅f x, f y⁆ | lie_hom.map_lie' f | lemma | lie_hom.map_lie | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero (f : L₁ →ₗ⁅R⁆ L₂) : f 0 = 0 | (f : L₁ →ₗ[R] L₂).map_zero | lemma | lie_hom.map_zero | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : L₁ →ₗ⁅R⁆ L₁ | { map_lie' := λ x y, rfl,
.. (linear_map.id : L₁ →ₗ[R] L₁) } | def | lie_hom.id | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.id"
] | The identity map is a morphism of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ((id : L₁ →ₗ⁅R⁆ L₁) : L₁ → L₁) = _root_.id | rfl | lemma | lie_hom.coe_id | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (x : L₁) : (id : L₁ →ₗ⁅R⁆ L₁) x = x | rfl | lemma | lie_hom.id_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) = 0 | rfl | lemma | lie_hom.coe_zero | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : L₁) : (0 : L₁ →ₗ⁅R⁆ L₂) x = 0 | rfl | lemma | lie_hom.zero_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : (L₁ →ₗ⁅R⁆ L₁)) : L₁ → L₁) = _root_.id | rfl | lemma | lie_hom.coe_one | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : L₁) : (1 : (L₁ →ₗ⁅R⁆ L₁)) x = x | rfl | lemma | lie_hom.one_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (L₁ →ₗ⁅R⁆ L₂) (L₁ → L₂) coe_fn | by rintro ⟨⟨f, _⟩⟩ ⟨⟨g, _⟩⟩ ⟨h⟩; congr | lemma | lie_hom.coe_injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : L₁ →ₗ⁅R⁆ L₂} (h : ∀ x, f x = g x) : f = g | coe_injective $ funext h | lemma | lie_hom.ext | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : L₁ →ₗ⁅R⁆ L₂} : f = g ↔ ∀ x, f x = g x | ⟨by { rintro rfl x, refl }, ext⟩ | lemma | lie_hom.ext_iff | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : L₁ →ₗ⁅R⁆ L₂} (h : f = g) (x : L₁) : f x = g x | h ▸ rfl | lemma | lie_hom.congr_fun | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : L₁ →ₗ⁅R⁆ L₂) (h₁ h₂ h₃) :
(⟨⟨f, h₁, h₂⟩, h₃⟩ : L₁ →ₗ⁅R⁆ L₂) = f | by { ext, refl, } | lemma | lie_hom.mk_coe | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : L₁ → L₂) (h₁ h₂ h₃) :
((⟨⟨f, h₁, h₂⟩, h₃⟩ : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) = f | rfl | lemma | lie_hom.coe_mk | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) : L₁ →ₗ⁅R⁆ L₃ | { map_lie' := λ x y, by { change f (g ⁅x, y⁆) = ⁅f (g x), f (g y)⁆, rw [map_lie, map_lie], },
..linear_map.comp f.to_linear_map g.to_linear_map } | def | lie_hom.comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.comp"
] | The composition of morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) (x : L₁) :
f.comp g x = f (g x) | rfl | lemma | lie_hom.comp_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) :
(f.comp g : L₁ → L₃) = f ∘ g | rfl | lemma | lie_hom.coe_comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map_comp (f : L₂ →ₗ⁅R⁆ L₃) (g : L₁ →ₗ⁅R⁆ L₂) :
(f.comp g : L₁ →ₗ[R] L₃) = (f : L₂ →ₗ[R] L₃).comp (g : L₁ →ₗ[R] L₂) | rfl | lemma | lie_hom.coe_linear_map_comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : L₁ →ₗ⁅R⁆ L₂) : f.comp (id : L₁ →ₗ⁅R⁆ L₁) = f | by { ext, refl, } | lemma | lie_hom.comp_id | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : L₁ →ₗ⁅R⁆ L₂) : (id : L₂ →ₗ⁅R⁆ L₂).comp f = f | by { ext, refl, } | lemma | lie_hom.id_comp | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse (f : L₁ →ₗ⁅R⁆ L₂) (g : L₂ → L₁)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) : L₂ →ₗ⁅R⁆ L₁ | { map_lie' := λ x y,
calc g ⁅x, y⁆ = g ⁅f (g x), f (g y)⁆ : by { conv_lhs { rw [←h₂ x, ←h₂ y], }, }
... = g (f ⁅g x, g y⁆) : by rw map_lie
... = ⁅g x, g y⁆ : (h₁ _),
..linear_map.inverse f.to_linear_map g h₁ h₂ } | def | lie_hom.inverse | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_map.inverse"
] | The inverse of a bijective morphism is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ring_module.comp_lie_hom : lie_ring_module L₁ M | { bracket := λ x m, ⁅f x, m⁆,
lie_add := λ x, lie_add (f x),
add_lie := λ x y m, by simp only [lie_hom.map_add, add_lie],
leibniz_lie := λ x y m, by simp only [lie_lie, sub_add_cancel, lie_hom.map_lie], } | def | lie_ring_module.comp_lie_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"add_lie",
"leibniz_lie",
"lie_add",
"lie_hom.map_add",
"lie_hom.map_lie",
"lie_lie",
"lie_ring_module"
] | A Lie ring module may be pulled back along a morphism of Lie algebras.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_ring_module.comp_lie_hom_apply (x : L₁) (m : M) :
by haveI | lie_ring_module.comp_lie_hom M f; exact
⁅x, m⁆ = ⁅f x, m⁆ :=
rfl | lemma | lie_ring_module.comp_lie_hom_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_ring_module.comp_lie_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lie_module.comp_lie_hom [module R M] [lie_module R L₂ M] :
@lie_module R L₁ M _ _ _ _ _ (lie_ring_module.comp_lie_hom M f) | { smul_lie := λ t x m, by simp only [smul_lie, lie_hom.map_smul],
lie_smul := λ t x m, by simp only [lie_smul], } | def | lie_module.comp_lie_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_hom.map_smul",
"lie_module",
"lie_ring_module.comp_lie_hom",
"lie_smul",
"module",
"smul_lie"
] | A Lie module may be pulled back along a morphism of Lie algebras.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_equiv (R : Type u) (L : Type v) (L' : Type w)
[comm_ring R] [lie_ring L] [lie_algebra R L] [lie_ring L'] [lie_algebra R L']
extends L →ₗ⁅R⁆ L' | (inv_fun : L' → L)
(left_inv : function.left_inverse inv_fun to_lie_hom.to_fun)
(right_inv : function.right_inverse inv_fun to_lie_hom.to_fun) | structure | lie_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"comm_ring",
"inv_fun",
"lie_algebra",
"lie_ring"
] | An equivalence of Lie algebras is a morphism which is also a linear equivalence. We could
instead define an equivalence to be a morphism which is also a (plain) equivalence. However it is
more convenient to define via linear equivalence to get `.to_linear_equiv` for free. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_equiv (f : L₁ ≃ₗ⁅R⁆ L₂) : L₁ ≃ₗ[R] L₂ | { ..f.to_lie_hom, ..f } | def | lie_equiv.to_linear_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | Consider an equivalence of Lie algebras as a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_to_lie_hom : has_coe (L₁ ≃ₗ⁅R⁆ L₂) (L₁ →ₗ⁅R⁆ L₂) | ⟨to_lie_hom⟩ | instance | lie_equiv.has_coe_to_lie_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_linear_equiv : has_coe (L₁ ≃ₗ⁅R⁆ L₂) (L₁ ≃ₗ[R] L₂) | ⟨to_linear_equiv⟩ | instance | lie_equiv.has_coe_to_linear_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_lie_hom (e : L₁ ≃ₗ⁅R⁆ L₂) : ((e : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) = e | rfl | lemma | lie_equiv.coe_to_lie_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_linear_equiv (e : L₁ ≃ₗ⁅R⁆ L₂) :
((e : L₁ ≃ₗ[R] L₂) : L₁ → L₂) = e | rfl | lemma | lie_equiv.coe_to_linear_equiv | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_mk (f : L₁ →ₗ⁅R⁆ L₂) (g h₁ h₂) :
(mk f g h₁ h₂ : L₁ ≃ₗ[R] L₂) = { inv_fun := g, left_inv := h₁, right_inv := h₂, .. f } | rfl | lemma | lie_equiv.to_linear_equiv_mk | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_equiv_injective : injective (coe : (L₁ ≃ₗ⁅R⁆ L₂) → (L₁ ≃ₗ[R] L₂)) | begin
intros f₁ f₂ h, cases f₁, cases f₂, dsimp at h, simp only at h,
congr, exacts [lie_hom.coe_injective h.1, h.2]
end | lemma | lie_equiv.coe_linear_equiv_injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_hom.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @injective (L₁ ≃ₗ⁅R⁆ L₂) (L₁ → L₂) coe_fn | linear_equiv.coe_injective.comp coe_linear_equiv_injective | lemma | lie_equiv.coe_injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : L₁ ≃ₗ⁅R⁆ L₂} (h : ∀ x, f x = g x) : f = g | coe_injective $ funext h | lemma | lie_equiv.ext | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : L₁) : (1 : (L₁ ≃ₗ⁅R⁆ L₁)) x = x | rfl | lemma | lie_equiv.one_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : L₁ ≃ₗ⁅R⁆ L₁ | 1 | def | lie_equiv.refl | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | Lie algebra equivalences are reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_apply (x : L₁) : (refl : L₁ ≃ₗ⁅R⁆ L₁) x = x | rfl | lemma | lie_equiv.refl_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : L₁ ≃ₗ⁅R⁆ L₂) : L₂ ≃ₗ⁅R⁆ L₁ | { ..lie_hom.inverse e.to_lie_hom e.inv_fun e.left_inv e.right_inv,
..e.to_linear_equiv.symm } | def | lie_equiv.symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_hom.inverse"
] | Lie algebra equivalences are symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm (e : L₁ ≃ₗ⁅R⁆ L₂) : e.symm.symm = e | by { ext, refl } | lemma | lie_equiv.symm_symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply (e : L₁ ≃ₗ⁅R⁆ L₂) : ∀ x, e (e.symm x) = x | e.to_linear_equiv.apply_symm_apply | lemma | lie_equiv.apply_symm_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (e : L₁ ≃ₗ⁅R⁆ L₂) : ∀ x, e.symm (e x) = x | e.to_linear_equiv.symm_apply_apply | lemma | lie_equiv.symm_apply_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (refl : L₁ ≃ₗ⁅R⁆ L₁).symm = refl | rfl | theorem | lie_equiv.refl_symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e₁ : L₁ ≃ₗ⁅R⁆ L₂) (e₂ : L₂ ≃ₗ⁅R⁆ L₃) : L₁ ≃ₗ⁅R⁆ L₃ | { ..lie_hom.comp e₂.to_lie_hom e₁.to_lie_hom,
..linear_equiv.trans e₁.to_linear_equiv e₂.to_linear_equiv } | def | lie_equiv.trans | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"lie_hom.comp",
"linear_equiv.trans"
] | Lie algebra equivalences are transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_trans_symm (e : L₁ ≃ₗ⁅R⁆ L₂) : e.trans e.symm = refl | ext e.symm_apply_apply | lemma | lie_equiv.self_trans_symm | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (e : L₁ ≃ₗ⁅R⁆ L₂) : e.symm.trans e = refl | e.symm.self_trans_symm | lemma | lie_equiv.symm_trans_self | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_apply (e₁ : L₁ ≃ₗ⁅R⁆ L₂) (e₂ : L₂ ≃ₗ⁅R⁆ L₃) (x : L₁) :
(e₁.trans e₂) x = e₂ (e₁ x) | rfl | lemma | lie_equiv.trans_apply | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans (e₁ : L₁ ≃ₗ⁅R⁆ L₂) (e₂ : L₂ ≃ₗ⁅R⁆ L₃) :
(e₁.trans e₂).symm = e₂.symm.trans e₁.symm | rfl | lemma | lie_equiv.symm_trans | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective (e : L₁ ≃ₗ⁅R⁆ L₂) : function.bijective ((e : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) | e.to_linear_equiv.bijective | lemma | lie_equiv.bijective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (e : L₁ ≃ₗ⁅R⁆ L₂) : function.injective ((e : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) | e.to_linear_equiv.injective | lemma | lie_equiv.injective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (e : L₁ ≃ₗ⁅R⁆ L₂) : function.surjective ((e : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) | e.to_linear_equiv.surjective | lemma | lie_equiv.surjective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective (f : L₁ →ₗ⁅R⁆ L₂)
(h : function.bijective f) : L₁ ≃ₗ⁅R⁆ L₂ | { to_fun := f,
map_lie' := f.map_lie,
.. (linear_equiv.of_bijective (f : L₁ →ₗ[R] L₂) h), } | def | lie_equiv.of_bijective | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [
"linear_equiv.of_bijective"
] | A bijective morphism of Lie algebras yields an equivalence of Lie algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lie_module_hom extends M →ₗ[R] N | (map_lie' : ∀ {x : L} {m : M}, to_fun ⁅x, m⁆ = ⁅x, to_fun m⁆) | structure | lie_module_hom | algebra.lie | src/algebra/lie/basic.lean | [
"algebra.module.equiv",
"data.bracket",
"linear_algebra.basic",
"tactic.noncomm_ring"
] | [] | A morphism of Lie algebra modules is a linear map which commutes with the action of the Lie
algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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