Facts about algebras involving bilinear maps and tensor products #

We move a few basic statements about algebras out of Algebra.Algebra.Basic, in order to avoid importing LinearAlgebra.BilinearMap and LinearAlgebra.TensorProduct unnecessarily.

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def LinearMap.mul (R : Type u_1) (A : Type u_2) [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] :

A →ₗ[R] A →ₗ[R] A

The multiplication in a non-unital non-associative algebra is a bilinear map.

A weaker version of this for semirings exists as AddMonoidHom.mul.

Equations

LinearMap.mul R A = LinearMap.mk₂ R (fun (x1 x2 : A) => x1 * x2) ⋯ ⋯ ⋯ ⋯

Instances For

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noncomputable def LinearMap.mul' (R : Type u_1) (A : Type u_2) [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] :

TensorProduct R A A →ₗ[R] A

The multiplication map on a non-unital algebra, as an R-linear map from A ⊗[R] A to A.

Equations

LinearMap.mul' R A = TensorProduct.lift (LinearMap.mul R A)

Instances For

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def LinearMap.mulLeft (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A) :

A →ₗ[R] A

The multiplication on the left in a non-unital algebra is a linear map.

Equations

LinearMap.mulLeft R a = (LinearMap.mul R A) a

Instances For

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def LinearMap.mulRight (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A) :

A →ₗ[R] A

The multiplication on the right in an algebra is a linear map.

Equations

LinearMap.mulRight R a = (LinearMap.mul R A).flip a

Instances For

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def LinearMap.mulLeftRight (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (ab : A × A) :

A →ₗ[R] A

Simultaneous multiplication on the left and right is a linear map.

Equations

LinearMap.mulLeftRight R ab = LinearMap.mulRight R ab.2 ∘ₗ LinearMap.mulLeft R ab.1

Instances For

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@[simp]

theorem LinearMap.mulLeft_toAddMonoidHom (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A) :

↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft a

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@[simp]

theorem LinearMap.mulRight_toAddMonoidHom (R : Type u_1) {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A) :

↑(LinearMap.mulRight R a) = AddMonoidHom.mulRight a

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@[simp]

theorem LinearMap.mul_apply' {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A) :

((LinearMap.mul R A) a) b = a * b

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@[simp]

theorem LinearMap.mulLeft_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A) :

(LinearMap.mulLeft R a) b = a * b

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@[simp]

theorem LinearMap.mulRight_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A) :

(LinearMap.mulRight R a) b = b * a

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@[simp]

theorem LinearMap.mulLeftRight_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b x : A) :

(LinearMap.mulLeftRight R (a, b)) x = a * x * b

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@[simp]

theorem LinearMap.mul'_apply {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {a b : A} :

(LinearMap.mul' R A) (a ⊗ₜ[R] b) = a * b

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@[simp]

theorem LinearMap.mulLeft_zero_eq_zero {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] :

LinearMap.mulLeft R 0 = 0

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@[simp]

theorem LinearMap.mulRight_zero_eq_zero {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] :

LinearMap.mulRight R 0 = 0

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def NonUnitalAlgHom.lmul (R : Type u_1) (A : Type u_2) [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] :

A →ₙₐ[R] Module.End R A

The multiplication in a non-unital algebra is a bilinear map.

A weaker version of this for non-unital non-associative algebras exists as LinearMap.mul.

Equations

NonUnitalAlgHom.lmul R A = { toFun := (LinearMap.mul R A).toFun, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯ }

Instances For

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@[simp]

theorem NonUnitalAlgHom.coe_lmul_eq_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] :

⇑(NonUnitalAlgHom.lmul R A) = ⇑(LinearMap.mul R A)

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theorem LinearMap.commute_mulLeft_right {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A) :

Commute (LinearMap.mulLeft R a) (LinearMap.mulRight R b)

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@[simp]

theorem LinearMap.mulLeft_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A) :

LinearMap.mulLeft R (a * b) = LinearMap.mulLeft R a ∘ₗ LinearMap.mulLeft R b

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@[simp]

theorem LinearMap.mulRight_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b : A) :

LinearMap.mulRight R (a * b) = LinearMap.mulRight R b ∘ₗ LinearMap.mulRight R a

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theorem LinearMap.map_mul_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) :

(∀ (x y : A), f (x * y) = f x * f y) ↔ (LinearMap.mul R A).compr₂ f = (LinearMap.mul R B ∘ₗ f).compl₂ f

A LinearMap preserves multiplication if pre- and post- composition with LinearMap.mul are equivalent. By converting the statement into an equality of LinearMaps, this lemma allows various specialized ext lemmas about →ₗ[R] to then be applied.

This is the LinearMap version of AddMonoidHom.map_mul_iff.

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def Algebra.lmul (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] :

A →ₐ[R] Module.End R A

The multiplication in an algebra is an algebra homomorphism into the endomorphisms on the algebra.

A weaker version of this for non-unital algebras exists as NonUnitalAlgHom.mul.

Equations

Algebra.lmul R A = { toFun := (LinearMap.mul R A).toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

Instances For

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@[simp]

theorem Algebra.coe_lmul_eq_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] :

⇑(Algebra.lmul R A) = ⇑(LinearMap.mul R A)

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theorem Algebra.lmul_injective {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] :

Function.Injective ⇑(Algebra.lmul R A)

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theorem Algebra.lmul_isUnit_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {x : A} :

IsUnit ((Algebra.lmul R A) x) ↔ IsUnit x

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@[simp]

theorem LinearMap.mulLeft_eq_zero_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) :

LinearMap.mulLeft R a = 0 ↔ a = 0

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@[simp]

theorem LinearMap.mulRight_eq_zero_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) :

LinearMap.mulRight R a = 0 ↔ a = 0

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@[simp]

theorem LinearMap.mulLeft_one {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] :

LinearMap.mulLeft R 1 = LinearMap.id

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@[simp]

theorem LinearMap.mulRight_one {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] :

LinearMap.mulRight R 1 = LinearMap.id

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@[simp]

theorem LinearMap.pow_mulLeft {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) (n : ℕ) :

LinearMap.mulLeft R a ^ n = LinearMap.mulLeft R (a ^ n)

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@[simp]

theorem LinearMap.pow_mulRight {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) (n : ℕ) :

LinearMap.mulRight R a ^ n = LinearMap.mulRight R (a ^ n)

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theorem LinearMap.toSpanSingleton_eq_algebra_linearMap {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] :

LinearMap.toSpanSingleton R A 1 = Algebra.linearMap R A

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@[deprecated mul_right_injective₀]

theorem LinearMap.mulLeft_injective {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] [NoZeroDivisors A] {x : A} (hx : x ≠ 0) :

Function.Injective ⇑(LinearMap.mulLeft R x)

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@[deprecated mul_left_injective₀]

theorem LinearMap.mulRight_injective {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] [NoZeroDivisors A] {x : A} (hx : x ≠ 0) :

Function.Injective ⇑(LinearMap.mulRight R x)

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@[deprecated mul_right_injective₀]

theorem LinearMap.mul_injective {R : Type u_1} {A : Type u_2} [CommSemiring R] [Ring A] [Algebra R A] [NoZeroDivisors A] {x : A} (hx : x ≠ 0) :

Function.Injective ⇑((LinearMap.mul R A) x)

Documentation

Mathlib.Algebra.Algebra.Bilinear

Facts about algebras involving bilinear maps and tensor products #