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.

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

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

Note that this only assumes SMulCommClass R A A, so that it also works for R := Aᵐᵒᵖ.

When A is unital and associative, this is the same as DistribMulAction.toLinearMap R A a

Equations

• LinearMap.mulLeft R a = { toFun := fun (x : A) => a * x, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem LinearMap.mulLeft_apply (R : Type u_1) {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] (a b : A) : (mulLeft R a) b = a * b

@[simp]

theorem LinearMap.mulLeft_toAddMonoidHom (R : Type u_1) {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] (a : A) : ↑(mulLeft R a) = AddMonoidHom.mulLeft a

@[simp]

theorem LinearMap.mulLeft_zero_eq_zero (R : Type u_1) (A : Type u_2) [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] : mulLeft R 0 = 0

def LinearMap.mulRight (R : Type u_1) {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] (b : A) : A →ₗ[R] A

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

Note that this only assumes IsScalarTower R A A, so that it also works for R := A.

When A is unital and associative, this is the same as DistribMulAction.toLinearMap R A (MulOpposite.op b).

Equations

• LinearMap.mulRight R b = { toFun := fun (x : A) => x * b, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem LinearMap.mulRight_apply (R : Type u_1) {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] (a b : A) : (mulRight R a) b = b * a

@[simp]

theorem LinearMap.mulRight_toAddMonoidHom (R : Type u_1) {A : Type u_2} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] (a : A) : ↑(mulRight R a) = AddMonoidHom.mulRight a

@[simp]

theorem LinearMap.mulRight_zero_eq_zero (R : Type u_1) (A : Type u_2) [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] : mulRight R 0 = 0

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) ⋯ ⋯ ⋯ ⋯

@[simp]

theorem LinearMap.mul_apply_apply (R : Type u_1) (A : Type u_2) [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (m x2✝ : A) : ((mul R A) m) x2✝ = m * x2✝

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)

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

@[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) : ((mul R A) a) b = a * b

@[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) : (mulLeftRight R (a, b)) x = a * x * b

@[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} : (mul' R A) (a ⊗ₜ[R] b) = a * b

@[simp]

theorem LinearMap.mulLeft_mul (R : Type u_1) (A : Type u_2) [Semiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] (a b : A) : mulLeft R (a * b) = mulLeft R a ∘ₗ mulLeft R b

@[simp]

theorem LinearMap.mulRight_mul (R : Type u_1) (A : Type u_2) [Semiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] (a b : A) : mulRight R (a * b) = mulRight R b ∘ₗ mulRight R a

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' := ⋯ }

@[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] : ⇑(lmul R A) = ⇑(LinearMap.mul R A)

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 (mulLeft R a) (mulRight R b)

theorem LinearMap.map_mul_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [NonUnitalSemiring A] [NonUnitalSemiring B] [Module R B] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [SMulCommClass R B B] [IsScalarTower R B B] (f : A →ₗ[R] B) : (∀ (x y : A), f (x * y) = f x * f y) ↔ (mul R A).compr₂ f = (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.

@[simp]

theorem LinearMap.mulLeft_inj {R : Type u_1} {A : Type u_2} [Semiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] {a b : A} : mulLeft R a = mulLeft R b ↔ a = b

@[simp]

theorem LinearMap.mulRight_inj {R : Type u_1} {A : Type u_2} [Semiring R] [NonAssocSemiring A] [Module R A] [IsScalarTower R A A] {a b : A} : mulRight R a = mulRight R b ↔ a = b

@[simp]

theorem LinearMap.mulLeft_one (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] : mulLeft R 1 = id

@[simp]

theorem LinearMap.mulLeft_eq_zero_iff (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] (a : A) : mulLeft R a = 0 ↔ a = 0

@[simp]

theorem LinearMap.pow_mulLeft (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] (a : A) (n : ℕ) : mulLeft R a ^ n = mulLeft R (a ^ n)

@[simp]

theorem LinearMap.mulRight_one (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] : mulRight R 1 = id

@[simp]

theorem LinearMap.mulRight_eq_zero_iff (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (a : A) : mulRight R a = 0 ↔ a = 0

@[simp]

theorem LinearMap.pow_mulRight (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (a : A) (n : ℕ) : mulRight R a ^ n = mulRight R (a ^ n)

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.lmul.

Equations

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

@[simp]

theorem Algebra.coe_lmul_eq_mul {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] : ⇑(lmul R A) = ⇑(LinearMap.mul R A)

theorem Algebra.lmul_injective {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] : Function.Injective ⇑(lmul R A)

theorem Algebra.lmul_isUnit_iff {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] {x : A} : IsUnit ((lmul R A) x) ↔ IsUnit x

theorem LinearMap.toSpanSingleton_eq_algebra_linearMap {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] : toSpanSingleton R A 1 = Algebra.linearMap R A