# Tensor Algebras #

Given a commutative semiring R, and an R-module M, we construct the tensor algebra of M. This is the free R-algebra generated (R-linearly) by the module M.

## Notation #

1. TensorAlgebra R M is the tensor algebra itself. It is endowed with an R-algebra structure.
2. TensorAlgebra.ι R is the canonical R-linear map M → TensorAlgebra R M.
3. Given a linear map f : M → A to an R-algebra A, lift R f is the lift of f to an R-algebra morphism TensorAlgebra R M → A.

## Theorems #

1. ι_comp_lift states that the composition (lift R f) ∘ (ι R) is identical to f.
2. lift_unique states that whenever an R-algebra morphism g : TensorAlgebra R M → A is given whose composition with ι R is f, then one has g = lift R f.
3. hom_ext is a variant of lift_unique in the form of an extensionality theorem.
4. lift_comp_ι is a combination of ι_comp_lift and lift_unique. It states that the lift of the composition of an algebra morphism with ι is the algebra morphism itself.

## Implementation details #

As noted above, the tensor algebra of M is constructed as the free R-algebra generated by M, modulo the additional relations making the inclusion of M into an R-linear map.

inductive TensorAlgebra.Rel (R : Type u_1) [] (M : Type u_2) [] [Module R M] :
Prop

An inductively defined relation on Pre R M used to force the initial algebra structure on the associated quotient.

Instances For
def TensorAlgebra (R : Type u_1) [] (M : Type u_2) [] [Module R M] :
Type (max u_1 u_2)

The tensor algebra of the module M over the commutative semiring R.

Equations
Instances For
instance instInhabitedTensorAlgebra (R : Type u_1) [] (M : Type u_2) [] [Module R M] :
Equations
instance instSemiringTensorAlgebra (R : Type u_1) [] (M : Type u_2) [] [Module R M] :
Equations
instance instAlgebra {R : Type u_3} {A : Type u_4} {M : Type u_5} [] [] [] [Algebra R A] [Module R M] [Module A M] [] :
Algebra R ()
Equations
• instAlgebra =
instance instSMulCommClassTensorAlgebraToSMulInstSemiringTensorAlgebraInstAlgebraToSMulInstAlgebra {R : Type u_3} {S : Type u_4} {A : Type u_5} {M : Type u_6} [] [] [] [] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] [] [] :
Equations
• =
instance instIsScalarTowerTensorAlgebraToSMulInstSemiringTensorAlgebraInstAlgebraToSMulInstAlgebra {R : Type u_3} {S : Type u_4} {A : Type u_5} {M : Type u_6} [] [] [] [] [SMul R S] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] [] [] [] :
Equations
• =
instance TensorAlgebra.instRingTensorAlgebraToCommSemiring (M : Type u_2) [] {S : Type u_3} [] [Module S M] :
Ring ()
Equations
theorem TensorAlgebra.ι_def (R : Type u_3) [] {M : Type u_4} [] [Module R M] :
= { toAddHom := { toFun := fun (m : M) => () (), map_add' := }, map_smul' := }
@[irreducible]
def TensorAlgebra.ι (R : Type u_3) [] {M : Type u_4} [] [Module R M] :

The canonical linear map M →ₗ[R] TensorAlgebra R M.

Equations
Instances For
theorem TensorAlgebra.ringQuot_mkAlgHom_freeAlgebra_ι_eq_ι (R : Type u_1) [] {M : Type u_2} [] [Module R M] (m : M) :
() () = () m
@[simp]
theorem TensorAlgebra.lift_symm_apply (R : Type u_1) [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] (F : →ₐ[R] A) :
.symm F =
def TensorAlgebra.lift (R : Type u_1) [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] :
(M →ₗ[R] A) ( →ₐ[R] A)

Given a linear map f : M → A where A is an R-algebra, lift R f is the unique lift of f to a morphism of R-algebras TensorAlgebra R M → A.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem TensorAlgebra.ι_comp_lift {R : Type u_1} [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] (f : M →ₗ[R] A) :
∘ₗ = f
@[simp]
theorem TensorAlgebra.lift_ι_apply {R : Type u_1} [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] (f : M →ₗ[R] A) (x : M) :
( f) (() x) = f x
@[simp]
theorem TensorAlgebra.lift_unique {R : Type u_1} [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] (f : M →ₗ[R] A) (g : →ₐ[R] A) :
= f g = f
@[simp]
theorem TensorAlgebra.lift_comp_ι {R : Type u_1} [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] (g : →ₐ[R] A) :
() = g
theorem TensorAlgebra.hom_ext {R : Type u_1} [] {M : Type u_2} [] [Module R M] {A : Type u_3} [] [Algebra R A] {f : →ₐ[R] A} {g : →ₐ[R] A} (w : ) :
f = g

See note [partially-applied ext lemmas].

theorem TensorAlgebra.induction {R : Type u_1} [] {M : Type u_2} [] [Module R M] {C : Prop} (algebraMap : ∀ (r : R), C (() r)) (ι : ∀ (x : M), C (() x)) (mul : ∀ (a b : ), C aC bC (a * b)) (add : ∀ (a b : ), C aC bC (a + b)) (a : ) :
C a

If C holds for the algebraMap of r : R into TensorAlgebra R M, the ι of x : M, and is preserved under addition and muliplication, then it holds for all of TensorAlgebra R M.

def TensorAlgebra.algebraMapInv {R : Type u_1} [] {M : Type u_2} [] [Module R M] :

The left-inverse of algebraMap.

Equations
• TensorAlgebra.algebraMapInv = 0
Instances For
theorem TensorAlgebra.algebraMap_leftInverse {R : Type u_1} [] (M : Type u_2) [] [Module R M] :
Function.LeftInverse TensorAlgebra.algebraMapInv (algebraMap R ())
@[simp]
theorem TensorAlgebra.algebraMap_inj {R : Type u_1} [] (M : Type u_2) [] [Module R M] (x : R) (y : R) :
(algebraMap R ()) x = (algebraMap R ()) y x = y
@[simp]
theorem TensorAlgebra.algebraMap_eq_zero_iff {R : Type u_1} [] (M : Type u_2) [] [Module R M] (x : R) :
(algebraMap R ()) x = 0 x = 0
@[simp]
theorem TensorAlgebra.algebraMap_eq_one_iff {R : Type u_1} [] (M : Type u_2) [] [Module R M] (x : R) :
(algebraMap R ()) x = 1 x = 1
instance TensorAlgebra.instNontrivialTensorAlgebra {R : Type u_1} [] (M : Type u_2) [] [Module R M] [] :

A TensorAlgebra over a nontrivial semiring is nontrivial.

Equations
• =
def TensorAlgebra.toTrivSqZeroExt {R : Type u_1} [] {M : Type u_2} [] [Module R M] [] [] :

The canonical map from TensorAlgebra R M into TrivSqZeroExt R M that sends TensorAlgebra.ι to TrivSqZeroExt.inr.

Equations
• TensorAlgebra.toTrivSqZeroExt = ()
Instances For
@[simp]
theorem TensorAlgebra.toTrivSqZeroExt_ι {R : Type u_1} [] {M : Type u_2} [] [Module R M] (x : M) [] [] :
TensorAlgebra.toTrivSqZeroExt (() x) =
def TensorAlgebra.ιInv {R : Type u_1} [] {M : Type u_2} [] [Module R M] :

The left-inverse of ι.

As an implementation detail, we implement this using TrivSqZeroExt which has a suitable algebra structure.

Equations
Instances For
theorem TensorAlgebra.ι_leftInverse {R : Type u_1} [] {M : Type u_2} [] [Module R M] :
Function.LeftInverse TensorAlgebra.ιInv ()
@[simp]
theorem TensorAlgebra.ι_inj (R : Type u_1) [] {M : Type u_2} [] [Module R M] (x : M) (y : M) :
() x = () y x = y
@[simp]
theorem TensorAlgebra.ι_eq_zero_iff (R : Type u_1) [] {M : Type u_2} [] [Module R M] (x : M) :
() x = 0 x = 0
@[simp]
theorem TensorAlgebra.ι_eq_algebraMap_iff {R : Type u_1} [] {M : Type u_2} [] [Module R M] (x : M) (r : R) :
() x = (algebraMap R ()) r x = 0 r = 0
@[simp]
theorem TensorAlgebra.ι_ne_one {R : Type u_1} [] {M : Type u_2} [] [Module R M] [] (x : M) :
() x 1
theorem TensorAlgebra.ι_range_disjoint_one {R : Type u_1} [] {M : Type u_2} [] [Module R M] :

The generators of the tensor algebra are disjoint from its scalars.

def TensorAlgebra.tprod (R : Type u_1) [] (M : Type u_2) [] [Module R M] (n : ) :
MultilinearMap R (fun (x : Fin n) => M) ()

Construct a product of n elements of the module within the tensor algebra.

See also PiTensorProduct.tprod.

Equations
Instances For
@[simp]
theorem TensorAlgebra.tprod_apply (R : Type u_1) [] (M : Type u_2) [] [Module R M] {n : } (x : Fin nM) :
() x = List.prod (List.ofFn fun (i : Fin n) => () (x i))
def FreeAlgebra.toTensor {R : Type u_1} [] {M : Type u_2} [] [Module R M] :

The canonical image of the FreeAlgebra in the TensorAlgebra, which maps FreeAlgebra.ι R x to TensorAlgebra.ι R x.

Equations
• FreeAlgebra.toTensor = () ()
Instances For
@[simp]
theorem FreeAlgebra.toTensor_ι {R : Type u_1} [] {M : Type u_2} [] [Module R M] (m : M) :
FreeAlgebra.toTensor () = () m