mathlib3 documentation

linear_algebra.tensor_algebra.basic

Tensor Algebras #

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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. tensor_algebra R M is the tensor algebra itself. It is endowed with an R-algebra structure.
  2. tensor_algebra.ι R is the canonical R-linear map M → tensor_algebra 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 tensor_algebra 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 : tensor_algebra 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 tensor_algebra.rel (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :

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

@[protected, instance]
def tensor_algebra (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
Type (max u_1 u_2)

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

Equations
Instances for tensor_algebra
@[protected, instance]
@[protected, instance]
def tensor_algebra.algebra (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
@[protected, instance]
def tensor_algebra.ring (M : Type u_2) [add_comm_monoid M] {S : Type u_1} [comm_ring S] [module S M] :
Equations
@[irreducible]
def tensor_algebra.ι (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] :

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

Equations
@[irreducible]
def tensor_algebra.lift (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra 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 tensor_algebra R M → A.

Equations
@[simp]
theorem tensor_algebra.ι_comp_lift {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) :
@[simp]
theorem tensor_algebra.lift_ι_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (x : M) :
@[simp]
theorem tensor_algebra.lift_unique {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (g : tensor_algebra R M →ₐ[R] A) :
@[simp]
theorem tensor_algebra.induction {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {C : tensor_algebra R M Prop} (h_grade0 : (r : R), C ((algebra_map R (tensor_algebra R M)) r)) (h_grade1 : (x : M), C ((tensor_algebra.ι R) x)) (h_mul : (a b : tensor_algebra R M), C a C b C (a * b)) (h_add : (a b : tensor_algebra R M), C a C b C (a + b)) (a : tensor_algebra R M) :
C a

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

@[simp]
theorem tensor_algebra.algebra_map_inj {R : Type u_1} [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] (x y : R) :
@[simp]
theorem tensor_algebra.algebra_map_eq_zero_iff {R : Type u_1} [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] (x : R) :
(algebra_map R (tensor_algebra R M)) x = 0 x = 0
@[simp]
theorem tensor_algebra.algebra_map_eq_one_iff {R : Type u_1} [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] (x : R) :
(algebra_map R (tensor_algebra R M)) x = 1 x = 1
def tensor_algebra.ι_inv {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] :

The left-inverse of ι.

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

Equations
@[simp]
theorem tensor_algebra.ι_inj (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] (x y : M) :
@[simp]
theorem tensor_algebra.ι_eq_zero_iff (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] (x : M) :
@[simp]
theorem tensor_algebra.ι_eq_algebra_map_iff {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] (x : M) (r : R) :
@[simp]
theorem tensor_algebra.ι_ne_one {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] [nontrivial R] (x : M) :

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

def tensor_algebra.tprod (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] (n : ) :
multilinear_map R (λ (i : fin n), M) (tensor_algebra R M)

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

See also pi_tensor_product.tprod.

Equations
@[simp]
theorem tensor_algebra.tprod_apply (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] {n : } (x : fin n M) :
(tensor_algebra.tprod R M n) x = (list.of_fn (λ (i : fin n), (tensor_algebra.ι R) (x i))).prod