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linear_algebra.clifford_algebra

Clifford Algebras

We construct the Clifford algebra of a module M over a commutative ring R, equipped with a quadratic_form Q.

Notation

The Clifford algebra of the R-module M equipped with a quadratic_form Q is denoted as clifford_algebra Q.

Given a linear morphism f : M → A from a semimodule M to another R-algebra A, such that cond : ∀ m, f m * f m = algebra_map _ _ (Q m), there is a (unique) lift of f to an R-algebra morphism, which is denoted clifford_algebra.lift Q f cond.

The canonical linear map M → clifford_algebra Q is denoted clifford_algebra.ι Q.

Theorems

The main theorems proved ensure that clifford_algebra Q satisfies the universal property of the Clifford algebra.

  1. ι_comp_lift is the fact that the composition of ι Q with lift Q f cond agrees with f.
  2. lift_unique ensures the uniqueness of lift Q f cond with respect to 1.

Additionally, when Q = 0 an alg_equiv to the exterior_algebra is provided as as_exterior.

Implementation details

The Clifford algebra of M is constructed as a quotient of the tensor algebra, as follows.

  1. We define a relation clifford_algebra.rel Q on tensor_algebra R M. This is the smallest relation which identifies squares of elements of M with Q m.
  2. The Clifford algebra is the quotient of the tensor algebra by this relation.

This file is almost identical to linear_algebra/exterior_algebra.lean.

inductive clifford_algebra.rel {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) :
tensor_algebra R Mtensor_algebra R M → Prop

rel relates each ι m * ι m, for m : M, with Q m.

The Clifford algebra of M is defined as the quotient modulo this relation.

@[instance]
def clifford_algebra.inhabited {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) :

def clifford_algebra {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) :
Type (max u_1 u_2)

The Clifford algebra of an R-module M equipped with a quadratic_form Q.

Equations
@[instance]
def clifford_algebra.inst {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) :

@[instance]
def clifford_algebra.ring {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) :

def clifford_algebra.ι {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) :

The canonical linear map M →ₗ[R] clifford_algebra Q.

Equations
@[simp]
theorem clifford_algebra.ι_square_scalar {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) (m : M) :

As well as being linear, ι Q squares to the quadratic form

@[simp]
theorem clifford_algebra.comp_ι_square_scalar {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] {Q : quadratic_form R M} {A : Type u_3} [semiring A] [algebra R A] (g : clifford_algebra Q →ₐ[R] A) (m : M) :

def clifford_algebra.lift {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) {A : Type u_3} [semiring A] [algebra R A] :
{f // ∀ (m : M), (f m) * f m = (algebra_map R A) (Q m)} (clifford_algebra Q →ₐ[R] A)

Given a linear map f : M →ₗ[R] A into an R-algebra A, which satisfies the condition: cond : ∀ m : M, f m * f m = Q(m), this is the canonical lift of f to a morphism of R-algebras from clifford_algebra Q to A.

Equations
@[simp]
theorem clifford_algebra.lift_symm_apply {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] (Q : quadratic_form R M) {A : Type u_3} [semiring A] [algebra R A] (F : clifford_algebra Q →ₐ[R] A) :

@[simp]
theorem clifford_algebra.ι_comp_lift {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] {Q : quadratic_form R M} {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), (f m) * f m = (algebra_map R A) (Q m)) :

@[simp]
theorem clifford_algebra.lift_ι_apply {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] {Q : quadratic_form R M} {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), (f m) * f m = (algebra_map R A) (Q m)) (x : M) :

@[simp]
theorem clifford_algebra.lift_unique {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] {Q : quadratic_form R M} {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), (f m) * f m = (algebra_map R A) (Q m)) (g : clifford_algebra Q →ₐ[R] A) :

@[simp]
theorem clifford_algebra.lift_comp_ι {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] {Q : quadratic_form R M} {A : Type u_3} [semiring A] [algebra R A] (g : clifford_algebra Q →ₐ[R] A) :

@[ext]
theorem clifford_algebra.hom_ext {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] {Q : quadratic_form R M} {A : Type u_3} [semiring A] [algebra R A] {f g : clifford_algebra Q →ₐ[R] A} :

def clifford_algebra.as_exterior {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M] [module R M] :

A Clifford algebra with a zero quadratic form is isomorphic to an exterior_algebra

Equations