# Documentation

The main result is CliffordAlgebra.gradedAlgebra, which says that the clifford algebra is a ℤ₂-graded algebra (or "superalgebra").

def CliffordAlgebra.evenOdd {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (i : ZMod 2) :

The even or odd submodule, defined as the supremum of the even or odd powers of (ι Q).range. evenOdd 0 is the even submodule, and evenOdd 1 is the odd submodule.

Equations
• = ⨆ (j : { n : // n = i }),
Instances For
theorem CliffordAlgebra.one_le_evenOdd_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :
theorem CliffordAlgebra.range_ι_le_evenOdd_one {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :
theorem CliffordAlgebra.ι_mem_evenOdd_one {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (m : M) :
theorem CliffordAlgebra.ι_mul_ι_mem_evenOdd_zero {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (m₁ : M) (m₂ : M) :
m₁ * m₂
theorem CliffordAlgebra.evenOdd_mul_le {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (i : ZMod 2) (j : ZMod 2) :
instance CliffordAlgebra.evenOdd.gradedMonoid {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :
Equations
• =
def CliffordAlgebra.GradedAlgebra.ι {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :
M →ₗ[R] DirectSum (ZMod 2) fun (i : ZMod 2) => ()

A version of CliffordAlgebra.ι that maps directly into the graded structure. This is primarily an auxiliary construction used to provide CliffordAlgebra.gradedAlgebra.

Equations
• One or more equations did not get rendered due to their size.
Instances For
theorem CliffordAlgebra.GradedAlgebra.ι_apply {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (m : M) :
= (DirectSum.of (fun (i : ZMod 2) => ()) 1) m,
theorem CliffordAlgebra.GradedAlgebra.ι_sq_scalar {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (m : M) :
= (algebraMap R (DirectSum (ZMod 2) fun (i : ZMod 2) => ())) (Q m)
theorem CliffordAlgebra.GradedAlgebra.lift_ι_eq {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (i' : ZMod 2) (x' : ()) :
() x' = (DirectSum.of (fun (i : ZMod 2) => ()) i') x'
instance CliffordAlgebra.gradedAlgebra {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :

The clifford algebra is graded by the even and odd parts.

Equations
theorem CliffordAlgebra.iSup_ι_range_eq_top {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :
⨆ (i : ), =
theorem CliffordAlgebra.evenOdd_isCompl {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) :
theorem CliffordAlgebra.evenOdd_induction {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) (n : ZMod 2) {motive : (x : ) → Prop} (range_ι_pow : ∀ (v : ) (h : v ^ n.val), motive v ) (add : ∀ (x y : ) (hx : ) (hy : ), motive x hxmotive y hymotive (x + y) ) (ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : ) (hx : ), motive x hxmotive ( m₁ * m₂ * x) ) (x : ) (hx : ) :
motive x hx

To show a property is true on the even or odd part, it suffices to show it is true on the scalars or vectors (respectively), closed under addition, and under left-multiplication by a pair of vectors.

theorem CliffordAlgebra.even_induction {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) {motive : (x : ) → Prop} (algebraMap : ∀ (r : R), motive (() r) ) (add : ∀ (x y : ) (hx : ) (hy : ), motive x hxmotive y hymotive (x + y) ) (ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : ) (hx : ), motive x hxmotive ( m₁ * m₂ * x) ) (x : ) (hx : ) :
motive x hx

To show a property is true on the even parts, it suffices to show it is true on the scalars, closed under addition, and under left-multiplication by a pair of vectors.

theorem CliffordAlgebra.odd_induction {R : Type u_1} {M : Type u_2} [] [] [Module R M] (Q : ) {P : (x : ) → Prop} (ι : ∀ (v : M), P ( v) ) (add : ∀ (x y : ) (hx : ) (hy : ), P x hxP y hyP (x + y) ) (ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : ) (hx : ), P x hxP ( m₁ * m₂ * x) ) (x : ) (hx : ) :
P x hx

To show a property is true on the odd parts, it suffices to show it is true on the vectors, closed under addition, and under left-multiplication by a pair of vectors.