The centre of a Clifford algebra

Let Q be a nondegenerate quadratic form on a finite-dimensional vector space over a field of characteristic different from two. In even dimension the Clifford algebra is central, as recorded by CliffordAlgebra.isCentral_of_even_finrank. This file gives the complementary odd-dimensional description: if v₁, …, vₙ is an anisotropic orthogonal basis and ω = v₁ ⋯ vₙ is its volume element, then

  center (CliffordAlgebra Q) = K[ω] = K ⊕ Kω.

The result is stated for any anisotropic orthogonal spanning list of odd length; such a list is a basis (QuadraticMap.nondegenerate_and_length_eq_finrank_of_pairwise_isOrtho), so the form is automatically nondegenerate and the dimension odd. Conversely, every nondegenerate quadratic form on a finite-dimensional space has such a list, by QuadraticMap.Nondegenerate.exists_list_pairwise_isOrtho. Reordering or rescaling the list changes ω by a nonzero scalar, hence leaves the generated subalgebra unchanged. Reading the description as a dimension count, the centre is two-dimensional, and in particular strictly larger than the scalars: an odd-dimensional Clifford algebra is never central over K. That last statement needs no nondegeneracy at all, and holds for an arbitrary quadratic form of odd rank.

Main results

• CliffordAlgebra.exists_central_even_add_odd_of_mem_center: when 2 is invertible, the even and odd components of a central element are central.
• CliffordAlgebra.add_smul_volume_injective: the coordinates in K ⊕ Kω are unique.
• CliffordAlgebra.mem_center_iff_exists_eq_add_smul_volume: a central element is of the form a + b • ω.
• CliffordAlgebra.center_eq_adjoin_volume: in odd dimension the centre is the subalgebra generated by the volume element.
• CliffordAlgebra.finrank_center_eq_two_of_odd_finrank: that centre is two-dimensional.
• CliffordAlgebra.not_isCentral_of_odd_finrank: an odd-dimensional Clifford algebra is not central over K, nondegenerate or not.

References

• C. Chevalley, The Algebraic Theory of Spinors (1954), Chapter II.
• T. Y. Lam, Introduction to Quadratic Forms over Fields (2005), Chapter V, §2.

theorem CliffordAlgebra.exists_central_even_add_odd_of_mem_center {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} [Invertible 2] {x : CliffordAlgebra Q} (hx : x ∈ Subalgebra.center R (CliffordAlgebra Q)) : ∃ e ∈ evenOdd Q 0, ∃ o ∈ evenOdd Q 1, e + o = x ∧ e ∈ Subalgebra.center R (CliffordAlgebra Q) ∧ o ∈ Subalgebra.center R (CliffordAlgebra Q)

The homogeneous components of a central element are central. When 2 is invertible, a central element of a Clifford algebra is the sum of a central element of even degree and a central element of odd degree.

theorem CliffordAlgebra.add_smul_volume_injective {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} [NeZero 2] {l : List V} (hlen : Odd l.length) (hω : (List.map (⇑(ι Q)) l).prod ≠ 0) : Function.Injective fun (p : K × K) => (algebraMap K (CliffordAlgebra Q)) p.1 + p.2 • (List.map (⇑(ι Q)) l).prod

The scalar and volume coordinates a + b • ω are unique whenever the volume element ω of a list of odd length is nonzero. Thus the notation K ⊕ Kω for their span is a genuine direct sum, not merely a sum of subspaces. Neither orthogonality nor anisotropy is needed; for an anisotropic list, ω is a unit by CliffordAlgebra.isUnit_prod_map_ι.

theorem CliffordAlgebra.mem_center_iff_exists_eq_add_smul_volume {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} [NeZero 2] {l : List V} (hl : List.Pairwise (QuadraticMap.IsOrtho Q) l) (hlen : Odd l.length) (hspan : Submodule.span K {x : V | x ∈ l} = ⊤) (hQl : ∀ v ∈ l, Q v ≠ 0) {x : CliffordAlgebra Q} : x ∈ Subalgebra.center K (CliffordAlgebra Q) ↔ ∃ (a : K) (b : K), x = (algebraMap K (CliffordAlgebra Q)) a + b • (List.map (⇑(ι Q)) l).prod

The centre in odd dimension is K ⊕ Kω. Let l be an anisotropic orthogonal spanning list of odd length and let ω be the ordered product of its Clifford generators. An element is central exactly when it is a scalar plus a scalar multiple of ω.

Such a list is a basis, so the form is nondegenerate and the dimension odd. Conversely, every nondegenerate form on a finite-dimensional space has such a list, by QuadraticMap.Nondegenerate.exists_list_pairwise_isOrtho.

theorem CliffordAlgebra.center_eq_adjoin_volume {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} [NeZero 2] {l : List V} (hl : List.Pairwise (QuadraticMap.IsOrtho Q) l) (hlen : Odd l.length) (hspan : Submodule.span K {x : V | x ∈ l} = ⊤) (hQl : ∀ v ∈ l, Q v ≠ 0) : Subalgebra.center K (CliffordAlgebra Q) = K[(List.map (⇑(ι Q)) l).prod]

The centre of an odd-dimensional Clifford algebra is generated by a volume element. For an anisotropic orthogonal spanning list l of odd length, its ordered Clifford product ω generates the full centre as a K-subalgebra. Equivalently, this subalgebra is the two-dimensional space K ⊕ Kω described by CliffordAlgebra.mem_center_iff_exists_eq_add_smul_volume.

The dimension of the centre

theorem CliffordAlgebra.finrank_center_eq_two_of_odd_finrank {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} [NeZero 2] (hQ : QuadraticMap.Nondegenerate) (hodd : Odd (Module.finrank K V)) : Module.finrank K ↥(Subalgebra.center K (CliffordAlgebra Q)) = 2

The centre of an odd-dimensional Clifford algebra is two-dimensional. For a nondegenerate quadratic form on a space of odd dimension over a field of characteristic different from two, the centre is the plane K ⊕ Kω of CliffordAlgebra.mem_center_iff_exists_eq_add_smul_volume, whose two coordinates are unique by CliffordAlgebra.add_smul_volume_injective.

Which rank-two algebra it is depends on the field. By CliffordAlgebra.center_eq_adjoin_volume it is K[ω] for a volume element whose square is the nonzero scalar d = (-1) ^ (n.choose 2) * Q v₁ ⋯ Q vₙ, so it is K × K when d is a square in K — as for ⟨1⟩ over ℝ, where d = 1 — and the quadratic field extension K[X] ⧸ (X ^ 2 - d) otherwise — as for ⟨1, 1, 1⟩ over ℚ, where d = -1.

theorem CliffordAlgebra.not_isCentral_of_odd_finrank {K : Type u} {V : Type v} [Field K] [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} [NeZero 2] (hodd : Odd (Module.finrank K V)) : ¬Algebra.IsCentral K (CliffordAlgebra Q)

An odd-dimensional Clifford algebra is not central over the base field. The quadratic form is arbitrary; no nondegeneracy is needed.

When Q is nondegenerate the centre is moreover exactly two-dimensional, by CliffordAlgebra.finrank_center_eq_two_of_odd_finrank; for a degenerate form it can be larger, as for the zero form on a space of dimension three, whose Clifford algebra is an exterior algebra with all of its even part central. Either way this is the contrast with the even-dimensional CliffordAlgebra.isCentral_of_even_finrank.