The Clifford algebra of an even-dimensional regular form is central simple

Let Q be a nondegenerate quadratic form on a finite-dimensional vector space V over a field K of characteristic different from 2. This file proves that when finrank K V is even, the Clifford algebra CliffordAlgebra Q is a central simple K-algebra (Lam, Introduction to Quadratic Forms over Fields, V.2.4; Chevalley, The Algebraic Theory of Spinors, II.2). It is finite-dimensional by CliffordAlgebra.instFinite, so its Brauer class is defined: this is the even half of the Clifford invariant of a regular quadratic form.

Both hypotheses matter. In odd dimension the Clifford algebra is classically not central, its centre being spanned by 1 and the volume element, and the central simple algebra attached to the form is its even subalgebra instead (Lam V.2.5); that case is not treated here. For the zero form on a nonzero space the Clifford algebra is the exterior algebra, which is not simple.

Main results

• CliffordAlgebra.isCentral_of_even_finrank: the Clifford algebra of a nondegenerate form in even dimension is central over K.
• CliffordAlgebra.isSimpleRing_of_even_finrank: it is a simple ring.

References

• T. Y. Lam, Introduction to Quadratic Forms over Fields, GSM 67, AMS (2005), Theorem V.2.4.
• C. Chevalley, The Algebraic Theory of Spinors (1954), Chapter II.

theorem CliffordAlgebra.isCentral_of_even_finrank {K : Type u} [Field K] {V : Type v} [AddCommGroup V] [Module K V] [FiniteDimensional K V] {Q : QuadraticForm K V} [NeZero 2] (hQ : QuadraticMap.Nondegenerate) (heven : Even (Module.finrank K V)) : Algebra.IsCentral K (CliffordAlgebra Q)

The Clifford algebra of a nondegenerate form in even dimension is central. Over a field of characteristic different from 2, if Q is nondegenerate on a finite-dimensional space of even dimension, then the centre of CliffordAlgebra Q is the base field.

theorem CliffordAlgebra.isSimpleRing_of_even_finrank {K : Type u} [Field K] {V : Type v} [AddCommGroup V] [Module K V] [FiniteDimensional K V] {Q : QuadraticForm K V} [NeZero 2] (hQ : QuadraticMap.Nondegenerate) (heven : Even (Module.finrank K V)) : IsSimpleRing (CliffordAlgebra Q)

The Clifford algebra of a nondegenerate form in even dimension is a simple ring. Over a field of characteristic different from 2, if Q is nondegenerate on a finite-dimensional space of even dimension, then CliffordAlgebra Q has no two-sided ideals other than ⊥ and ⊤. Together with CliffordAlgebra.isCentral_of_even_finrank and CliffordAlgebra.instFinite, it is a finite-dimensional central simple K-algebra.