Classification of Clifford algebras

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cℓ2,0(R) and Cℓ1,1(R), which are both isomorphic to the ring of two-by-two matrices over the real numbers. In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cℓ2,0(R) and Cℓ1,1(R), which are both isomorphic to the ring of two-by-two matrices over the real numbers. The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, are not used here. This article uses the (+) sign convention for Clifford multiplication so that for all vectors v ∈ V, where Q is the quadratic form on the vector space V. We will denote the algebra of n×n matrices with entries in the division algebra K by Mn(K) or M(n, K). The direct sum of two such identical algebras will be denoted by Mn(K) ⊕ Mn(K) = Mn2(K), which is isomorphic to Mn(K ⊕ K). Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group.