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 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 Cℓ_2,0(R) and Cℓ_1,1(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 M_n(K) or M(n,K). The direct sum of two such identical algebras will be denoted by M_n²(K) = M_n(K) ⊕ M_n(K).

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