Classification of Clifford Algebras

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 such algebras, 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.