Clifford algebra

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras. A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K. The Clifford algebra Cℓ(V, Q) is the 'freest' algebra generated by V subject to the condition where the product on the left is that of the algebra, and the 1 is its multiplicative identity. The idea of being the 'freest' or 'most general' algebra subject to this identity can be formally expressed through the notion of a universal property, as done below. Clifford algebras can be identified by the label Cℓp,q(R), indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −1, and where R indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers. The free algebra generated by V may be written as the tensor algebra ⊕n≥0 V ⊗ ... ⊗ V, that is, the sum of the tensor product of n copies of V over all n, and so a Clifford algebra would be the quotient of this tensor algebra by the two-sided ideal generated by elements of the form v ⊗ v − Q(v)1 for all elements v ∈ V. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. uv). Its associativity follows from the associativity of the tensor product. The Clifford algebra has a distinguished subspace V. Such a subspace cannot in general be uniquely determined given only a K-algebra isomorphic to the Clifford algebra. If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form