Clifford Algebra - Introduction and Basic Properties

Introduction and Basic Properties

Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. 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 definition of a Clifford algebra endows it with more structure than a "bare" K-algebra: specifically it has a designated or privileged subspace that is isomorphic to 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

where ⟨u, v⟩ = (Q(u + v) − Q(u) − Q(v))/2 is the symmetric bilinear form associated with Q, via the polarization 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.

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char(K) = 2 it is not true that a quadratic form determines a symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed.