Clifford Algebra

Given a vector space with a quadratic form , the Clifford algebra is the associative algebra generated by subject to , the universal algebra in which vectors square to their length. Polarising gives the defining anticommutation

so orthogonal vectors anticommute. It is the freest such algebra, since any linear map into an algebra with factors through , and comparing Clifford algebras by maps looser than these rigid algebra homomorphisms is what measuring coalgebras make precise. Its invertible elements, the versors, implement the geometric transports used downstream, so a single algebra interpolates the classical numeric algebras rather than requiring a separate arithmetic for each.

Example. It interpolates the classical algebras

, , and . The complex numbers and quaternions are the Clifford algebras of small negative-definite forms, with the anticommuting unit vectors.

Property. Geometric product = interior + exterior

As a vector space is -graded into even and odd parts; the product splits as a symmetric metric contraction plus an antisymmetric wedge, . Bivectors under the commutator are the Lie algebra of rotations, and the even part’s units form the Spin group acting on spinors, the route by which Clifford algebra encodes Euclidean geometry algebraically.

Property. Involutions and the scalar product

Three grade-wise sign involutions organise the algebra: the grade involution ( on grade ), the reversion (, reversing the order of vectors in a blade), and the Clifford conjugation . They give a scalar product (the grade-0 part), a norm , and, where is scalar, an inverse . The blade basis is orthogonal under , positive-definite in Euclidean signature.