Things to Know About Clifford Algebras

I collect here some pointers to answers I have found and also that I want to find.

Clifford algebras (CAs) are determined by a real vector space with a quadratic form on that space. In Clifford’s day that quadratic form was required to be positive definite but 20th century physics found it useful to relax that requirement and allow non-degenerate quadratic forms. (Q is degenerate if there is a non-zero vector z such that for all x Q(x+z) = Q(x).) Two n dimensional real vector spaces with quadratic forms of the same signature are isomorphic. Two CAs over such spaces are isomorphic as well. For n≥0 any two n-dimensional vector spaces are isomorphic to each other and thus there is just one CA of for any particular signature up to isomorphism. The map between the two underlying spaces determines the map between the CAs.

The automorphisms of a CA are closely related. In addition to the automorphisms of the base vector space, there are two discrete automorphisms:

• If the determinant of the matrix that transforms the underlying vector space is negative then the topology of the space of automorphism of both V and C are not connected.
• There is also the anti automorphism consisting of negating the odd terms ... (find a better way to describe this.).

Clifford group

reflections