Math 210C. Clifford algebras and spin groups Clifford algebras were discovered by Clifford in the late 19th century as part of his search

Clifford algebras were discovered by Clifford in the late 19th century as part of his search for generalizations of quaternions. He considered an algebra generated by V = R subject to the relation v = −||v|| for all v ∈ V . (For n = 2 this gives the quaternions via i = e1, j = e2, and k = e1e2.) They were rediscovered by Dirac. In this handout we explain some general features of Clifford algebras beyond the setting of R, including its role in the definition of spin groups. This may be regarded as a supplement to the discussion in 6.1–6.19 in Chapter I of the course text, putting those constructions into a broader context. Our discussion is generally self-contained, but we punt to the course text for some arguments.