Transcription of Math 210C. Clifford algebras and spin groups
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Math 210C. Clifford algebras and spin groups
Math 210C. Clifford algebras and spin groupsClifford algebras were discovered by Clifford in the late 19th century as part of his searchfor generalizations of quaternions. He considered an algebra generated byV=Rnsubject tothe relationv2= ||v||2for allv V. (Forn= 2 this gives the quaternions viai=e1,j=e2,andk=e1e2.) They were rediscovered by Dirac. In this handout we explain some generalfeatures of Clifford algebras beyond the setting ofRn, including its role in the definition ofspin groups . This may be regarded as a supplement to the discussion in in ChapterI of the course text, putting those constructions into a broader context. Our discussion isgenerally self-contained, but we punt to the course text for some spaces and associated orthogonal groupsLetVbe a finite-dimensional nonzero vector space over a fieldkand letq:V kbeaquadratic form; ,q(cv) =c2q(v) for allv Vandc k, and the symmetric functionBq:V V kdefined byBq(v,w) :=q(v+w) q(v) q(w) isk-bilinear.
Clifford algebras For our initial construction of the Cli ord algebra associated to (V;q) we make no non-degeneracy hypothesis; the best properties occur only when (V;q) is non-degenerate, but for the purpose of some early examples the case q= 0 (with V 6= 0!) is worth keeping in mind.
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