Clifford Algebra I: Constructing Pin and Spin - Wiliam Gollinger скачать в хорошем качестве

Clifford Algebra I: Constructing Pin and Spin - Wiliam Gollinger

Abstract: We present the basic definitions and properties of the Clifford algebra associated to a quadratic form q on a vector space V. This Z/2-graded algebra acts on its underlying vector space by the (twisted) adjoint representation, and there is a subgroup of units which in particular preserves an embedded copy of V. We identify special subgroups called Pin and Spin, which surject onto O(V,q) and SO(V,q) respectively when we work over nice enough fields, such as the real and complex numbers. We will conclude by focusing specifically on the real case. This talk is meant to be a preliminary introduction in a series on the topic. In subsequent talks (hopefully) we will study the classification and representation theory of Clifford algebras over R and C, and relate the representations to K-Theory via the Atiyah-Bott-Shapiro isomorphism. Since this is ostensibly a Physics seminar one objective will be to elucidate what is meant by "The Spinor Bundle". The "We promise this applies to Physics" seminar series YouTube Playlist:    • The "We promise this applies to Physics" S...