Clifford Algebras and Lie Theory

• Main Author: Meinrenken, Eckhard. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2013.
• Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 58
• Subjects: Mathematics.; Algebra.; Topological Groups.; Global differential geometry.; Mathematical physics.; Topological Groups, Lie Groups.; Associative Rings and Algebras.; Mathematical Applications in the Physical Sciences.; Differential Geometry.; Mathematical Methods in Physics.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-3-642-36216-3

 This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan<U+0019>s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci<U+0019>s proof of the PoincařBirkhoff<U+0013>Witt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo<U+0019>s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant<U+0019>s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his <U+001c>Clifford algebra analogue of the Hopf<U+0013>Koszul<U+0013>Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.