Overview
- Convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics
- Included are many developments from the last 15 years, drawn in part from the author's research
- Largely self-contained exposition
- Includes supplementary material: sn.pub/extras
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About this book
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’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–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’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’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–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.
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Keywords
Table of contents (11 chapters)
Authors and Affiliations
About the author
Main areas of research are symplectic geometry, with applications to Lie theory and mathematical physics.
Professor at the University of Toronto since 1998.
Honors include: Fellowship of the Royal Society of Canada (since 2008), Steacie Fellowship (2007), McLean Award (2003), Andre Aisenstadt Prize (2001).
Invited speaker at the 2002 ICM in Beijing.
Bibliographic Information
Book Title: Clifford Algebras and Lie Theory
Authors: Eckhard Meinrenken
DOI: https://doi.org/10.1007/978-3-642-36216-3
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2013
Hardcover ISBN: 978-3-642-36215-6Published: 16 March 2013
Softcover ISBN: 978-3-642-54466-8Published: 07 May 2014
eBook ISBN: 978-3-642-36216-3Published: 28 February 2013
Edition Number: 1
Number of Pages: XX, 321
Topics: Topological Groups, Lie Groups, Associative Rings and Algebras, Mathematical Applications in the Physical Sciences, Differential Geometry, Mathematical Methods in Physics