Clifford Algebra, Geometric Algebra, and Applications by Douglas Lundholm, Lars Svensson
About this book :-
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.
This document provides an abstract and table of contents for a set of lecture notes on Clifford algebra, geometric algebra, and their applications. The notes were originally prepared for an advanced undergraduate/PhD course and introduce Clifford algebra through both a tensor algebra construction called geometric algebra as well as a more general combinatorial approach. A wide range of applications are presented, including vector space geometry, projective geometry, discrete geometry, classification of real and complex geometric algebras, groups, Euclidean/conformal geometry, and representation theory.
Clifford originally introduced the notion nowadays known as Clifford algebra (but which he himself called geometric algebra) as a generalization of the complex numbers to arbitrarily many imaginary units. The conceptual framework for this was laid by Grassmann already in 1844, but it is only in recent times that one has fully begun to appreciate the algebraisation of geometry in general that the constructions of Clifford and Grassmann result in. Among other things, one obtains an algebraic description of geometric operations in vector spaces such as orthogonal complements, intersections, and sums of subspaces, which gives a way of proving geometric theorems that lies closer to the classical synthetic method of proof than for example Descartes's coordinate geometry. This formalism gives in addition a natural language for the formulation of classical physics and mechanics.
Book Detail :-
Title:
Clifford Algebra, Geometric Algebra, and Applications by Douglas Lundholm, Lars Svensson
Publisher:
arXiv
Year:
2009
Pages:
117
Type:
PDF
Language:
English
ISBN-10 #:
N\A
ISBN-13 #:
N\A
License:
Linked Content Owned by Author
Amazon:
Amazon
About Author :-
The author
Douglas Lundholm
is Professor of Mathematics at Department of Mathematics, Stockholm, Sweden. His research interests is Mathematical physics, including: spectral theory of quantum mechanical systems (typically involving super symmetry - such as super membrane matrix models, or exotic particle statistics), Clifford (geometric) algebras and their applications, quantum gravity and quantum geometry.
Book Contents :-
1. Introduction
2. Foundations
3. Vector Space Geometry
4. Discrete Geometry
5. Classification of Real and Complex Geometric Algebras
6. Groups
7. Euclidean and Conformal Geometry
8. Representation Theory
9. Spinors
10. Some Clifford analysis on Rn
11. The Algebra of Minkowski Spacetime
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