Clifford algebra, geometric algebra, and applications

Abstract

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.

117 pages, 14 figures; some sections are currently incomplete

References in corpus (4)

• Gravity, Gauge Theories and Geometric Algebra
• Geometric (Clifford) algebra and its applications
• On the Geometry of Supersymmetric Quantum Mechanical Systems
• Notions of determinant, spectrum, and Hermitian conjugation of Clifford algebra elements

Cited by in corpus (8)

• On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension
• Geometric extensions of many-particle Hardy inequalities
• Inverse of multivector: Beyond p+q=5 threshold
• The exterior algebra and central notions in mathematics
• Less is more: Selecting informative and diverse subsets with balancing constraints
• Compositions of invertibility preserving maps for some monoids and their application to Clifford algebras
• Second Order Linear Elliptic Equations and Hodge-Dirac Operators
• Graded sets, graded groups, and Clifford algebras