Section I. Geometric Algebra
Section I is concerned with purely algebraic matters. The sequence of papers
on projective geometry, linear algebra and Lie groups make important
improvements and extensions of the concepts and methods in the book
Clifford Algebra to Geometric Calculus (CA to GC). They have many current applications in computer science. The third paper is especially important, as it provides a general framework for computations in linear algebra without matrices. It
thus provides a new framework for Lie groups and their representations.
: The claim that Clifford algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry. Clifford algebra is used to develop a coordinate-free algebraic formulation of projective geometry. Major theorems of projective geometry are reduced to algebraic identities which apply as well to metrical geometry. Improvements in the formulation of linear algebra are suggested to simplify its intimate relation to projective geometry. Relations among Clifford algebras of different dimensions are interpreted geometrically as "projective and conformal splits." The conformal split is employed to simplify and elucidate the pin and spin representations of the conformal group for arbitrary dimension and signature.
: Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to projective geometry and facilitates connections with the rest of mathematics.
: Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations. As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformalgroup is identified.
: The discovery of Mathematical Viruses is announced here for the first time. Such viruses are a serious threat to the general mental health of the mathematical community. Several viruses inimical to the unity of mathematics are identified, and their deleterious characteristics are described. A strong dose of geometric algebra and calculus is the best medicine for both prevention and cure.
: It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can berepresented as a spin group . Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.
