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.
