Color algebras and affine connections on S6

Abstract

The classification of algebras having a given Lie group as their automorphism group has recently surged as an important problem in the study of general structure theory of algebras, particle physics, differential geometry, and invariant system theory [2-4, 11, 14, 17-191. Since the automorphism group Aut A of a real or complex algebra A is a Lie group whose Lie algebra is the derivation algebra Der A of A [Zl 1, this is equivalent to the problem of classifying algebras having a given Lie algebra as their derivation algebra. Symmetries that frequently appear in physics are represented by automorphisms of certain nonassociative algebras A, where the largeness of Aut A or Der A reflects the symmetries. Such algebras with the symmetry group X(3, C), SU(3), or the compact exceptional Lie group G2 have extensively been investigated in particle physics, geometry of affrne connections on a reductive homogeneous space, and for the study of ~nite-dimensional real division algebras [Z, 4, 14, 191. The color algebra of Domokos and Kiivesi-Domokos [4], o&onion and para-octonion algebras, and the complex or real Okubo algebra [S] are examples of such algebras. A well-known theorem of F. Adams asserts that if the sphere S” admits