TANUKI-CD
Research program on Cayley-Dickson algebras
Preprint CC BY · · arXiv:2609.11588v2 · Updated (v2)
A uniform bilinear algorithm for Cayley–Dickson multiplication over unital commutative rings with 2 invertible using O(N log N) ring operations and O(N) auxiliary storage with an MIT-licensed implementation fastCD for the standard real tower.
Automorphisms, derivations, centers, nuclei, and subalgebra structure. How do these structures change with dimension, and which properties persist throughout the Cayley–Dickson tower?
Determinants, characteristic and minimal polynomials, ranks, kernels, and operator spectra. How do the determinants and characteristic polynomials of multiplication operators on An factor, and what do these factorizations reveal about the underlying algebra?
Bilinear complexity, tensor rank, recursive multiplication, and arithmetic operation counts. What is the least number of scalar operations needed to multiply in An, and which algorithms attain or approach this bound?
Zero-divisor families, annihilator dimensions, and the geometry of zero-divisor loci. How can zero divisors in An be classified through their annihilators, and what geometric structures organize these families?
Twisted group algebras, Clifford algebras, exceptional Lie groups, and projective geometry. Which structures associated with the classical division algebras extend to higher Cayley–Dickson algebras, and how do these relationships change along the tower?