Large Annihilators in Cayley–Dickson Algebras

Abstract

Cayley–Dickson algebras are nonassociative ℝ-algebras that generalize the well-known algebras ℝ, ℂ, ℍ, and 𝕆. We study zero-divisors in these algebras. In particular, we show that the annihilator of any element of the 2 ⁿ -dimensional Cayley–Dickson algebra has dimension at most 2 ⁿ  − 4n + 4. Moreover, every multiple of 4 between 0 and this upper bound occurs as the dimension of some annihilator. Although a complete description of zero-divisors seems to be out of reach, we can describe precisely the elements whose annihilators have dimension 2 ⁿ  − 4n + 4.

Key Words:

• Cayley–Dickson algebra
• Nonassociative algebra
• Zero-divisor

Mathematics Subject Classification:

• 17A99

ACKNOWLEDGMENTS

The authors would like to thank Dan Christensen for his assistance with several computer calculations.

This research was conducted during the period the first author served as a Clay Mathematics Institute Research Fellow.

Notes

Communicated by E. I. Zelmanov.