Page 612 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 612
ALGEBRA
Real Cayley-Dickson algebras: doubly alternative elements and zero
divisor graphs
Svetlana Zhilina, s.a.zhilina@gmail.com
Lomonosov Moscow State University, Russian Federation
Due to the non-alternativity of Cayley-Dickson algebras with dimension at least 16, there appear
zero divisors which are hard to study and to classify, except for some particular cases. At
present most of the authors restrict their attention to the algebras of the main sequence where
all parameters which determine the Cayley-Dickson construction are assumed to be equal to −1.
We mention in particular the works by Moreno and Biss, Dugger, and Isaksen. Moreno’s key
idea was to study doubly alternative zero divisors, that is, such elements that their components
are both alternative elements of the previous algebra. This notion was then extended to the
Cayley-Dickson split-algebras and led to similar results.
The talk is devoted to zero divisors in arbitrary real Cayley-Dickson algebras whose compo-
nents satisfy some additional conditions on the norm and alternativity. We are interested in the
patterns which they form in orthogonality and zero divisor graphs, and these patterns appear to
be hexagonal. In the case of the algebras of the main sequence, these hexagons can be extended
to the so-called double hexagons. Moreover, the vertices of a double hexagon have a convenient
multiplication table which has a block structure.
610
Real Cayley-Dickson algebras: doubly alternative elements and zero
divisor graphs
Svetlana Zhilina, s.a.zhilina@gmail.com
Lomonosov Moscow State University, Russian Federation
Due to the non-alternativity of Cayley-Dickson algebras with dimension at least 16, there appear
zero divisors which are hard to study and to classify, except for some particular cases. At
present most of the authors restrict their attention to the algebras of the main sequence where
all parameters which determine the Cayley-Dickson construction are assumed to be equal to −1.
We mention in particular the works by Moreno and Biss, Dugger, and Isaksen. Moreno’s key
idea was to study doubly alternative zero divisors, that is, such elements that their components
are both alternative elements of the previous algebra. This notion was then extended to the
Cayley-Dickson split-algebras and led to similar results.
The talk is devoted to zero divisors in arbitrary real Cayley-Dickson algebras whose compo-
nents satisfy some additional conditions on the norm and alternativity. We are interested in the
patterns which they form in orthogonality and zero divisor graphs, and these patterns appear to
be hexagonal. In the case of the algebras of the main sequence, these hexagons can be extended
to the so-called double hexagons. Moreover, the vertices of a double hexagon have a convenient
multiplication table which has a block structure.
610
