Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar)

Editors: David Towers & Ivan Kaygorodov

1. Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras

Adam Chapman ; Solomon Vishkautsan.
Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when $f(x)$ is linear or monic quadratic, but it is false in general. Similar questions about the connections between $f$ and its companion $C_f(x)=f(x)\cdot \overline{f(x)}$ are studied. Finally, we compute the left eigenvalues of $2\times 2$ octonion matrices.

2. Set-theoretical solutions to the pentagon equation: a survey

Marzia Mazzotta.
This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of solutions and raise some questions.

3. On the free metabelian Novikov and metabelian Lie-admissible algebras

A. Dauletiyarova ; K. Abdukhalikov ; B. K. Sartayev.
In this paper, we consider Lie-admissible algebras, which are free Novikov and free Lie-admissible algebras with an additional metabelian identity. We construct a linear basis for both free metabelian Novikov and free metabelian Lie-admissible algebras. Additionally, we describe a space of symmetric polynomials for both the free metabelian Novikov algebra and the free metabelian Lie-admissible algebra.