On the classification of simple antiflexible algebras

Abstract:

In this paper, we begin a classification of simple totally antiflexible algebras (finite dimensional) over splitting fields of characteristic $\ne 2,3$. For such an algebra $A$ let $P$ be the largest associative ideal in ${A^ + }$ and let $N$ be the radical of $P$. We say that $A$ is of type $(m,n)$ if $N$ is nilpotent of class $m$ with $\dim A = n$. Define ${N_i} = {N_{i - 1}} \cdot N,{N_1} = N$, then $A$ is said to be of type $(m,n,{d_1},{d_2}, \cdots ,{d_q})$ if $A$ is of type $(m,n),\dim ({N_i} - {N_{i - 1}}) = {d_i}$ for $1 \leqslant i \leqslant q$ and $\dim ({N_i} - {N_{i + 1}}) = 1$ for $q < i < m$. We then determine all nodal simple totally antiflexible algebras of types $(n,n),(n - k,n,k + 1),(n - 2,n)$ (over fields of characteristic $\ne 2,3$) and of type (3, 6) (over the field of complex numbers). We also give preliminary results for nodal simple totally antiflexible algebras of type $(n - k,n,k,2)$ and of type $(m,n,{d_1}, \cdots ,{d_q})$ in general with $m > 2$ (the case $m = 2$ has been classified by D. J. Rodabaugh).