An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Bénéteau, L. ; Hashish, M. Abou

Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 2, pp. 237-255.

Résumé

The Generalized Elliptic Curves $(\operatorname{GECs})$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of ``points'' from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $\operatorname{GEC}$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $\operatorname{GEC}$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(\operatorname{CML})$. If in addition $x^2=x$, we have Hall $\operatorname{GECs}$ and $(Q,*)$ is an exponent $3$ $\operatorname{CML}$. Any finite terentropic $\operatorname{GEC}$ admits a direct decomposition in primary components and only the $3$-component may eventually be non entropic, in which case its order is at least $81$. It turns out that there are fifteen order $81$ terentropic $\operatorname{GECs}$ (including just three non-entropic $\operatorname{GECs}$). In class $2$ $\operatorname{CMLs}$ the associator enjoys some pseudo-linearity: $(x*x',y,z)=(x,y,z)*(x',y,z)$. We are thus led to searching representatives in the set $\operatorname{AT}(n,m,K)$ of image-rank $m$ alternate trilinear mappings from $(V(n,K))^3$ to $V(m,K)$ up to changes of basis in these $K$-vector spaces. Denote by $\alpha (n,m,K)$ the cardinal number of the sets of representatives. We establish that $\alpha (5,2,K)\le 5$ whenever each field-element is quadratic; moreover $\alpha (5,2,\Bbb F_{3})=6$ and $\alpha(6,2,\Bbb F_{3})\geq 13$. We obtained a transfer theorem providing a one-to-one correspondence between the classes from $\operatorname{AT}(n,m,\Bbb F_{3})$ and the rank $n+1$ class $2$ Hall $\operatorname{GECs}$ of $3$-order $n+m$. Now $\alpha (7,1,\operatorname{GF}(3^s))=11$ for any $s$. We derive a complete classification and explicit descriptions of the eleven Hall $\operatorname{GECs}$ whose rank and $3$-order both equal $8$. One of these has for automorphism group some extension of the Chevalley group $G_{2}(\Bbb F_{3})$.

Détail

Classification : 11G07, 14H52, 20N05, 46G25
Keywords: totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves; extended triple systems; alternate trilinear mappings