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

Bénéteau, Lucien, and Hashish, M. Abou. "An alternative way to classify some Generalized Elliptic Curves and their isotopic loops." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 237-255. <http://eudml.org/doc/249353>.

@article{Bénéteau2004,
abstract = {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: $(x*x^\{\prime \},y,z)=(x,y,z)*(x^\{\prime \},y,z)$. 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 $\alpha (5,2,\mathbb \{F\}_\{3\})=6$ and $\alpha (6,2,\mathbb \{F\}_\{3\})\ge 13$. We obtained a transfer theorem providing a one-to-one correspondence between the classes from $\operatorname\{AT\}(n,m,\mathbb \{F\}_\{3\})$ 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 $G_\{2\}(\mathbb \{F\}_\{3\})$.},
author = {Bénéteau, Lucien, Hashish, M. Abou},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves; extended triple systems; alternate trilinear mappings; totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves},
language = {eng},
number = {2},
pages = {237-255},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An alternative way to classify some Generalized Elliptic Curves and their isotopic loops},
url = {http://eudml.org/doc/249353},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Bénéteau, Lucien
AU - Hashish, M. Abou
TI - An alternative way to classify some Generalized Elliptic Curves and their isotopic loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 237
EP - 255
AB - 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: $(x*x^{\prime },y,z)=(x,y,z)*(x^{\prime },y,z)$. 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 $\alpha (5,2,\mathbb {F}_{3})=6$ and $\alpha (6,2,\mathbb {F}_{3})\ge 13$. We obtained a transfer theorem providing a one-to-one correspondence between the classes from $\operatorname{AT}(n,m,\mathbb {F}_{3})$ 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 $G_{2}(\mathbb {F}_{3})$.
LA - eng
KW - totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves; extended triple systems; alternate trilinear mappings; totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves
UR - http://eudml.org/doc/249353
ER -