Trilinear alternating forms and related CMLs and GECs
The classification of trivectors(trilinear alternating forms) depends essentially on the dimension $n$ of the base space. This classification seems to be a difficult problem (unlike in the bilinear case). For $n\leq 8 $ there exist finitely many trivector classes under the action of the general line...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Isfahan
2023-12-01
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| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | https://ijgt.ui.ac.ir/article_26353_f99f3afe2da545dfb861114d76e0f418.pdf |
| Summary: | The classification of trivectors(trilinear alternating forms) depends essentially on the dimension $n$ of the base space. This classification seems to be a difficult problem (unlike in the bilinear case). For $n\leq 8 $ there exist finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\bar{K}$(the algebraic closure of $K$) to $K.$ In this paper, we are interested in the classification of trivectors of an eight dimensional vector space over a finite field of characteristic $3,$ $% K=\mathbb{F}_{3^{m}}.$ We obtain a $31$ inequivalent trivectors, $20$ of which are full rank. Having its motivation in the theory of the generalized elliptic curves and commutative moufang loop, this research studies the case of the forms over the 3 elements field. We use a transfer theorem providing a one-to-one correspondence between the classes of trilinear alternating forms of rank $8$ over a finite field with $3$ elements $\mathbb{F}_{3}$ and the rank $9$ class $2$ Hall generalized elliptic curves (GECs) of $3$-order $9$ and commutative moufang loop (CMLs). We derive a classification and explicit descriptions of the $31$ Hall GECs whose rank and $3$-order both equal $9$ and the number of order $3^{9}$-CMLs. |
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| ISSN: | 2251-7650 2251-7669 |