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Trilinear alternating forms and related CMLs and GECs | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 12، شماره 4، اسفند 2023، صفحه 227-235 اصل مقاله (445.23 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2022.131611.1760 | ||
| نویسندگان | ||
| Noureddine Midoune* 1؛ Mohamed Anouar Rakdi2 | ||
| 1Department of Mathematics, University of MSILA, P.O.Box 166, Msila, Algeria | ||
| 2Department of Mathematics, University of MSILA, P.O.Box 166 Msila, Algeria | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Commutative moufang loops؛ Generalized elliptic curves؛ Trivectors؛ Classification | ||
| مراجع | ||
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[1] M. Abou Hashish and L. Bénéteau, An alternative way to classify some generalized elliptic curves and their isotopic loops, Comment. Math.Univ.Carolinae, 45 (2004) 237–255. [2] L. Bénéteau and J. Lacaze, Symplectic trilinear form and related designs and quasigroups, Comm. Algebra, 16 (1988) 1035–1051. [3] F. Buekenhout, Generalized elliptic cubic curves, Part 1, Finite Geometries, Dev. Math., 3, Kluwer Acad. Publ., Dordrecht, (2001) 35–48. [4] O. Chein, H. O. Pflugfelder and J. D. H. Smith, Quasigroups and Loops, Theory and Applications, Sigma Series in PureMathematics, 8 (1990). [5] A. M. Cohen and A. G. Helminck, Trilinear alternating forms on a vector space of dimension 7, Comm. Algebra, 16 (1988) 1–25. [6] D. Djokovic, Classification of trivectors of an eight dimensional real vector space, Linear Multilinear Algebr., 13 (1983) 3–39. [7] G. B. Gurevitch, Foundations of the Theory of Algebraic Invariants, P. Noordhoff Ltd., Groningen, the Netherlands, (1964). [8] J. Hora and P. Pudlák, Classification of 8-dimensional trilinear alternating forms over GF(2), Comm. Algebra, 43 (2015) 3459–3471. [9] N. Midoune and L. Noui, Trilinear alternating forms on a vector space of dimension 8 over a finite field, Linear Multilinear Algebra, 61 (2013) 15–21. [10] L. Noui, Transvecteur de rang 8 sur un corps algébriquement clos, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997) 611–614. | ||
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