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On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras | ||
| International Journal of Group Theory | ||
| مقاله 7، دوره 7، شماره 2، شهریور 2018، صفحه 45-50 اصل مقاله (177.49 K) | ||
| نوع مقاله: Ischia Group Theory 2016 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.21481 | ||
| نویسنده | ||
| Nil Mansuroğlu* | ||
| Ahi Evran University | ||
| چکیده | ||
| Let $L$ be a free Lie algebra of rank $r\geq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$, we determine the dimension of $[L_2,L_2,L_1]$. Moreover, by this method, we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$. | ||
| کلیدواژه‌ها | ||
| Free Lie algebra؛ homogeneous and fine homogeneous components؛ free centre-by-metabelian Lie algebra؛ second derived ideal | ||
| مراجع | ||
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[1] L. G. Kovács and R. Stöhr, Free centre-by-metabelian Lie algebras in characteristic 2, Bull. London math. soc., 46 (2014) 491–502.
[2] Yu. V. Kuz’min, Free centre-by-metabelian groups. Lie algebras and D-groups, Math. USSR Izv., 11 (1997) 1–30.
[3] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Presentations of Groups in terms of Generators and Relations, 2nd revised ed. Dover Publications, Inc. New York, 1976.
[4] N. Mansuro˘glu, Products of homogeneous subspaces in free Lie algebra, MSc thesis, University of Manchester, 2010.
[5] N. Mansuro˘glu, structure of second derived ideal in free centre-by-metabelian Lie rings, PhD thesis, University of Manchester, 2014.
[6] N. Mansuro˘glu, R. Stöhr, On the dimension of products of homogeneous subspaces in free Lie algebras, Internat. J. Algebra Comput., 23 (2013) 205–213.
[7] N. Mansuro˘ glu, R. Stöhr, Free centre-by-metabelian Lie rings, Quart. J. Math., (2013) 1–25. [8] R. Stöhr and M. Vaughan-Lee, Products of homogeneous subspaces in free Lie algebras, Internat. J. Algebra Comput., 19 (2009) 699–703. [9] E. Witt, Treue Darstellungen Liescher Ringe, J. Reine Angew. Math., 177 (1937) 152–160. | ||
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آمار تعداد مشاهده مقاله: 669 تعداد دریافت فایل اصل مقاله: 522 |
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