Mansuroğlu, N. (2018). On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras. International Journal of Group Theory, 7(2), 45-50. doi: 10.22108/ijgt.2017.21481

Nil Mansuroğlu. "On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras". International Journal of Group Theory, 7, 2, 2018, 45-50. doi: 10.22108/ijgt.2017.21481

Mansuroğlu, N. (2018). 'On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras', International Journal of Group Theory, 7(2), pp. 45-50. doi: 10.22108/ijgt.2017.21481

Mansuroğlu, N. On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras. International Journal of Group Theory, 2018; 7(2): 45-50. doi: 10.22108/ijgt.2017.21481

On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras

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 $2$.

[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.