The dimension of products of n homogeneous components in free lie algebras

Yıl 2019, Cilt: 68 Sayı: 2, 1774 - 1779, 01.08.2019

Nil Mansuroğlu

https://doi.org/10.31801/cfsuasmas.457500

Öz

Let L be a free Lie algebra of finite rank r≥2 over a field F and we let L_{m_{i}} denote the degree m_{i} homogeneous component of L. Ralph Stöhr and Micheal Vaughan-Lee derived formulae for the dimension of the subspaces [L_{m₁},L_{m₂}] for all m₁ and m₂. Then, the author and R. Stöhr obtained formulae for the dimension of the products [L_{m₁},L_{m₂},L_{m₃}] under certain conditions on m₁,m₂,m₃. In this paper, we study on products of n homogeneous components in free Lie algebra and we derive formulae for the dimension of such products.

Anahtar Kelimeler

Free Lie algebras, homogeneous component

Kaynakça

• Magnus W, Karrass A, Solitar D. Combinatorial Group Theory, Presentations of Groups interms of Generators and Relations, New York, NY, USA Dover Publications, 2nd revised ed.,1976.
• Mansuroğlu N, Stöhr R. On the dimension of products of homogeneous subspaces in free Liealgebras. Internat. J. Algebra Comput., (2013), 1(23):205-213.
• Shirshov A.I. Subalgebras of free Lie algebras. Mat. Sb., (1953), 33:441-452, (in Russian).
• Shirshov A.I. Selected works of A. I. Shirshov. Translated from the Russian by Murray Bremnerand Mikhail V. Kotchetov, Edited by Leonid A. Bokut, Victor Latyshev, Ivan Shestakov andEfim Zelmanov, Contemporary Matematicians, Birkhuser Verlag, Basel, viii+242 pp, 2009.
• Stöhr R, Vaughan-Lee M. Products of homogeneous subspaces in free Lie algebras, Internat.J. Algebra Comput., (2009), 5(19):699-703.
• Witt E. Treue Darstellungen Liescher Ringe. J. Reine Angew. Math., (1937), 177:152-160.
• Witt E. Die Unterringe der freien Lieschen Ringe, Math. Z., (1956), 64:195-216.