Abelian product of free Abelian and free Lie algebras

Abstract

Let $F_{n}$ be a free Lie Algebra of finite rank $n$ and $A$ be a free abelian Lie algebra of finite rank $m\geq 0$ . We investigate the properties of the generating sets and subalgebras of the abelian product $A\ast _{ab}F_{n}$. Moreover these properties are used to solve the membership problem for $A\ast _{ab}F_{n}$.

Keywords

• Generating set,Abelian product,Free Lie Algebra

References

1. Bahturin Yu., Olshanski A., Filtrations and Distortion in infinite dimensional algebras, arxiv:1002.0015v1[math.RA](29 January 2010)62pp.
2. Bokut L.A., Kukin G.P., Algorithmic and Combinatorial Algebra, Kluwer Academic Publ.,Dordrecht, 1994.
3. Delgado J.,Ventura E., Algorithmic problems for free abelian times free group , J. Algebra. (2013)http://dx.doi.org.//101016/J.Algebra.2013.04.033.
4. Ekici N., Ögüslü N.S., Test rank of an abelian product of a free Lie algebra and a free abelian Lie algebra, Proc. Indian Acad. Sci. (Math. Sci.) 121,3,291-300,2011.
5. Gupta C.K., Timoshenko E.I., Test rank of a Soluble product of free abelian groups, Sbornik Math. 199,4,495-510,2008.
6. Gupta C.K., Timoshenko E.I., Criterion for invertibility of endomorphisms and test rank of metabelian product of abelian groups, Alg. Logic,43,5,565-581,2003.
7. Kukin, G.P., Primitive elements of free Lie algebras, Algebrai Logika 9, 4,458-472,1970.