Parafree metabelian Lie algebras which are determined by parafree Lie algebras

Abstract

Let L be a Lie algebra. Denote by δ^{k}(L) the k-th term of the derived series of L and by Δ_{w}(L) the intersection of the ideals I of L such that L/I is nilpotent. We prove that if P is a parafree Lie algebra, then the algebra Q=(P/δ^{k}(P))/Δ_{w}(P/δ^{k}(P)), k≥2 is a parafree solvable Lie algebra. Moreover we show that if Q is not free metabelian, then P is not free solvable for k=2.

Keywords

• Parafree Lie algebras,metabelian Lie algebras,solvable Lie algebras

References

1. Baumslag, G., Groups with the same lower central sequence as a relatively free group I. The groups, Trans. Amer. Math. Soc., 129(1967), 308-321.
2. Baumslag, G., Groups with the same lower central sequence as a relatively free group. II Properties, Trans. Amer. Math. Soc., 142(1969), 507-538.
3. Baumslag, G., Parafree groups, Progress in Math., 248(2005), 1-14.
4. Baumslag, G., and Cleary, S., Parafree one-relator groups, J. Group Theory, 9(2006), 191-201.
5. Baumslag, G., and Cleary, S. And Havas, G., Experimenting with infinite group, Experimental Math., 13(2004), 495-502.
6. Baur, H., Parafreie Liealgebren und homologie, Diss. Eth Nr. 6126, (1978), 60 pp.
7. Baur, H., A note on parafree Lie algebras, Commun. in Alg., 8(1980), No.10 953-960.
8. Ekici, N. and Velioğlu, Z., Unions of Parafree Lie Algebras, Algebra, 2014(2014), Article ID 385397.

9. Ekici, N. and Velioğlu, Z., Direct Limit of Parafree Lie Algebras, Journal of Lie Theory 25(2015), No. 2 477-484.