∆-PRIMITIVE ELEMENTS OF FREE METABELIAN LIE ALGEBRAS

Year 2013, Volume: 1 Issue: 2, 51 - 55, 01.12.2013

Dilek Ersalan Zerrinesmerligil

Abstract

Let L2be a free metabelian Lie algebra of rank two. We obtaina characterization of ∆-primitive elements of L2. In particular, we prove thatevery endomorphism Φ of L2is an automorhism if and only if Φ takes a ∆primitive element of L2to another ∆-primitive element. Also we show that ina free metabelian Lie algebra of odd rank there are no ∆-primitive elements

Keywords

Primitive element , Free metabelian Lie algebra

References

• Drensky, V.; Automorphisms of relatively free algebras, Commun. In Algebra 18(1990) No.12, 4323-4351. Fox, R.H.; Free differential calculus I.Derivation in free group ring, Ann.of Math.2(1953)No.57, 547-560.
• Mikhalev, A.A., Zolotykh, A.A.; The rank of element of a free Lie (p)superalgebra, Dok- lady Akad. Nauk 334(1994), No.6, 690-693; Translation in Russian Acad. Sci. Dokl. Math. 49(1994), no.1,189-193.
• Mikhalev, A.A., Zolotykh, A.A.; Rank and primitivity of elements of free color Lie (p)superalgebras, Int.J. of Algebra and Comp., 4(1994) No.4, 617-655.
• Mikhalev, A.A., Yu, J.T.; Primitive, almost primitive, test and ∆-primitive elements of free algebras with the Nielsen-Schreier Property, J. of Algebra, 228(2000),603-623.
• Shmelkin, A.L., Wreath products of Lie algebras and their application in the theory of groups(Russian), Trudy Moskov. Math. Obshch., 29(1973) 247-260.Tanslation:Trans. Moscow Math. Soc. 29(1973), 239-252.
• Shpilrain, V., On generators of L/R2Lie algebras, Proc. Amer. Math. Soc. 119(1993), No.4, 1039-1043.
• Shpilrain, V., Yu, J.T.; On generators of polynomial algebras in two commuting or non- commuting variables, J. Pure Appl. Algebra 132(1998), 309-315.
• Shpilrain, V.; Generalized primitive elements of a free group, Arch. Math. 71(1998), 270-278.
• Umirbaev, U., U., Primitive elements of free groups, Russian Math. Surveys 49(1994), No.2, 184-1
• Yunus, I.A.; On the Fox problem for Lie algebras, Uspekhi Mat. Nauk 39(1984), 251-252.
• English transl., Russian Math. Surveys 39(1984), no.3, 221-222.
• Department of Mathematics, C¸ ukurova University , Adana-TURKEY E-mail address: [e-mail], [e-mail]