Normal automorphisms of free metabelian Leibniz algebras

Year 2024, Volume: 73 Issue: 1, 147 - 152, 16.03.2024

Zeynep Yaptı Özkurt

https://doi.org/10.31801/cfsuasmas.1265768

Abstract

Let $\mathfrak{M}$ be a free metabelian Leibniz algebra generating set $X=\{x_{1},...,x_{n}\}$ over the field $\mathfrak{K}$ of characteristic $0$. An automorphism $ \phi $ of $\mathfrak{M}$ is said to be normal automorphism if each ideal of $\mathfrak{M}$ is invariant under $ \phi $. In this work, it is proven that every normal automorphism of $\mathfrak{M}$ is an IA-automorphism and the group of normal automorphisms coincides with the group of inner automorphisms.

Keywords

Leibniz algebra, normal automorphism, inner automorphism

References

• Abdykhalykov, A. T., Mikhalev, A. A., Umirbaev, U. U., Automorphisms of two-generated free Leibniz algebras, Comm. Algebra, 29(7) (2001), 2953-2960. https://doi.org/10.1081/AGB-4998
• Bloh, A., A generalization of the concept of a Lie algebra, Dokl. Akad. Nauk SSSR, 165(3) (1965), 471-473. English translation: Sov. Math., Dokl., 6 (1965), 1450-1452.
• Drensky, V., Cattaneo, G. M. P., Varieties of metabelian Leibniz algebras, Journal of Algebra and Its Appl., 1(1) (2002), 31-50. https://doi.org/10.1142/S0219498802000033
• Endimioni, G., Normal automorphisms of a free metabelian nilpotent group, Glasgow Math. J., 52 (2010), 169-177. https://doi.org/10.1017/S0017089509990267
• Jarden, M., Ritter, J., Normal automorphisms of absolute Galois groups of p-adic fields, Duke Math. J., 47 (1980). https://doi.org/10.1215/S0012-7094-80-04705-5
• Fındık, S¸., Normal and normally outer automorphisms of free metabelian nilpotent Lie algebras, Serdica Math. J., 36 (2010), 171-210.
• Fındık, S¸., Özkurt, Z., Symmetric polynomials in Leibniz algebras and their inner automorphisms, Turkish Journal of Mathematics, 44 (2020), 2306-2311. https://doi.org/10.3906/mat-2006-44
• Loday, J. L., Une version non commutative des alg`ebres de Lie: les algebres de Leibniz, Enseign. Math., 39 (1993), 269-293.
• Loday, J. L., Pirashvili, T., Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296 (1993), 139-158. https://doi.org/10.1007/BF01445099
• Lubotzky, A., Normal automorphisms of free groups, J. Algebra, 63 (1980), 494. https://doi.org/10.1016/0021-8693(80)90086-1
• Mikhalev, A. A., Umirbaev, U. U., Subalgebras of free Leibniz algebras, Communications in Algebra, 26 (1998), 435-446. https://doi.org/10.1080/00927879808826139
• Öğüşü, N. S¸., Normal automorphisms of the metabelian product of free abelian Lie algebra, Algebra and Discrete Mathematics, 30(2) (2020), 230-234. http://dx.doi.org/10.12958/adm1258
• Özkurt, Z., Orbits and test elements in free Leibniz algebras of rank two, Communications in Algebra, 43(8) (2015), 3534-3544. https://doi.org/10.1080/00927872.2014.982806
• Romankov, V. A., Normal automorphisms of discrete groups, Sib. Mat. Zh., 24(4) (1983), 138-149. English translation: Siberian Math.J., 24 (1983), 604-614.
• Romanovskii, N. S., Normal automorphisms of free solvable pro-p-groups, Algebra Logika, 36(4) (1997), 441-453, English translation: Algebra Log., 36(4) (1997), 257-263. https://doi.org/10.1007/BF02261748
• Shahryari, M., Hall bases for free Leibniz algebras, Bull. Iranian Math. Soc., 45(2) (2019), 617-625. https://doi.org/10.1007/s41980-018-0153-3
• Taş Adiyaman, T., Özkurt, Z., Automorphisms of free metabelian Leibniz algebras of rank three, Turk. J. Math., 43(5) (2019), 2262-2274. https://doi.org/10.3906/mat-1903-104
• Taş, Adiyaman, T., Özkurt, Z., Automorphisms of free metabelian Leibniz algebras. Comm. Algebra, 49(10) (2021), 4348-4359. https://doi.org/10.1080/00927872.2021.1919690
• Timoshenko, E. I., Normal automorphisms of a soluble product of abelian groups, Sib. Mat. Zh., 56(1) (2015), 227-236. English translation: Siberian Math. J., 56(1) (2015), 191-198. https://doi.org/10.1134/S0037446615010188