jum Journal of Universal Mathematics 2618-5660 2618-5660 Gökhan ÇUVALCIOĞLU 10.33773/jum.1428190 Algebra and Number Theory Cebir ve Sayı Teorisi NECESSARY CONDITION FOR $IA$-AUTOMORPHISMS IN LEIBNIZ ALGEBRAS https://orcid.org/0000-0001-9703-3463 Yaptı Özkurt Zeynep ÇUKUROVA ÜNİVERSİTESİ 07 31 2024 7 2 75 84 01 30 2024 07 16 2024 Copyright © 2018, Journal of Universal Mathematics 2018 Journal of Universal Mathematics

Let $F$ be the free Leibniz algebra generated by the set $%X=\{x_{1},...,x_{n}\}$ over the field $K$ of characteristic $0$. consider $R$ as anideal of $F$. This study initially derives an explicit matrix representation for the $IA$-automorphisms of the Leibniz algebra $F/R^{\prime }$. Subsequently, we establish a necessary condition for an $IA$%-endomorphism of $F/R^{\prime }$ to be an $IA$-automorphism. This method is explicitly based on Dieudonn\'{e} determinant.

 Leibniz algebra Automorphisms Dieudonné Determinant Cukurova University BAP Coordination Council FBA-2020-13173 A. T. Abdykhalykov, A. A. Mikhalev, U. U. Umirbaev, Automorphism of two-generated free Leibniz algebras. Commun. Algebra, Vol.29, No.7, pp.2953–2960 (2001). Y. Bahturin, S. Nabiyev, 9Automorphisms and Derivations of Abelian Extensions of Some Lie Algebras, Abh. Math. Sem. Univ. Hamburg, Vol.62, pp.43-57 (1992). Y. Bahturin, V. Shipilrain, , On Lie Algebras with wild Automorphisms, Results Math., Vol.28, pp.209-213 (1995). P.K. Draxl, Skew Fields, London Mathematical Society Lecture Notes Series, Vol.83, Cambridge University Press (1983). V. Drensky, A.I. Papıstas, Automorphisms Of Free Left Nilpotent Leibniz Algebras, Communications in Algebra, Vol.33, pp.2957-2975 (2005). V. Drensky, G . M. Piacentini Cattaneo, Varieties Of Metabelian Leibniz Algebras, J.Algebra Appl., Vol.1, pp.31-50 (2002). J.-L. Loday, T. Pirashvili, Universal Enveloping Algebras of Leibniz algebras and (co)Homology, Math. Ann., Vol.296, pp.139-158 (1993). A. A. Mikhalev, U.U. Umirbaev, Subalgebras of free Leibniz algebras, Commun. Algebra Vol.26, pp.435–446 (1998). V. Shpilrain, Non-commutative Determinants and Automorphisms of Groups, Comm. Algebra, Vol.25, pp.559-574 (1997). A.L. Smel’kin, Wreath products of Lie Algebras and Their Application in the Theory of Groups, Trans. Mosc. Math. Soc., Vol.29, pp.247-260 (1973). T. Oki, Computing valuations of the Dieudonn´e determinants, J. Symbolic Comput. Vol.116, pp.284–323 (2023). Z. Özkurt, Orbits and test elements in free Leibniz algebras of rank two, Commun. Alg., Vol.43, No.8, pp.3534-3544 (2023). Z. Özkurt, Generating sets of F/R′ Leibniz algebras, Georgian Mathematical Journal, Vol.30, No.5, pp.769–773 (2023). Z. Özkurt, N.Ekici, An Application of the Dieudonne Determinant: Detecting non-tame Automorphisms, Journal of Lie Theory, Vol.18, pp.205-214 (2008). T. Ta，S. Adıyaman, Z. Özkurt, Automorphisms of free metabelian Leibniz algebras of rank three, Turkish J. Math., Vol.43, No. 5, pp.2262–2274 (2019). T. Ta，S. Adıyaman, Z. Özkurt, , Automorphisms of free metabelian Leibniz algebras, Comm. Algebra, Vol.49, No.10, pp.4348–4359 (2021).