Bases of fixed point subalgebras on nilpotent Leibniz algebras

Yıl 2024, Cilt: 26 Sayı: 1, 272 - 278, 19.01.2024

Zeynep Yaptı Özkurt

https://doi.org/10.25092/baunfbed.1332488

Öz

Let K be a field of characteristic zero, X={x_(1,) x_2,…,x_n} and R_m={r_(1,) ,…,r_m} be two sets of variables, F be the free left nitpotent Leibniz algebra generated by X, and K[R_m ] be the commutative polynomial algebra generated by R_m over the base field K. The fixed point subalgebra of an automorphism φ is the subalgebra of F consisting of elements that are invariant under the automorphism. In this work, we consider specific automorphisms of F and determine the fixed point subalgebras of these automorphisms. Then, we find bases of these fixed point subalgebras. In addition, we get generators of these subalgebras as a free K[R_m ] -module.

Anahtar Kelimeler

nilpotent Leibniz algebra, fixed point, automorphism

Nilpotent Leibniz cebirlerinde sabit nokta altcebirlerinin bazları

Öz

K karakteristiği 0 olan bir cisim, X={x_(1,) x_2,…,x_n} ve R_m={r_(1,) ,…,r_m} iki değişkenler kümesi, F, K cismi üzerinde X tarafından üretilen bir serbest sol nilpotent Leibniz cebiri ve K[R_m ], K cismi üzerinde R_m tarafından üretilen komutatif polinomlar cebiri olsunlar. F nin bir φ otomorfizminin sabit nokta altcebiri, F nin bu otomorfizm altında invaryant kalan elemanlarını içeren altcebiridir. Bu çalışmada F nin bazı özel otomorfizmleri ele alınarak bu otomorfizmlerin sabit nokta altcebirleri belirlenmiştir. Sonra, bu sabit nokta altcebirlerinin baz kümeleri elde edilmiştir. Daha sonra bu altcebirlerin serbest K[R_m ]-modülü olarak üreteçleri verilmiştir.

Anahtar Kelimeler

nilpotent Leibniz cebirleri, sabit nokta, otomorfizm

Kaynakça

• Bloh, A., On a generalization of Lie algebra notion, Mathematical in USSR Doklady, 165 (3), 471-473, (1965).
• Loday, J. L., Une version noncommutative des algebres de Lie: les algebres de Leibniz, Enseignement Mathématique 39, 269-293, (1993).
• Loday, J. L., Pirashvili, T., Universal enveloping algebras of Leibniz algebra and (co)Homology, Mathematical Annalen 296, 139-158, (1993).
• Mikhalev, A. A., Umirbaev, U. U., Subalgebras of free Leibniz algebras, Communications in Algebra, 26, 435-446, (1998).
• Drensky, V., Piacentini Cattaneo G. M., Varieties of metabelian Leibniz algebras, Journal of Algebra and its Applications 1, 31-50, (2002).
• Abanina, L. E., Mishchenko, S. P., The variety of Leibniz algebras defined by the identity , Serdica Mathematical Journal 29, 291–300, (2003).
• Agore, A. L., Militaru, G., Itˆo’s theorem and metabelian Leibniz algebras, Linear Multilinear Algebra 63(11), 2187-2199, (2005).
• Özkurt, Z., Orbits and test elements in free Leibniz algebras of rank two, Communications in Algebra 43 (8), 3534-3544, (2015).
• Taş Adıyaman, T., Özkurt, Z., Automorphisms of free metabelian Leibniz algebras of rank three, Turkish Journal of Mathematics 43 (5), 2262-2274, (2019).
• Taş Adıyaman, T., Özkurt, Z., Automorphisms of free metabelian Leibniz algebras, Communications in Algebra 49 (10), 4348-4359, (2021).
• Fındık, Ş., Özkurt, Z., Symmetric polynomials in Leibniz algebras and their inner automorphisms, Turkish Journal of Mathematics 44 (6), 2306-2311, (2020).
• Drensky, V., Papıstas, A. I., Automorphisms of free left nilpotent Leibniz algebras”, Communications in Algebra, 33, 2957-2975, (2005).
• Formanek, E., Noncommutative invariant theory, in group actions on rings, Contemporary Mathematics 43, 87–119, (1985),
• Bryant, R. M., On the fixed points of a finite group acting on a free Lie algebra, Journal of London Mathematical Society 43(2), 215–224, (1991)
• Drensky, V., Fixed algebras of residually nilpotent Lie algebras, Proceedings of American Mathematical Society 120, 1021–1028, (1994).
• Bryant, R. M., Papistas, A.I., On the fixed points of a finite group acting on a relatively free Lie algebra, Glasgow Mathematical Journal 42, 167–181, (2000)
• Ekici, N., Parlak Sönmez D., Fixed points of IA-endomorphisms of a free metabelian Lie Algebra, Proceedings Indian Academy of Science 121(4), 405–416, (2011).