Automorphisms of free metabelian Leibniz algebras of rank three

Adiyaman, Tuba Taş; Özkurt, Zeynep

doi:10.3906/mat-1903-104

Cited by 7 publications

(3 citation statements)

References 8 publications

(12 reference statements)

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“…Additionally, Abanina and Mishchenko investigated the variety of left nilpotent Leibniz algebras of class 3 defined by the polynomial identity [𝑥1,[𝑥2,[𝑥3,𝑥4]]]=0 [6]. On the relatively free Leibniz algebras, for more details see the works [7][8][9][10][11]. In [12], Drensky and Papistas obtained a generating set of the automorphism group of free nilpotent Leibniz algebras and they show that the fixed points subalgebra is not finitely generated.…”

Section: Introductionmentioning

confidence: 99%

Bases of fixed point subalgebras on nilpotent Leibniz algebras

YAPTI ÖZKURT

2024

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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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.

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Section: Introductionmentioning

confidence: 99%

Bases of fixed point subalgebras on nilpotent Leibniz algebras

YAPTI ÖZKURT

2024

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…The kernel of π is called the IA-automorphism group and denoted by IAut(M). In [17,18], the author and Taş Adıyaman described a generating set for IAut(M) of rank three and n, respectively. Recently, symmetric 148 Z. ÖZKURT polynomials of M were considered in [7].…”

Section: Introductionmentioning

confidence: 99%

Normal automorphisms of free metabelian Leibniz algebras

YAPTI ÖZKURT

2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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Let $\mathfrak{M}$ be a free metabelian Leibniz algebra generating set $% X=\{x_{1},...,x_{n}\}$ over the field $K$ 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.

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“…Leibniz algebras are related with many branches of mathematics. See the papers [1,[11][12][13]16] for more details.…”

Section: Introductionmentioning

confidence: 99%

Symmetric polynomials in Leibniz algebras and their inner automorphisms

Findik¹,

Özkurt²

2020

Turk J Math

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Let Ln be the free metabelian Leibniz algebra generated by the set Xn = {x1,. .. , xn} over a field K of characteristic zero. This is the free algebra of rank n in the variety of solvable of class 2 Leibniz algebras. We call an element s(Xn) ∈ Ln symmetric if s(x σ(1) ,. .. , x σ(n)) = s(x1,. .. , xn) for each permutation σ of {1,. .. , n}. The set L Sn n of symmetric polynomials of Ln is the algebra of invariants of the symmetric group Sn. Let K[Xn] be the usual polynomial algebra with indeterminates from Xn. The description of the algebra K[Xn] Sn is well known, and the algebra (L n) Sn in the commutator ideal L n is a right K[Xn] Sn-module. We give explicit forms of elements of the K[Xn] Sn-module (L n) Sn. Additionally, we determine the description of the group Inn(L Sn n) of inner automorphisms of the algebra L Sn n. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank n .

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Automorphisms of free metabelian Leibniz algebras of rank three

Abstract: In this work, we determine the structure of the automorphism group of the free metabelian Leibniz algebra of rank three over a field K of characteristic zero.

Cited by 7 publications

References 8 publications

Bases of fixed point subalgebras on nilpotent Leibniz algebras

Bases of fixed point subalgebras on nilpotent Leibniz algebras

Normal automorphisms of free metabelian Leibniz algebras

Symmetric polynomials in Leibniz algebras and their inner automorphisms

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