On Theta Pairs and Theta Completions for Proper Subalgebras in Leibniz Algebras
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On Theta Pairs and Theta Completions for Proper Subalgebras in Leibniz Algebras Leila Goudarzi1
· Zahra Riyahi2
Received: 1 April 2020 / Revised: 3 August 2020 / Accepted: 21 August 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we investigate the structure of Leibniz algebras, using the concepts of the θ -pair and the θ -completion that are analogous to the corresponding ones in group theory and Lie algebras. In fact, using these concepts for proper subalgebras, we give some characterizations for the solvability and the supersolvability of Leibniz algebras. Keywords θ -pair · θ -completion · Leibniz algebra · Solvable · Supersolvable Mathematics Subject Classification 17A32
1 Introduction The concept of θ -pairs of a maximal subgroup of a finite group was first introduced by Mukherjee and Bhattacharya [18]. This concept was motivated by the concept of the index complex and the completion defined by Deskins in [9,10], and then was generalized to the Lie algebras. It has been shown that this concept is not only of intrinsic interest, but also of significant role in characterizing properties of algebraic structures, such as nilpotency, solvability and supersolvability of groups and Lie algebras. For a deeper discussion of θ -pairs in the group theory, we refer the reader to Ballester-Bolinches and Yaoqing [1], Beidleman and Smith [4], Li and Li [13], Shirong [21], Xiuyun [23] and Yaoqing [24].
Communicated by Malihe Yousofzadeh.
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Leila Goudarzi [email protected]; [email protected] Zahra Riyahi [email protected]
1
Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran
2
Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran
123
Bulletin of the Iranian Mathematical Society
In the theory of Lie algebras, Towers [22] introduced the notions of the completion and the index complex for maximal subalgebras of a Lie algebra and investigated the influence of the maximal subalgebras on the structure of a Lie algebra. Salemkar et al. in [19,20] presented θ -pairs for maximal and proper subalgebras, respectively. They gave some characterizations of nilpotent, solvable and supersolvable Lie algebras and asserted that some theorems proved by Mukherjee and Bhattacharya [18] and Li [14] in group theory also hold for Lie algebras. In continuation, the first author in the joint paper [12] defined the concept of an ideal index for proper subalgebras and generalized the concept of the ideal index from maximal subalgebras to all subalgebras of a finite dimensional Lie algebra. Moreover, she defined the notion of θ -completions for maximal subalgebras, and expressed the connection between some of these concepts in [11]. In the last decades, some authors extended some properties of Lie algebras to Leibniz algebras, which are a non-anti-commutative version of Lie algebras (see [15,16]). In 2015 and 2018, the notions of the θ -pair and the completion for maximal subalgebras and the ideal index for proper subalgebras of a Leibniz algebra
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