HNN-extensions of Leibniz algebras
type: Item
instance of: scholarly article
title: HNN-extensions of Leibniz algebras
zbMATH Open document ID: 1473.17009
author: Yanyan Li
published in: Journal of Algebra
publication date: 2019-06-26T00:00:00Z
review text: A \emph{Leibniz algebra} [\textit{J.-L. Loday}, Enseign. Math. (2) 39, No. 3--4, 269--293 (1993; Zbl 0806.55009)] is a vector space \(L\) over a field \(\mathbb{K}\) satisfying the \emph{Leibniz identity}: \[[[x,y],z]=[[x,z],y]+[x,[y,z]], \quad x, y, z \in L.\] A \emph{diassociative algebra} or \emph{dialgebra} [\textit{J.-L. Loday}, C. R. Acad. Sci., Paris, Sér. I 321, No. 2, 141--146 (1995; Zbl 0845.16036)] is a \(\mathbb{K}\)-linear space, equipped with two associative \(\mathbb{K}\)-linear products \(\dashv, \vdash: D\times D\to D\) called respectively left product and right product, such that the products satisfy the following laws: \[\begin{cases}x\vdash( y\vdash z)=(x\dashv y)\vdash z& \\ x\vdash (y\dashv z)=(x\vdash y)\dashv z&\\ x\dashv (y\dashv z)=x\dashv(y\vdash z)&\\ \end{cases}\] for all \(x,y,z\in D,\) Given a Leibniz algebra \(L\), a subalgebra \(A\) of \(L\) and a derivation \(d: A\to L,\) the authors define the HNN-extension corresponding to \(A\) and \(d\) by the presentation \[L^*_d:=\left\langle L,t : d(a)=[a,t], a\in A\right\rangle.\] similarly, given a dialgebra \(D\), a subalgebra \(A\) of \(D\) and a derivation \(d: A\to D,\) the authors define the HNN-extension corresponding to \(A\) and \(d\) as the Leibniz algebra given by the presentation \[D^*_d:=\left\langle D,t ~|~a\dashv t-t\vdash a=d(a), a\in A\right\rangle.\] Leibniz algebras were introduced as D-algebras [\textit{A. Bloh}, ``On a generalization of the concept of Lie algebra'', Dokl. Akad. Nauk SSSR 165, 471--473 (1965); translated in Soviet Math. Dokl. 6, 1450--1452 (1965)], then rediscovered by J. L. Loday when he obtained a new chain complex, called the loday complex, by lifting the Chevalley-Eilenberg boundary map in the exterior module of a Lie algebra to the tensor module. Leibniz algebras are often considered as non commutative Lie algebras, since the Leibniz identity is equivalent to the Jacobi identity when the two-sided ideal \(\{a\in L~|~[a,a]=0\}\) coincides with \(L.\) …
Mathematics Subject Classification ID: 17A32
Mathematics Subject Classification ID: 17A36
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 16S15
zbMATH DE Number: 7073743
zbMATH Keywords: Leibniz algebras
zbMATH Keywords: dialgebras
zbMATH Keywords: HNN-extensions
zbMATH Keywords: Gröbner-Shirshov bases
Wikidata QID: Q123335400
reviewed by: Guy R. Biyogmam
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.05.014
OpenAlex ID: W2947167962
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DOI: 10.1016/J.JALGEBRA.2019.05.014
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description: scientific article; zbMATH DE number 7073743