HNN-extensions of Leibniz algebras
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 \(L.\) For this reason, a significant amount of research attempts to extend results on groups and Lie algebras to Leibniz algebras. In this paper, the authors extend certain results of HNN-extensions for groups and Lie algebras to Leibniz algebras. The Higman-Neumann-Neumann extensions (HNN-extensions) were introduced by Graham Higman, B. H. Neumann, and Hanna Neuman in [\textit{G. Higman} et al., J. Lond. Math. Soc. 24, 247--254 (1950; Zbl 0034.30101)] in their study of embeddability of groups. This study has been extended to associative algebras in [\textit{W. Dicks}, J. Algebra 81, 434--487 (1983; Zbl 0513.16023); \textit{A. I. Lichtman} and \textit{M. Shirvani}, Proc. Am. Math. Soc. 125, No. 12, 3501--3508 (1997; Zbl 0897.17004)] and to Lie algebras in [\textit{A. I. Lichtman} and \textit{M. Shirvani}, Proc. Am. Math. Soc. 125, No. 12, 3501--3508 (1997; Zbl 0897.17004); \textit{A. Wasserman}, Isr. J. Math. 106, 79--92 (1998; Zbl 0930.17006)]. In both cases the main results are respectively that any countably generated associative algebra can be embedded into a two-generated associative algebra, and any Lie algebra over a field of at most countable rank can be embedded into a Lie algebra with two generators over the same field. In the case of Lie algebras, the proof employs the Gröbner-Shirshov bases theory (see [\textit{A. I. Shirshov}, Selected works of A. I. Shirshov. Translated by Murray Bremner and Mikhail V. Kotchetov. Edited by Leonid A. Bokut, Victor Latyshev, Ivan Shestakov and Efim Zelmanov. Basel: Birkhäuser (2009; Zbl 1188.01028)]). In this paper the authors construct HNN-extensions of dialgebras and Leibniz algebras, and prove an analogue of these results for Leibniz algebras. More precisely, they prove that every dialgebra embeds inside its HNN-extensions (Theorem 5.1) and every Leibniz algebra embeds into any of its HNN-extensions (Theorem 6.2). Moreover, using HNN-extensions corresponding to anti-derivations, they prove that every Leibniz algebra with at most countable dimension embeds into a two-generator Leibniz algebra (Theorem 7.3).
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