Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules
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Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules José Manuel Casas1
· Hajar Ravanbod2
Received: 6 May 2019 / Revised: 21 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019
Abstract In this article, we investigate the interplay between stem covers, the Schur multiplier of Leibniz crossed modules and the non-abelian exterior product of Leibniz algebras. Explicitly, we obtain a six-term exact sequence associated with a central extension of Leibniz crossed modules, which is useful to characterize stem covers. We show the existence of stem covers and determine the structure of all stem covers of Leibniz crossed modules. Also, we give the connection between the stem cover of a Lie crossed module in the categories of Lie and Leibniz crossed modules, respectively. Keywords Leibniz algebra · Leibniz crossed module · Schur multiplier · Stem cover · Stem extension Mathematics Subject Classification 17A32 · 17B55 · 18G05
1 Introduction Leibniz algebras are algebraic structures introduced by Bloh in [1,2] as a non-skew symmetric generalization of Lie algebras. In the 1990s, Loday rediscovered and developed them [22,23] when he handled periodicity phenomena in algebraic K-theory [24].
Communicated by Peyman Niroomand. First author was supported by Agencia Estatal de Investigación (Spain), Grant MTM2016-79661-P (AEI/FEDER, UE, support included).
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José Manuel Casas [email protected] Hajar Ravanbod [email protected]
1
Dpto. Matemática Aplicada, Universidade de Vigo, E. E. Forestal, Campus Universitario A Xunqueira, 36005 Pontevedra, Spain
2
Faculty of Mathematical Sciences, Shahid Beheshti University, G. C., Tehran, Iran
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J. M. Casas, H. Ravanbod
This structure is not only important by algebraic reasons, but also for its applications in other branches such as geometry or physics (see, for instance, [11,18,20,25]). A Leibniz algebra is a K-vector space q equipped with a bilinear map [−, −] : q × q −→ q satisfying the Leibniz identity [x, [y, z]] = [[x, y], z] − [[x, z], y], for all x, y, z ∈ q. If we assume [x, x] = 0 for all x ∈ q, then q is a Lie algebra. An active line of research consists in the extension of properties from Lie algebras to Leibniz algebras. As an example of these generalizations, stem covers and stem extensions of a Leibniz algebra were studied in [10]; in [19] was extended to Leibniz algebras the notion of non-abelian tensor product of Lie algebras introduced by Ellis in [14]; in [12], authors investigated the interplay between the non-abelian tensor and exterior products of Leibniz algebras with the low-dimensional Leibniz homology of Leibniz algebras. Crossed modules of groups were described for the first time by Whitehead in the late 1940s [32] as an algebraic model for path-connected CW spaces whose homotopy groups are trivial in dimensions greater than 2. Crossed modules of different algebraic objects can be regarded as algebraic structures that generalize simultaneously the notions of normal subobjec
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