Liu, Jiefeng; Sheng, Yunhe; Wang, Qi On non-abelian extensions of Leibniz algebras. (English) Zbl 1427.17044 Commun. Algebra 46, No. 2, 574-587 (2018). Summary: In this paper, we study non-abelian extensions of Leibniz algebras using two different approaches. First we construct two Leibniz 2-algebras using biderivations of Leibniz algebras and show that under a condition on centers, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. Furthermore, under this condition, non-abelian extensions are classified by homotopy classes of Leibniz 2-algebra morphisms. Then we give a description of non-abelian extensions of Leibniz algebras in terms of Maurer-Cartan elements in a differential graded Lie algebra. Cited in 10 Documents MSC: 17B99 Lie algebras and Lie superalgebras 20K40 Homological and categorical methods for abelian groups 55U15 Chain complexes in algebraic topology Keywords:Leibniz algebras; Leibniz 2-algebras; Maurer-Cartan elements; non-abelian extensions PDFBibTeX XMLCite \textit{J. Liu} et al., Commun. Algebra 46, No. 2, 574--587 (2018; Zbl 1427.17044) Full Text: DOI arXiv References: [1] Alekseevsky, D.; Michor, P. W.; Ruppert, W. [2] Alekseevsky, D.; Michor, P. W.; Ruppert, W., Extensions of super Lie algebras, J. Lie Theory, 15, 1, 125-134 (2005) · Zbl 1098.17014 [3] Ammar, M.; Poncin, N., Coalgebraic approach to the Loday infinity category, stem differential for 2n-ary graded and homotopy algebras, Ann. Inst. Fourier (Grenoble)., 60, 1, 355-387 (2010) · Zbl 1208.53084 [4] Baez, J.; Crans, A. S., Higher-dimensional algebra VI: Lie 2-algebras, Theory and Appl. Categ., 12, 492-528 (2004) · Zbl 1057.17011 [5] Balavoine, D., Contemp. Math., 202, 207-234 (1997), Providence, RI: Amer, Providence, RI · Zbl 0883.17004 [6] Brahic, O., Extensions of Lie brackets, J. Geom. Phys., 60, 2, 352-374 (2010) · Zbl 1207.58018 [7] Casas, J. M., Crossed extensions of Leibniz algebras, Commun. Algebra, 27, 12, 6253-6272 (1999) · Zbl 0992.17002 [8] Casas Mirás, J. M., Central extensions of Leibniz algebras, Extracta Math, 13, 3, 393-397 (1998) · Zbl 1054.17500 [9] Casas, J. M.; Casado, R. F.; García-Martínez, X.; Khmaladze, E. [10] Casas, J. M.; Corral, N., On universal central extensions of Leibniz algebras, Commun. Algebra, 37, 6, 2104-2120 (2009) · Zbl 1196.17004 [11] Casas, J. M.; Faro, E.; Vieites, A. M., Abelian extensions of Leibniz algebras, Commun. Algebra, 27, 6, 2833-2846 (1999) · Zbl 0936.17002 [12] Casas, J. M.; Khmaladze, E.; Ladra, M., Low-dimensional non-abelian Leibniz cohomology, Forum Math., 25, 3, 443-469 (2013) · Zbl 1353.17004 [13] Cigoli, A. S.; Metere, G.; Montoli, A., Obstruction theory in action accessible categories, J. Algebra, 385, 27-46 (2013) · Zbl 1285.18006 [14] Demir, I.; Misra, K. C.; Stitzinger, E., On classification of four-dimensional nilpotent Leibniz algebras, Commun. Algebra, 45, 3, 1012-1018 (2017) · Zbl 1418.17007 [15] Eilenberg, S.; Maclane, S., Cohomology theory in abstract groups. II. Group extensions with non-abelian kernel, Ann. Math., 48, 326-341 (1947) · Zbl 0029.34101 [16] Frégier, Y., Non-abelian cohomology of extensions of Lie algebras as Deligne groupoid, J. Algebra, 398, 243-257 (2014) · Zbl 1367.17015 [17] Fialowski, A.; Mandal, A., About Leibniz algebra deformations of a Lie algebra, J. Math. Phys., 49, 9, 093511, 11 (2008) · Zbl 1152.81434 [18] Fialowski, A.; Penkava, M., Extensions of (super) Lie algebras, Commun. Contemp. Math., 11, 5, 709-737 (2009) · Zbl 1255.17010 [19] Gnedbaye, A. V., A non-Abelian tensor product of Leibniz algebras, Ann. Inst. Fourier, 49, 1149-1177 (1999) · Zbl 0936.17004 [20] Hu, N.; Pei, Y. Liu, A cohomological characterization of Leibniz central extensions of Lie algebras, Proc. Am. Math. Soc., 136, 2, 437-447 (2008) · Zbl 1140.17003 [21] Hochschild, G., Cohomology classes of finite type and finite dimensional kernels for Lie algebras, Am. J. Math., 76, 763-778 (1954) · Zbl 0057.27204 [22] Inassaridze, N.; Khmaladze, E.; Ladra, M., Non-abelian cohomology and extensions of Lie algebras, J. Lie Theory, 18, 413-432 (2008) · Zbl 1179.17019 [23] Khmaladze, E., On non-abelian Leibniz cohomology. Translated from Sovrem. Mat. Prilozh., Vol. 83, 2012, J. Math. Sci. (N. Y.), 195, 4, 481-485 (2013) · Zbl 1345.17003 [24] Livernet, M., Homologie des algébres stables de matrices sur une \(A_∞\)-algébre, C. R. Acad. Sci. Paris S é r. I Math., 329, 2, 113-116 (1999) · Zbl 0968.17001 [25] Loday, J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 39, 2, 269-293 (1993) · Zbl 0806.55009 [26] Loday, J.-L.; Pirashvili, T., Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296, 139-158 (1993) · Zbl 0821.17022 [27] Sheng, Y.; Liu, Z., Leibniz 2-algebras and twisted Courant algebroids, Commun. Algebra, 41, 5, 1929-1953 (2013) · Zbl 1337.17006 [28] Sheng, Y.; Zhu, C., Integration of Lie 2-algebras and their morphisms, Lett. Math. Phys., 102, 2, 223-244 (2012) · Zbl 1335.17008 [29] Uchino, K., Derived brackets and sh Leibniz algebras, J. Pure Appl. Algebra, 215, 1102-1111 (2011) · Zbl 1219.17003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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