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Classifications of some classes of Zinbiel algebras. (English) Zbl 1241.17003

The notion of Zinbiel algebras was introduced by J.-L. Loday [Math. Scand. 77, No. 2, 189–196 (1995; Zbl 0859.17015)] as Koszul dual to the category of Leibniz algebras (the word “Zinbiel” is the reverse of the word “Leibniz”). These algebras are characterized by the identity \([x,[y,z]] = [[x,y],z] - [[x,z],y]\). In the paper [A. S. Dzhumadil’daev and K. M. Tulenbaev, J. Dyn. Control Syst. 11, No. 2, 195–213 (2005; Zbl 1063.17002)] it was proved that every finite dimensional complex Zinbiel algebra is nilpotent, moreover the classification of complex Zinbiel algebras of dimension \(\leq 3\) was obtained. In the present paper the authors give the classification of complex Zinbiel algebras of the dimension 4.

MSC:

17A32 Leibniz algebras
17A60 Structure theory for nonassociative algebras
17A99 General nonassociative rings
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