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

Adashev, J. Q.; Khudoyberdiyev, A. Kh.; Omirov, B. A.

J. Gen. Lie Theory Appl. 4, Article ID S090601, 10 p. (2010).

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.

Reviewer: Sh. A. Ayupov (Tashkent)

MSC:

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

Keywords:

Zinbiel algebras; Leibniz algebras; Koszul duality; nilpotent algebras

Citations:

Zbl 0859.17015; Zbl 1063.17002