Symmetric Zinbiel superalgebras. (English) Zbl 1523.17003
The category of Zinbiel algebras was defined by J.-L. Loday [Math. Scand. 77, No. 2, 189–196 (1995; Zbl 0859.17015)]. The Zinbiel operad is the Koszul dual of the Leibniz operad. A Zinbiel algebra is a vector space \(A\) equipped with a binary operation \(\prec\) satisfying the relation \[ (x\prec y)\prec z=x\prec (y\prec z+z\prec y)\] for all \(x,y,z \in A\). The symmetrized operation \(x y := x\prec y+y\prec x\) is commutative and associative. Therefore we get a functor from the category of Zinbiel algebras to the category of commutative algebras. The commutative algebra associated to the free Zinbiel algebra is the shuffle algebra.
In this paper the authors defined and study the notion of symmetric Zinbiel (super)algebras. The authors prove that each symmetric Zinbiel algebra is a 2-step nilpotent or 3-step nilpotent algebra. Moreover, the authors prove a classification theorem for two-generated symmetric Zinbiel algebras and they describe symmetric Zinbiel superalgebras with two odd generators. Next, the authors investigate mono and binary symmetric Zinbiel algebras. It is proved that each quadratic Zinbiel algebra is 2-step nilpotent. In the last part of the paper the authors introduce and study the notion of double extensions of quadratic Zinbiel algebras.
In this paper the authors defined and study the notion of symmetric Zinbiel (super)algebras. The authors prove that each symmetric Zinbiel algebra is a 2-step nilpotent or 3-step nilpotent algebra. Moreover, the authors prove a classification theorem for two-generated symmetric Zinbiel algebras and they describe symmetric Zinbiel superalgebras with two odd generators. Next, the authors investigate mono and binary symmetric Zinbiel algebras. It is proved that each quadratic Zinbiel algebra is 2-step nilpotent. In the last part of the paper the authors introduce and study the notion of double extensions of quadratic Zinbiel algebras.
Reviewer: Ioannis Dokas (Athína)
Citations:
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