J.Q. Adashev, A.Kh. Khuhoyberdiyev, and B.A. Omirov
This work is dedicated to the 60-th anniversary of Professor Shestakov I.P.
Abstract. In this work the description up to isomorphism of complex natu-
rally graded quasi-ﬁliform Zinbiel algebras is obtained.
In the present paper we investigate algebras which are Koszul dual to Leibniz
algebras. The Leibniz algebras were introduced in the work [L2] and they present
a ”non commutative” (to be more precise, a ”non antisymmetric”) analogue of Lie
algebras. Many works, including [L2]-[O1], were devoted to the investigation of
cohomological and structural properties of Leibniz algebras. Ginzburg and Kapra-
nov introduced and studied the concept of Koszul dual operads [G]. Following this
concept, it was shown in [L1] that the category of dual algebras to the category of
Leibniz algebras is deﬁned by the identity:
(x ◦ y) ◦ z = x ◦ (y ◦ z) + x ◦ (z ◦ y).
In this paper, dual Leibniz algebras will called Zinbiel algebras (Zinbiel is ob-
tained from Leibniz written in inverse order). Some interesting properties of Zinbiel
algebras were obtained in [A], [D1], and [D2]. In particular, the nilpotency of an
arbitrary ﬁnite-dimensional complex Zinbiel algebra was proved in [D2], and zero-
ﬁliform and ﬁliform Zinbiel algebras were classiﬁed in [A]. The classiﬁcation of
complex Zinbiel algebras up to dimension 4 is obtained in works [D2] and [O2]. In
this work, we present the classiﬁcation of complex naturally graded quasi-ﬁliform
Examples of Zinbiel algebras can be found in [A], [D2] and [L1].
We consider below only complex algebras and, for convenience, we will omit
zero products the algebra’s multiplication table.
1991 Mathematics Subject Classiﬁcation. Primary 17A32.
Key words and phrases. Zinbiel algebra, Leibniz algebra, nilpotency, nul-ﬁliform algebra,
ﬁliform and quasi-ﬁliform algebras.
The third author was supported by DFG project 436 USB 113/10/0-1 project (Germany).
Volume 483, 2009
c 2009 American Mathematical Society