Classiﬁcation of complex naturally graded quasi-ﬁliform

Zinbiel algebras

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.

1. Introduction

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

Zinbiel algebras.

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).

1

Contemporary Mathematics

Volume 483, 2009

c 2009 American Mathematical Society

1

Zinbiel algebras

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.

1. Introduction

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

Zinbiel algebras.

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).

1

Contemporary Mathematics

Volume 483, 2009

c 2009 American Mathematical Society

1