Classification of a subclass of low-dimensional complex filiform Leibniz algebras

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1 Linear Multilinear lgebra ISSN: (Print) (Online) Journal homepage: Classification of a subclass of low-dimensional complex filiform Leibniz algebras I.S. Rakhimov & S.K. Said Husain To cite this article: I.S. Rakhimov & S.K. Said Husain (20) Classification of a subclass of lowdimensional complex filiform Leibniz algebras, Linear Multilinear lgebra, 59:, 9-54, DOI: 0.080/ To link to this article: Published online: 25 Feb 20. Submit your article to this journal rticle views: 84 View related articles Citing articles: 2 View citing articles Full Terms & Conditions of access use can be found at Download by: [The UC Irvine Libraries] Date: 28 September 207, t: :02

2 Linear Multilinear lgebra Vol. 59, No., March 20, 9 54 Classification of a subclass of low-dimensional complex filiform Leibniz algebras I.S. Rakhimov* ab S.K. Said Husain ab a Institute for Mathematical Research (INSPEM), Laboratory of Theoretical Studies, Universiti Putra Malaysia, 4400, Serdang, Selangor Darul Ehsan, Malaysia; b Faculty of Science, Department of Mathematics, Universiti Putra Malaysia, 4400, Serdang, Selangor Darul Ehsan, Malaysia Communicated by S. Kirkl (Received 5 September 2009; final version received November 2009) We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-lie filiform Leibniz algebras. The isomorphism criteria in terms of invariant functions are given. Keywords: filiform Leibniz algebra; adapted basis; invariant; isomorphism MS Subject Classifications: 72; 7B0(primary); 50(secondary). Introduction Leibniz algebras were introduced by Loday [9]. skew-symmetric Leibniz algebra is a Lie algebra. The main motivation of Loday to introduce this class of algebras was the search of an obstruction to the periodicity in algebraic K-theory. Besides this purely algebraic motivation some relationships with classical geometry, noncommutative geometry physics have been recently discovered. The (co)homology theory, representations related problems of Leibniz algebras were studied by Loday Pirashvili [], Frabetti [6] others. good survey of all these related problems is [0]. The problems related to the group theoretical realizations of Leibniz algebras are studied by Kinyon Weinstein [8] others. Deformation theory of Leibniz algebras related physical applications of it is initiated by Fialowski et al. [5]. This article is devoted to the classification problem of filiform Leibniz algebras. The notion of filiform Leibniz algebra was introduced by yupov Omirov [2]. ccording to yupov Gomez Omirov theorem, the class of all filiform Leibniz algebras is separated in four subclasses which are invariant with respect to the action of the linear group. One of these classes is the class of filiform Lie algebras. In this case there is a classification in small dimensions (Gomez Khakimdjanov) there is a classification of filiform Lie algebras admitting a non-trivial Malcev Torus (Goze Khakimdjanov). Two of the other three classes come out from naturally graded non-lie filiform Leibniz algebras. For this case in [] [4] a method of *Corresponding author. risamiddin@gmail.com ISSN print/issn online ß 20 Taylor & Francis DOI: 0.080/

3 40 I.S. Rakhimov S.K. Said Husain classification based on algebraic invariants was proposed. The paper [] has dealt with the classification of one of these classes in low-dimensions based on results of []. This article is devoted to the classification of the second class in low dimensions. The third class, which comes out from naturally graded filiform Lie algebras, was treated in [2]. Notice that this class contains the class of all filiform Lie algebras. The article is organized as follows. Section 2 is a brief introduction to filiform Leibniz algebras. In Section, the concepts of adapted basis adapted transformation are given. Then we describe an action of the adapted transformations group on a filiform Leibniz algebra [7]. Section 4 contains main results of the article consisting of a complete classification of a subclass of low-dimensional filiform Leibniz algebras. Here, for 5-6-dimensional cases, we give only final results since the proofs in these cases are similar to those in 7-dimensional case (for the last instance we give a complete proof of a generic case in Section 4.. In the discrete orbits cases (Proposition 4.0) we give a base change leading to appropriate canonical representative). 2. Preliminaries Let L be a Leibniz algebra of dimension n. In some basis {e, e 2,..., e n }, the structure of the Leibniz algebra is defined by the structure tensor ¼fij k g, where ½e i, e j Š¼ij ke k: The components ij k satisfy the Leibniz identity: l jk m il l ij m lk þ l ik m lj ¼ 0, i, j, k, m ¼, 2,..., n: When passing to another basis in L, the structure constants are naturally transformed (in accordance with a tensor law). We denote by LB n the set of all possible structure tensors ¼fij k g corresponding to all possible n-dimensional Leibniz algebras (over a fixed field K ). It is clear that LB n can be regarded as an algebraic subset of K n. Consider the natural action of the group GL n (K) onlb n.itis generated by the linear action of GL n (K) on the coordinates with respect to a basis {e, e 2,..., e n } of the Leibniz algebra L. The orbits of the action of GL n (K )onlb n consist of all mutually isomorphic Leibniz algebras. The stationary subgroup of an arbitrary point under this action is naturally identified with the automorphism group ut(l) of the Leibniz algebra L corresponding to this point of the space of Leibniz algebras. The descending central sequence of a Leibniz algebra L is defined as {C i (L)}, i 2 N, where C (L) ¼ L C iþ (L) ¼ [C i (L), L]. Definition 2. Leibniz algebra L is said to be filiform, if dim C i (L) ¼ n i, where n ¼ dim L 2 i n. Let Leib n denote the class of all n-dimensional filiform Leibniz algebras. It is clear that a filiform Leibniz algebra is nilpotent. Let L be a nilpotent Leibniz algebra. Consider L i ¼ C i (L)/C iþ (L), i n, grl ¼ L L 2 L n. Then [L i, L j ] 7 L iþj we obtain the graded algebra grl. Definition 2.2 Leibniz algebra L 0 is said to be naturally graded if L 0 is isomorphic to grl, for some nilpotent Leibniz algebra L.

4 Linear Multilinear lgebra 4 Later on, all algebras are supposed to be over the field of complex numbers C omitted products of basis vectors are supposed to be zero. The following theorem summarizes the results of [2,4]. THEOREM 2. ny complex (n þ )-dimensional naturally graded filiform Leibniz algebra is isomorphic to one of the following pairwise non-isomorphic algebras: NGF ¼ ½e 0, e 0 Š¼e 2, ½e i, e 0 Š¼e iþ, i n, NGF 2 ¼ ½e 0, e 0 Š¼e 2, ½e i, e 0 Š¼e iþ, 2 i n, NGF ¼ ( ½e i, e 0 Š¼ ½e 0, e i Š¼e iþ, i n, ½e i, e n i Š¼ ½e n i, e i Š¼ð Þ iþ e n, i n, 2f0, g for odd n ¼ 0 for even n: It is clear that NGF is a Lie algebra, however neither NGF nor NGF 2 is a Lie algebra. The above theorem means that the natural gradation of a Leibniz algebra may be an algebra from one of NGF i for i ¼, 2,. The following result of [2,7] describes the class of complex filiform Leibniz algebras whose natural gradation is one of NGF i for i ¼, 2,. THEOREM 2.2 ny (n þ )-dimensional complex non-lie filiform Leibniz algebra can be included in one of the following three classes: 8 ½e 0, e 0 Š¼e 2, >< ½e i, e 0 Š¼e iþ, i n, FLeib nþ ¼ ½e 0, e Š¼ e þ 4 e 4 þþ n e n þ e n, >: ½e j, e Š¼ e jþ2 þ 4 e jþ þþ nþ j e n, j n 2,, 4,..., n, 2 C: 8 ½e 0, e 0 Š¼e 2, ½e i, e 0 Š¼e iþ, 2 i n, >< ½e SLeib nþ ¼ 0, e Š¼ e þ 4 e 4 þþ n e n, ½e, e Š¼e n, >: ½e j, e Š¼ e jþ2 þ 4 e jþ þþ nþ j e n, 2 j n 2,, 4,..., n, 2 C: 8 ½e i,e 0 Š¼e iþ, i n, ½e 0,e i Š¼ e iþ, 2 i n, ½e 0,e 0 Š¼b 0,0 e n, ½e 0,e Š¼ e 2 þ b 0, e n, >< ½e,e Š¼b, e n, TLeib nþ ¼ ½e i,e j Š¼a i,j e iþjþ þþa n ðiþjþþ i,j e n þ b i,j e n, i5j n 2, ½e i,e j Š¼ ½e j,e i Š, i5j n, ½e i,e n i Š¼ ½e n i,e i Š¼ð Þ i b i,n i e n, i n, where a k i,j,b i,j 2C, b i,n i ¼ b, whenever i n, >: b2f0,g, for odd n b ¼ 0, for even n: basis, leading to this representation, is said to be adapted.

5 42 I.S. Rakhimov S.K. Said Husain These three classes have no non-trivial intersection they are invariant with respect to the adapted base change. We have denoted the classes as FLeib nþ, SLeib nþ TLeib nþ, respectively. Hence, the classification problem of Leib n has been reduced to the classification problem within each of the subclasses FLeib n, SLeib n TLeib n. Isomorphism criteria, classifications invariants of FLeib n TLeib n have been studied in [,7,2,]. In this article, we focus on the second class of algebras of the above theorem. Elements of SLeib nþ will be denoted by L(, 4,..., n, ), pointing out the dependence of them on parameters, 4,..., n,. The (n þ )-dimensional stard algebra L(0, 0, 0,..., 0) is denoted by G s nþ :. Simplification of base change isomorphism criterion for SLeib ny Here we simplify the action of GL n ( transport of structure ) on SLeib n. ll the results of this section have appeared elsewhere, particularly in [4] [7]. Let L be a Leibniz algebra defined on a vector space V {e 0, e,..., e n }bean adapted basis of L. Definition. basis transformation f 2 GL(V) is said to be adapted for the structure of L, if the basis {f(e 0 ), f(e ),..., f(e n )} is adapted. The closed subgroup of GL(V ), spanned by the adapted transformations, is denoted by GL ad. Definition.2 The following types of basis transformations of SLeib nþ are said to be elementary: 8 f ðe 0 Þ¼e 0 >< f ðe first type ðb, nþ ¼ Þ¼e þ be n, >: f ðe iþ Þ¼½f ðe i Þ, f ðe 0 ÞŠ, 2 i n, f ðe 2 Þ¼½f ðe 0 Þ, f ðe 0 ÞŠ 8 f ðe 0 Þ¼e 0 þ ae k >< f ðe second type ða, kþ ¼ Þ¼e f ðe iþ Þ¼½fðe i Þ, f ðe 0 ÞŠ, 2 i n, 2 k n, >: f ðe 2 Þ¼½fðe 0 Þ, f ðe 0 ÞŠ 8 f ðe 0 Þ¼ae 0 þ be >< third type ða, b, d Þ¼ >: where a, b, d 2 C. PROPOSITION. f ðe Þ¼de bd a e n, ad 6¼ 0 f ðe iþ Þ¼½f ðe i Þ, f ðe 0 ÞŠ, 2 i n, f ðe 2 Þ¼½f ðe 0 Þ, f ðe 0 ÞŠ () n adapted transformation f of SLeib nþ can be represented in the form f ¼ ðb n, nþða n, nþða n, n 2Þða 2,2Þða 0, a, b Þ:

6 Linear Multilinear lgebra 4 (2) The transformations (b, n), (a, n) (a, k), where 2 k n 2, a 2 C preserve the structure constants of algebras from SLeib nþ. Proof The proof is straightforward. g Since a composition of adapted transformations is adapted, the proposition above means that the transformation (b n, n) (a n, n) (a n, n 2) (a 2,2) does not change the structure constants of algebras from SLeib nþ. Thus, the action of GL ad on SLeib nþ can be reduced to the action of elementary transformation of type three. Let R m a ðxþ :¼ ½½...½x, aš, aš,..., aš fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} m-times, R 0 aðxþ :¼ x: Now due to Proposition. it is easy to see that for SLeib nþ the adapted base change has the form [7]: e 0 0 ¼ e 0 þ Be, e 0 ¼ De BD e n, e ¼ ð þ BÞe 2 þ Bð e þþ n e n ÞþBð n þ BÞe n,! e 0 k ¼ Xk 2 k i B i R i e ðe k i ÞþB k R k ðe 0 Þ, i¼0 C k i k where k n, D 2 C such that D 6¼ 0. Now, we remind an isomorphism criterion for SLeib nþ. Introduce the following series of functions: tð y; zþ ¼ t ð y; z, z 4,..., z n, z nþ Þ ¼ z t Xt C k 2 k yz tþ2 k þ Ck k y2 k¼ þ C k 4 k y þ C k yk 2 Xt X i 2 i 2 ¼kþ i ¼kþ X t i k ¼2k 2 i k 4 ¼2k 2 Xt i ¼kþ2 e z tþ i z i þ k z tþ i2 z i2 þ i z i k þ X i k z i2 þ i z i þ5 2k þ y k X i 2 i ¼2k 2 X t i k 2 ¼2k i k ¼2k z tþ ik z ik þ i k 4 X i k 2... X i 2 i ¼2k z tþ ik 2! z ik 2 þ i k z i2 þ i z i þ4 2k k ð y; zþ, where t n, nþð y; zþ ¼z nþ : The isomorphism criterion for SLeib nþ, first appeared in [7] later on in [4], it was given in more invariant form as follows. THEOREM. Two algebras L() L( 0 ) from SLeib nþ, where ¼ (, 4,..., n, ), 0 ¼ ( 0, 0 4,..., 0 n, 0 ), are isomorphic, if only if

7 44 I.S. Rakhimov S.K. Said Husain there exist complex numbers, B D, such that D 6¼ 0 the following conditions hold: 0 t ¼ D B t 2 t ;, t n, ðþ 0 n ¼ D B n 2 þ B n ; ð2þ 0 ¼ D 2 n 2 nþ B ; : ðþ Hereinafter, for the simplification purpose in the above case for transition from (n þ )-dimensional filiform Leibniz algebra L() to (n þ )-dimensional filiform Leibniz algebra L( 0 ) we write 0 ¼ %, B, D ; : %, B, D ; where ¼ %, B, D ;, % 2, B, D ; % t ðx, y, u; zþ ¼x t u tþ2 ð y; zþ for t n 2, % n ðx, y, u; zþ ¼x n 5 u 2 nþð y; zþ:,...,% n, B, D ; Here are the main properties of the operator %, used in this article: 0. %(, 0, ; ) is the identity operator. 2 0 :%ð 2, B 2 2, D 2 2 ; %ð, B, D ; ÞÞ ¼ %ð 2, B 2 þb 2 D 2, D D 2 2 ; Þ 0 : If 0 ¼ %ð, B, D ; Þ then ¼ %ð, B D, D ; 0 Þ: From here on, we assume that n 4 since there are complete classifications of complex nilpotent Leibniz algebras in dimensions at most four []. In this article we proceed from the viewpoint of [4]. Let N n st for the adapted number of isomorphism classes in SLeib n. Later on, if no confusion is possible, we write xy for [x, y] as well., 4. Classification For the simplification purpose, we establish the following notations: ¼ , 2 ¼ , 0 ¼ , ¼ :

8 Linear Multilinear lgebra Dimension 5 The class SLeib 5 is represented as a disjoint union of its subsets as follows: SLeib 5 ¼ [6 U i, where i ¼ U ¼fLðÞ2SLeib 5 : 6¼ 0, 2 2 6¼ 0g, U 2 ¼fLðÞ2SLeib 5 : 6¼ 0, 2 2 ¼ 0, 4 6¼ 0g, U ¼fLðÞ2SLeib 5 : ¼ 0, 6¼ 0g, U 4 ¼fLðÞ2SLeib 5 : ¼ 0, ¼ 0, 4 ¼ 0g, U 5 ¼fLðÞ2SLeib 5 : 6¼ 0, 2 2 ¼ 0, 4 ¼ 0g, U 6 ¼fLðÞ2SLeib 5 : ¼ 0, ¼ 0, 4 6¼ 0g: Now we consider the isomorphism problem for each of these subsets separately. PROPOSITION 4. (i) Two algebras L() L( 0 ) from U are isomorphic, if only if 2 ¼ 0 : ð4þ (ii) Orbits in U can be parametrized as L(, 0, ), 2 C n {2}. PROPOSITION 4.2 The subsets U 2, U, U 4, U 5 U 6 are single orbits under the action of G ad with the representatives L(,, 2), L(0, 0, ), L(0,, 0), L(, 0, 2) L(0, 0, 0), respectively. THEOREM 4. Let L be an element of SLeib 5. Then, it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras: () Lð0, 0, 0Þ ¼G s 5 : e 0e 0 ¼ e 2, e i e 0 ¼ e iþ,2i. (2) L(0,, 0): G s 5, e 0e ¼ e 4 : () L(0, 0, ): G s 5, e e ¼ e 4 : (4) L(,, 2): G s 5, e 0e ¼ e þ e 4, e e ¼ 2e 4, e 2 e ¼ e 4 : (5) L(, 0, ): G s 5, e 0e ¼ e, e e ¼ e 4, e 2 e ¼ e 4, 2 C. Note 4. The orbit U 5 with the representative L(, 0, 2) can be included in the parametric family of orbits L(, 0, ) at ¼ 2. The adapted number of isomorphism classes N 5 ¼ Dimension 6 This section concerns 6-dimensional case. The set SLeib 6 can be represented as a disjoint union of its subsets as follows: SLeib 6 ¼ [9 i ¼ U i,

9 46 I.S. Rakhimov S.K. Said Husain where PROPOSITION 4. U ¼fLðÞ2SLeib 6 : 6¼ 0, 6¼ 0g, U 2 ¼fLðÞ2SLeib 6 : 6¼ 0, ¼ 0, 6¼ 0g, U ¼fLðÞ2SLeib 6 : 6¼ 0, ¼ 0, ¼ 0g, U 4 ¼fLðÞ2SLeib 6 : ¼ 0, 4 6¼ 0, 6¼ 0g, U 5 ¼fLðÞ2SLeib 6 : ¼ 0, 4 6¼ 0, ¼ 0, 5 6¼ 0g, U 6 ¼fLðÞ2SLeib 6 : ¼ 0, 4 6¼ 0, ¼ 0, 5 ¼ 0g, U 7 ¼fLðÞ2SLeib 6 : ¼ 0, 4 ¼ 0, 6¼ 0g, U 8 ¼fLðÞ2SLeib 6 : ¼ 0, 4 ¼ 0, ¼ 0, 5 6¼ 0g, U 9 ¼fLðÞ2SLeib 6 : ¼ 0, 4 ¼ 0, ¼ 0, 5 ¼ 0g: (i) Two algebras L() L( 0 ) from U are isomorphic, if only if 2 4 þ 2 2 ¼ þ 02 0 : ð5þ (ii) Orbits in U can be parametrized as L(, 0,, ), 2 C. PROPOSITION 4.4 The subsets U 2, U, U 4, U 5, U 6, U 7, U 8 U 9 are single orbits under the action of G ad with the representatives L(, 0,, 0), L(, 0, 0, 0), L(0,, 0, ), L(0,,, 0), L(0,, 0, 0), L(0, 0, 0, ), L(0, 0,, 0) L(0, 0, 0, 0), respectively. THEOREM 4.2 Let L be an element of SLeib 6. Then, it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras: Þ Lð0, 0, 0, 0Þ ¼G s 6 : e 0e 0 ¼ e 2, e i e 0 ¼ e iþ,2i4. 2) L(0, 0,, 0): G s 6, e 0e ¼ e 5 : ) L(0, 0, 0, ): G s 6, e e ¼ e 5 : 4) L(0,, 0, 0): G s 6, e 0e ¼ e 4, e 2 e ¼ e 5 : 5) L(0,,, 0): G s 6, e 0e ¼ e 4 þ e 5, e 2 e ¼ e 5 : 6) L(0,, 0, ): G s 6, e 0e ¼ e 4, e e ¼ e 5, e 2 e ¼ e 5 : 7) L(, 0, 0, 0): G s 6, e 0e ¼ e, e 2 e ¼ e 4, e e ¼ e 5 : 8) L(, 0,, 0): G s 6, e 0e ¼ e þ e 5, e 2 e ¼ e 4, e e ¼ e 5 : 9) L(, 0,, ): G s 6, e 0e ¼ e þ e 5, e e ¼ e 5, e 2 e ¼ e 4, e e ¼ e 5, 2 C: The adapted number of isomorphism classes N 6 ¼ Dimension 7 In this section one considers SLeib 7. The following is a representation of SLeib 7 as a disjoint union of its subsets: SLeib 7 ¼ [8 U i, i ¼

10 Linear Multilinear lgebra 47 where U ¼fLðÞ2SLeib 7 : 6¼ 0, 6¼ 0, 2 4 þ 2 4 6¼ 0g, U 2 ¼fLðÞ2SLeib 7 : 6¼ 0, 6¼ 0, 2 4 þ 2 4 ¼ 0g, U ¼fLðÞ2SLeib 7 : 6¼ 0, ¼ 0, 6¼ 0, 2 þ 2 4 6¼ 0g, U 4 ¼fLðÞ2SLeib 7 : ¼ 0, 4 6¼ 0, 5 6¼ 0g, PROPOSITION 4.5 U 5 ¼fLðÞ2SLeib 7 : ¼ 0, 4 6¼ 0, 5 ¼ 0, 2 4 6¼ 0g, U 6 ¼fLðÞ2SLeib 7 : 6¼ 0, ¼ 0, 6¼ 0, 2 þ 2 4 ¼ 0g, U 7 ¼fLðÞ2SLeib 7 : 6¼ 0, ¼ 0, ¼ 0, 2 6¼ 0g, U 8 ¼fLðÞ2SLeib 7 : 6¼ 0, ¼ 0, ¼ 0, 2 ¼ 0g, U 9 ¼fLðÞ2SLeib 7 : ¼ 0, 4 6¼ 0, 5 ¼ 0, 2 4 ¼ 0, 6 6¼ 0g, U 0 ¼fLðÞ2SLeib 7 : ¼ 0, 4 6¼ 0, 5 ¼ 0, 2 4 ¼ 0, 6 ¼ 0g, U ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 6¼ 0, 6 6¼ 0, 6¼ 0g, U 2 ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 6¼ 0, 6 6¼ 0, ¼ 0g, U ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 6¼ 0, 6 ¼ 0, 6¼ 0g, U 4 ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 6¼ 0, 6 ¼ 0, ¼ 0g, U 5 ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 ¼ 0, 6 6¼ 0, 6¼ 0g, U 6 ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 ¼ 0, 6 6¼ 0, ¼ 0g, U 7 ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 ¼ 0, 6 ¼ 0, 6¼ 0g, U 8 ¼fLðÞ2SLeib 7 : ¼ 0, 4 ¼ 0, 5 ¼ 0, 6 ¼ 0, ¼ 0g: (i) Two algebras L() L( 0 ) from U are isomorphic, if only if ð 2 4 þ 2 4 Þ 2 ¼ 0 ð þ Þ 2, ð6þ 2 ð 2 4 þ 2 4 Þ 2 ¼ 0 ð þ Þ 2 : ð7þ (ii) The subset U is a union of orbits with representatives L(, 0,,, 2 ), 2 C, 2 2 C. Proof (i) ): Let L() L( 0 ) be isomorphic. Then, due to Theorem., there are complex numbers, B D:D6¼ 0, such that the action of the adapted group G ad

11 48 I.S. Rakhimov S.K. Said Husain can be expressed by the following system of equalities: 0 ¼ D, ð8þ 0 4 ¼ D B 2, ð9þ 0 5 ¼ D 5 5 B 4 þ 5 B! 2, ð0þ ¼ 4 D B þ 6 6 B 5 þ 2 B B B! 4, ðþ 0 ¼ D 2 4 : ð2þ Then, it is easy to see that 0 ¼ D þ ¼ ð Þ. Now we need just substitute it into 2 4 þ 2 4 D þ to get the required equalities. ð Þ 2 ð þ Þ 2 (: Let the equalities (6) (7) hold. We put 0 ¼ 2 4 þ 2 4, ðþ B 0 ¼ 4ð 2 4 þ 2 4 Þ 2, ð4þ D 0 ¼ ð 2 4 þ 2 4 Þ 2, ð5þ ¼ þ , ð6þ 0 B 0 0 ¼ 0 4 ð þ Þ, 2 0 ð7þ Then, 0 ¼ %ð 0, B 0 Section ), where 0 ¼ L, 0, 0, D 0 D 0 0 ¼ ð þ Þ 2 0 0, Þ 0 0 ¼ %ð 0 0 : ð8þ, B , D0 0, 0 Þ (see the convention in 0 0 ð 2 4 þ 2 4 Þ 2, 2 ð 2 4 þ 2 4 Þ 2,

12 Linear Multilinear lgebra 49! 0 0 ¼ L, 0, 0 ð þ Þ 2, 0 ð þ Þ 2 : Then, the equalities (6) (7) imply that 0 ¼ 0 0. Now we make use of the properties 0 0 of % find the complex numbers, B D: D6¼ 0: ¼ 0 0, ð9þ 0 Thus we get B ¼ 0 ð 2 4 þ 2 4 Þ 2 ð þ Þ 2 B ¼ B 0D 0 0 B0 0 D D0 0 ð20þ D ¼ D 0 D 0 : ð2þ 0 ¼ 0 0 ð 2 4 þ 2 4 Þ ð þ Þ, 4 ð þ Þ ð 2 4 þ 2 4 Þ D ¼ 0 ð 2 4 þ 2 4 Þ 2 2 ð þ Þ 2 : ð24þ n easy computation shows that, for the, B D found above, we get the corresponding system of equalities (8) (2): D D ¼ 0, D B 2 ¼ 0 4, 5 5 B 4 þ 5 B! 2 ¼ 0 5, ð22þ ð2þ D 4 B þ 6 6 B 5 þ 2 B B B 4! ¼ 0 6, D 2 4 ¼ 0, meaning that L() L( 0 ) are isomorphic.

13 50 I.S. Rakhimov S.K. Said Husain (ii) Evidently, for any, 2 2 C: 2 6¼ 0 there exists an algebra L(, 4, 5, 6, ) from U such that ¼ 2 ð 2 4 þ 2 4 Þ 2 2 ¼ ð 2 4 þ 2 4 Þ 2 : The proof is complete. PROPOSITION 4.6 (i) Two algebras L() L( 0 ) from U 2 are isomorphic, if only if g ¼ 0 0 : ð25þ (ii) The subset U 2 is a union of continuous family of orbits with representatives L(, 0,, 0, ), 2 C. PROPOSITION 4.7 (i) Two algebras L() L( 0 ) from U are isomorphic, if only if ð 2 þ 2 4 Þ 2 ¼ 0 ð þ Þ 2 : (ii) U can be represented as a union of orbits the orbits are parametrized as L(, 0, 0,, ), 2 C. PROPOSITION 4.8 (i) Two algebras L() L( 0 ) from U 4 are isomorphic, if only if 2 ¼ 0 : 4 ð27þ 4 (ii) Orbits in U 4 can be parametrized as L(0,,, 0, ), 2 C. PROPOSITION 4.9 (i) Two algebras L() L( 0 ) from U 5 are isomorphic, if only if 2 ¼ 0 : 4 4 ð28þ (ii) Orbits in U 5 can be parametrized as L(0,, 0, 0, ), 2 C n {}. PROPOSITION 4.0 The subsets U 6, U 7, U 8, U 9, U 0, U, U 2, U, U 4, U 5, U 6, U 7, U 8 are single orbits under the action of G ad with the representatives L(, 0, 0, 0, ), L(, 0, 0,, 0), L(, 0, 0, 0, 0), L(0,, 0,, ), L(0,, 0, 0, ), L(0, 0,, 0, ), L(0, 0,,, 0), L(0, 0,,, ), L(0, 0,, 0, 0), L(0, 0, 0, 0, ), L(0, 0, 0,, 0), L(0, 0, 0,, ) L(0, 0, 0, 0, 0), respectively. Proof The subsets U 6,..., U 8 can be represented as orbits with respect to the action of G ad. Below the corresponding actions leading to the canonical ð26þ

14 Linear Multilinear lgebra 5 representatives are indicated: U 6 : For L(, 4, 5, 6, ) 2 U 6 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð, 0, 0, 0, Þ, qffiffiffi qffiffiffi where ¼ 2, B ¼ D ¼ : U 7 : For L(, 4, 5, 6, ) 2 U 7 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð, 0, 0,, 0Þ, qffiffiffiffiffi qffiffiffiffiffi qffiffiffiffiffi where ¼ 2, B ¼ D ¼ : U 8 : For L(, 4, 5, 6, ) 2 U 8 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð, 0, 0, 0, 0Þ, where is a nonzero complex number, B ¼ 4 D ¼ : U 9 : For L(, 4, 5, 6, ) 2 U 9 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0,, 0,, Þ, qffiffiffi qffiffiffi where ¼ 6 4, B is any complex number D ¼ U 0 : For L(, 4, 5, 6, ) 2 U 0 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0,, 0, 0, Þ, where is a nonzero complex number, B is any complex number D ¼ 4. U : For L(, 4, 5, 6, ) 2 U %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0,, 0, Þ, where ¼ p 5 ffiffi, B ¼ 6 D ¼ 5. 2 U 2 : For L(, 4, 5, 6, ) 2 U 2 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0,,, 0Þ, ð29þ ð0þ ðþ ð2þ ðþ ð4þ ð5þ where ¼ 6 5, B is any complex number D ¼ U : For L(, 4, 5, 6, ) 2 U %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0,,, Þ, where, B ¼ D are nonzero complex numbers. U 4 : For L(, 4, 5, 6, ) 2 U 4 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0,, 0, 0Þ, ð6þ ð7þ where is any nonzero complex number, D ¼ 5 4, B is any complex number.

15 52 I.S. Rakhimov S.K. Said Husain U 5 : For L(, 4, 5, 6, ) 2 U 5 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0, 0, 0, Þ, qffiffiffiffi where is any nonzero complex number, B ¼ 6 D ¼ 6. U 6 : For L(, 4, 5, 6, ) 2 U 6 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0, 0,, 0Þ, ð8þ ð9þ where is any nonzero complex number, D ¼ 5 6 B 2 C. U 7 : For L(, 4, 5, 6, ) 2 U 7 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0, 0,, Þ, where, B ¼ D are any nonzero complex numbers. U 8 : For L(, 4, 5, 6, ) 2 U 8 %, B, D ; Lð, 4, 5, 6, Þ ¼ Lð0, 0, 0, 0, 0Þ, where, D are any nonzero complex numbers B 2 C. g Note that in the basis changing of the subsets U 6, U 7, U 9, U U 5 above, the value of the roots can be taken an arbitrary. We summarize the previous results in the following classification theorem. THEOREM 4. Let L be an element of SLeib 7. Then, it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras: ðþ Lð0, 0, 0, 0, 0Þ ¼G s 7 : e 0e 0 ¼ e 2, e i e 0 ¼ e iþ,2 i 5. (2) L(0, 0, 0,,): G s 7, e 0e ¼ e 6, e e ¼ e 6 : () L(0, 0, 0,, 0): G s 7, e 0e ¼ e 6 : (4) L(0, 0, 0, 0, ): G s 7, e e ¼ e 6 : (5) L(0, 0,, 0, 0): G s 7, e 0e ¼ e 5, e 2 e ¼ e 6 : (6) L(0, 0,,, ): G s 7, e 0e ¼ e 5 þ e 6, e e ¼ e 6, e 2 e ¼ e 6 : (7) L(0, 0,,, 0): G s 7, e 0e ¼ e 5 þ e 6, e 2 e ¼ e 6 : (8) L(0, 0,, 0, ): G s 7, e 0e ¼ e 5, e e ¼ e 6, e 2 e ¼ e 6 : (9) L(0,, 0,, ): G s 7, e 0e ¼ e 4 þ e 6, e e ¼ e 6, e 2 e ¼ e 5, e e ¼ e 6 : (0) L(, 0, 0,, 0): G s 7, e 0e ¼ e þ e 4, e 2 e ¼ e 4, e e ¼ e 6, e 4 e ¼ e 6 : () L(0,, 0, 0, ): G s 7, e 0e ¼ e 4 þ e 5 þ e 6, e e ¼ e 6, e 2 e ¼ e 5 þ e 6, e e ¼ e 6, 2 C: (2) L(0,,, 0, ): G s 7, e 0e ¼ e 4 þ e 5, e e ¼ e 6, e 2 e ¼ e 5 þ e 6, e e ¼ e 6, 2 C: () L(, 0, 0,, ): G s 7, e 0e ¼ e þ e 6, e e ¼ e 6, e 2 e ¼ e 4, e e ¼ e 5, e 4 e ¼ e 6, 2 C. (4) L(, 0,, 0, ): G s 7, e 0e ¼ e þ e 5, e e ¼ e 6, e 2 e ¼ e 4 þ e 6, e e ¼ e 5, e 4 e ¼ e 6, 2 C: ð40þ ð4þ

16 Linear Multilinear lgebra 5 (5) L(, 0,,, 2 ): G s 7, e 0e ¼ e þ e 5 þ e 6, e e ¼ 2 e 6, e 2 e ¼ e 4 þ e 6, e e ¼ e 5, e 4 e ¼e 6,, 2 2 C. The adapted number of isomorphism classes N 7 ¼ 5. Note 4.2 The orbits U 6, U 8 U 0 can be included in parametric family of orbits with the representatives L(, 0,,, 2 ), L(, 0, 0,, ), L(0,, 0, 0, ) at the value of parameters { ¼ 0, 2 ¼ }, ¼ 0 ¼, respectively. 5. Conclusion To classify SLeib n, we split it into its subsets then classify algebras from each of these subsets. In this procedure, some of the subsets turn out to be a union of infinitely many orbits other to be just a single orbit. In each case, we indicate the corresponding canonical representatives of the orbits. We expect the formula N n ¼ n 2 7n þ 5 in dimension n for the number of isomorphism classes. The formula has been confirmed up to n ¼ 9. cknowledgements The authors acknowledge MOSTI (Malaysia) for the financial support. They are also grateful to Profs B.. Omirov U.D. Bekbaev for helpful discussions. The authors express gratitude to the referee for valuable suggestions comments in the original version of this article. References [] S. lbeverio, B.. Omirov, I.S. Rakhimov, Varieties of nilpotent complex Leibniz algebras of dimension less than five, Commun. lgebra (2005), pp [2] Sh.. yupov B.. Omirov, On some classes of nilpotent Leibniz algebras, Sib. Math. J. 42 (200), pp [] U.D. Bekbaev I.S. Rakhimov, On classification of finite dimensional complex filiform Leibniz algebras (part ) (2006). vailable at arxiv;math. R/ [4] U.D. Bekbaev I.S. Rakhimov, On classification of finite dimensional complex filiform Leibniz algebras (part 2) (2007). vaiable at arxiv; v [math.r]. [5]. Fialowski,. Mal, G. Mukherjee, Versal deformations of Leibniz algebras (2007). vailable at arxiv, math.q. [6]. Frabetti, Leibniz homology of dialgebras of matrices, J. Pure ppl. lgebra 29 (998), pp [7] J.R. Gomez B.. Omirov, On classification of complex filiform Leibniz algebras (2006). vailable at arxiv;math/06275 v [math.r..]. [8] M.K. Kinyon. Weinstein, Leibniz algebras, Courant algebroids, multiplications on reductive homogeneous spaces, mer. J. Math. 2 (200), pp [9] J.-L. Loday, Une version non commutative des alge bres de Lie: Les alge bres de Leibniz, L Ens. Math. 9 (99), pp [0] J.-L. Loday,. Frabetti, F. Chapoton, F. Goichot, Dialgebras related operads, Lecture Notes Math. IV (200), p. 76.

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