On isomorphism classes and invariants of a subclass of low-dimensional complex filiform Leibniz algebras

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1 Linear and Multilinear lgebra ISSN: (Print) (Online) Journal homepage: On isomorphism classes and invariants 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 (2011) On isomorphism classes and invariants of a subclass of low-dimensional complex filiform Leibniz algebras, Linear and Multilinear lgebra, 9:2, , DOI: / To link to this article: Published online: 18 Feb Submit your article to this journal rticle views: 82 View related articles Citing articles: 12 View citing articles Full Terms & Conditions of access and use can be found at Download by: [The UC Irvine Libraries] Date: 28 September 2017, t: 1:01

2 Linear and Multilinear lgebra Vol. 9, No. 2, February 2011, On isomorphism classes and invariants of a subclass of low-dimensional complex filiform Leibniz algebras I.S. Rakhimov* and S.K. Said Husain Department of Mathematics, Faculty of Science, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia 00, Serdang, Selangor Darul Ehsan, Malaysia Communicated by W.-F. Ke (Received 26 July 2009; final version received 16 September 2009) The article aims to study the classification problem of low-dimensional complex filiform Leibniz algebras. It is known that filiform Leibniz algebras come out from two sources. The first source is a naturally graded non-lie filiform Leibniz algebra, and another one is a naturally graded filiform Lie algebra. In this article, we classify a subclass of the class of filiform Leibniz algebras appearing from the naturally graded non-lie filiform Leibniz algebra. We give complete classification and isomorphism criteria in dimensions 7. The method of classification is purely algorithmic. The isomorphism criteria are given in terms of invariant functions. Keywords: filiform Leibniz algebra; invariant function; isomorphism MS Subject Classifications: 172; 17B0(primary); 10(secondary) 1. Introduction In the early 1990s, Loday [10] introduced Leibniz algebra as a non-associative algebra with multiplication, satisfying the Leibniz identity: ½x, ½ y, zšš ¼ ½½x, yš, zš ½½x, zš, yš: This identity and the classical Jacobi identity are equivalent, when the multiplication is skewsymmetric. Leibniz algebras appear to be related, in a natural way, to several topics, such as differential geometry and homological algebra, classical algebraic topology, algebraic K-theory, loop spaces, noncommutative geometry and quantum physics, as a generalization of the corresponding applications of Lie algebras to these topics. Most papers deal with the homological problems of Leibniz algebras. In [11] Loday and Pirashvili have described the free Leibniz algebras, paper [1] by Mikhalev and Umirbaev is devoted to solution of the non-commutative analogue of the Jacobian conjecture in the affirmative for free Leibniz algebras, in the spirit of the corresponding result of Reutenauer [1], Shpilrain [16] and Umirbaev [17]. The problems concerning Cartan subalgebras and solvability were studied by yupov and Omirov [1]. The notion of simple Leibniz algebra was suggested by Dzhumadil daev and bdukassymova [7], who obtained some results concerning special cases of simple Leibniz algebras. *Corresponding author. risamiddin@gmail.com ISSN print/issn online ß 2011 Taylor & Francis DOI: /

3 206 I.S. Rakhimov and S.K. Said Husain Unfortunately, up to now, there is no paper that has included complete discussion on comparisons of structural theory of Lie and Leibniz algebras (one means results like Levi-Malcev decomposition theorem, Lie Engel theorem, Malcev reduction theorem, the analogue of Killing form, Dinkin diagrams, root space decompositions, the Serre presentation, the theory of highest weight representations, the Weyl character formula and much more). Papers [2,,1] and preprints [,6,9] concern the classification problem of filiform Leibniz algebras. The organization of this article is as follows. Section 2 consists of some facts on filiform Leibniz algebras. They can be found in []. In Section, the reader will be reminded regarding adapted basis and adapted transformations. Then we describe the isomorphism action of the adapted transformations group. Section deals with the classification problem of low-dimensional filiform Leibniz algebras. Here for - and 6-dimensional cases we give just final result since the proofs in these cases are similar to those of 7-dimensional case (for the last case in Section. we give complete proof only for generic case, since other cases can be managed by a minor adaption). In discrete orbits cases (Proposition.1) we give a base change, leading to appropriate canonical representative. 2. Preliminaries Let V be a vector space of dimension n over an algebraically closed field K (chark ¼ 0). Bilinear maps V V! V form a vector space Hom(V V, V) of dimensional n, which can be considered together with its natural structure of an affine algebraic variety over K and denoted by lg n (K ) ffi K n.nn-dimensional algebra structure L over K on V can be considered as an element (L) of lg n (K ) via the bilinear mapping : V V! V defining a binary algebraic operation on V: let {e 1, e 2,..., e n } be a basis of the algebra L. Then, the table of multiplication of L is represented by point ðij k Þ of this affine space as follows: ðe i, e j Þ¼ Xn ij k e k, i, j ¼ 1, 2,..., n: k¼1 Here ij k is said to be structure constants of L. The linear reductive group GL n(k ) acts on lg n (K ) by (g * )(x, y) ¼ g((g 1 (x), g 1 ( y)))( transport of structure ). Two algebra structures 1 and 2 on V are isomorphic if and only if they belong to the same orbit under this action. Definition 2.1 n algebra L over a field K is called a Leibniz algebra, if its bilinear operation [,] satisfies the following Leibniz identity: ½x, ½ y, zšš ¼ ½½x, yš, zš ½½x, zš, yš: Let Leib n (K ) be the subvariety of lg n (K ), consisting of all n-dimensional Leibniz algebras over K. It is invariant under the above mentioned action of GL n (K ). s a subset of lg n (K ) the set Leib n (K ) is specified by a system of equations with respect to structure constants k ij : X ð l jk il m ij l m lk þ l ik m lj Þ¼0, i, j, k ¼ 1, 2,..., n:

4 Linear and Multilinear lgebra 207 It is easy to see that, if [,] in Leibniz algebra is anticommutative, then it is a Lie algebra. Therefore, Leibniz algebra is a noncommutative generalization of Lie algebra. For the Lie algebras case, several classifications of low-dimensional cases have been given. Except for simple Lie algebras, the classifications problem of all Lie algebras in common remains a big problem. Malcev [12] reduced the classification of solvable Lie algebras to the classification of nilpotent Lie algebras. pparently, the first non-trivial classification of some classes of low-dimensional nilpotent Lie algebras are due to Umlauf. In his thesis [18], he presented the redundant list of nilpotent Lie algebras of dimension less than seven. He also gave the list of nilpotent Lie algebras of dimension less than 10 admitting a so-called adapted basis (now, the nilpotent Lie algebra with this property is called filiform Lie algebra). The importance of filiform Lie algebras in the study of the variety of nilpotent Lie algebras laws was shown by Vergne [19]. Paper [8] concerns the classification problem of low-dimensional filiform Lie algebras. In [,6], a method of classification of fixed dimensional filiform Leibniz algebras has been proposed. This article is an implementation of the method in lowdimensional cases. Throughout the following sections, all algebras are assumed to be over the field of complex numbers C. Let L be a Leibniz algebra. We put L 1 ¼ L, L kþ1 ¼½L k, LŠ, k 2 N: Definition 2.2 Leibniz algebra L is said to be nilpotent, if there exists an integer s 2 N, such that L s ¼f0g: The smallest integer s for that L s ¼ 0 is called the nilindex of L. Definition 2. n n-dimensional Leibniz algebra L is said to be filiform, if diml i ¼ n i, where 2 i n: It is clear that a filiform Leibniz algebra is nilpotent. The class of filiform Leibniz algebras in dimension n is denoted by Leib n. In [9], Go mez and Omirov split Leib n into three subclasses as follows. THEOREM 2.1 ny (n þ 1)-dimensional complex filiform Leibniz algebra admits a basis {e 0, e 1,..., e n } called adapted, such that the table of multiplication of the algebra has one of the following forms, where omitted products of basis vectors are supposed to be zero: 8 ½e 0, e 0 Š¼e 2, >< ½e i, e 0 Š¼e iþ1, 1 i n 1, FLeib nþ1 ¼ ½e 0, e 1 Š¼ e þ e þþ n 1 e n 1 þ e n, ½e j, e 1 Š¼ e jþ2 þ e jþ þþ nþ1 j e n, 1 j n 2, >:,,..., n, 2 C:

5 208 I.S. Rakhimov and S.K. Said Husain 8 ½e 0, e 0 Š¼e 2, >< ½e i, e 0 Š¼e iþ1, 2 i n 1, ½e SLeib nþ1 ¼ 0, e 1 Š¼ e þ e þþ n e n, ½e 1, e 1 Š¼e n, >: ½e j, e 1 Š¼ e jþ2 þ e jþ þþ nþ1 j e n, 2 j n 2,,,..., n, 2 C: 8 ½e i, e 0 Š¼e iþ1, 1 i n 1, ½e 0, e i Š¼ e iþ1, 2 i n 1, ½e 0, e 0 Š¼b 0,0 e n, >< ½e 0, e 1 Š¼ e 2 þ b 0,1 e n, ½e T Leib nþ1 ¼ 1, e 1 Š¼b 1,1 e n, ½e i, e j Š¼ ½e j, e i Š2span C fe iþjþ1, e iþjþ2,..., e n j 1 i n, 2 j n 1 ig, ½e i, e n i Š¼ ½e n i, e i Š¼ð 1Þ i b i,n i e n, >: where a k i,j, b i,j 2 C and b i,n i ¼ b whenever 1 i n 1, and b ¼ 0 for even n: In [9], the base change for this kind algebra has been reduced to the so-called adapted base change. The subclasses are stable with respect to the adapted base change. Hence, the isomorphisms problem inside of each class can be studied separately. This article deals with F Leib nþ1. Results on SLeib nþ1 and TLeib nþ1 have appeared elsewhere, particularly in [6,1]. Elements of F Leib nþ1 will be denoted by L(,,..., n, ), which means that they are defined by the parameters,,..., n,.. On adapted changing of basis and isomorphism criterion for F Leib ny1 In this section, we simplify the isomorphic action of GL n ( transport of structure ) on F Leib n. ll the details can be found in [,9]. Let L be an element of F Leib nþ1, V be the underlying vector space and {e 0, e 1,..., e n } be the adapted basis of L. Definition.1 The basis transformation f 2 GL(V) is said to be adapted for the structure of L, if the basis { f (e 0 ), f (e 1 ),..., f (e n )} is adapted. The closed subgroup of GL(V) spanned by the adapted transformations is denoted by GL ad.ingl ad we consider the following types of basis transformations of F Leib nþ1 called elementary: 8 >< f ðe 0 Þ¼e 0 þ ae k, f ðe first type, ða, b, kþ ¼ 1 Þ¼e 1 þ be k, >: f ðe iþ1 Þ¼½fðe i Þ, f ðe 0 ÞŠ, 1 i n 1, 2 k n, f ðe 2 Þ¼½fðe 0 Þ, f ðe 0 ÞŠ: 8 f ðe 0 Þ¼ae 0 þ be 1, >< f ðe second type ða, bþ ¼ 1 Þ¼ðaþbÞe 1 þ bð n Þe n 1, aða þ bþ 6¼ 0, >: f ðe iþ1 Þ¼½fðe i Þ, f ðe 0 ÞŠ, 1 i n 1, f ðe 2 Þ¼½fðe 0 Þ, f ðe 0 ÞŠ: The proof of the following proposition is straightforward. PROPOSITION.1 (a) If f is an adapted transformation of F Leib nþ1 then f ¼ ða n, b n, nþða n 1, a n 1, n 1Þða 2, a 2,2Þða 0, a 1 Þ:

6 Linear and Multilinear lgebra 209 (b) Transformations of the form (a, b, n), (a, a, n) and (a, a, k), where 2 k n 2, a 2 C preserve the structure constants of FLeib nþ1. Since a composition of adapted transformations is adapted, the proposition above means that the transformation (a n, b n, n) (a n 1, a n 1, n 1) (a 2, a 2,2) does not change the structure constants of F Leib nþ1 and thus the action of GL ad on F Leib nþ1 second type. Let R m a can be reduced to the action of elementary transformation of the ðxþ :¼ ½½...½x, aš, aš,..., aš fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} m-times, and R 0 aðxþ :¼ x: Now, due to Proposition.1(b), it is easy to see that, for F Leib nþ1, the adapted change of basis has the following forms: e 0 ¼ e 0 þ Be 1, e 0 1 ¼ðþBÞe 1 Bð n Þe n 1, e 0 2 ¼ ð þ BÞe 2 þ ð þ BÞð e þþ n 1 e n 1 ÞþBð n þ BÞe n, e 0 k ¼ð þ BÞ X k 2 i¼0 C k 1 i k 1! k 1 i B i R i e 1 ðe k i ÞþB k 1 R k 1 ðe 0 Þ, where k n and, B 2 C such that ( þ B) 6¼ 0. Now we restate the isomorphism criterion for F Leib nþ1. First, we introduce the following series of functions: t ð y; zþ ¼ t ð y; z, z,..., z n, z nþ1 Þ ¼ ð1 þ yþz t Xt 1 þ C k k 1 y þ C 1 k 1 yk 2 k¼ Xt X i 2 ðc k 2 k 1 yz tþ2 k þ C k k 1 y2 i 2 ¼kþ i 1 ¼kþ X t X i k i k ¼2k 2 i k ¼2k 2 X t e 1 Xt i 1 ¼kþ2 z tþ i2 z i2 þ i 1 z i1 k þ X i 2 i 1 ¼2k 2 X i k 2 z tþ i1 z i1 þ1 k z tþ ik z ik þ i k z i2 þ i 1 z i1 þ 2k þ y k 1 z tþ ik 2 i k 2 ¼2k 1 i k ¼2k 1 i 1 ¼2k 1 z ik 2 þ i k z i2 þ i 1 z i1 þ 2kÞ k ð y; zþ, for t n: nþ1 ð y; zþ ¼ nþ1 ð y; z, z,..., z n, z nþ1 Þ ¼ z nþ1 þ yz n ð1þyþ Xn 1 ðc k 2 k 1 yz nþ2 k þ C k k 1 y2 Xn i 1 ¼kþ2 Xn X i 2 k¼ z nþ i1 z i1 þ1 k þ C k k 1 y i 2 ¼kþ i 1 ¼kþ X t þ C 1 k 1 yk 2 z nþ i2 z i2 þ i 1 z i1 k þ X i k i k ¼2k 2 i k ¼2k 2 X n X i 2 i 1 ¼2k 2 X i k 2 z i2 þ i 1 z i1 þ 2k þ y k 1 i k 2 ¼2k 1 i k ¼2k 1 z ik 2 þ i k z i2 þ i 1 z i1 þ 2kÞ k ð y; zþ : X i 2 z nþ ik z ik þ i k X i 2 i 1 ¼2k 1 z nþ ik 2

7 210 I.S. Rakhimov and S.K. Said Husain The isomorphism criterion for FLieb nþ1 proven in [9] is then spelled out as follows. THEOREM.1 [] Two algebras L() and L( 0 ) from F Leib nþ1, where ¼ (,,..., n, ) and 0 ¼ ( 0, 0,..., 0 n, 0 ), are isomorphic, if and only if there exist complex numbers and B, such that ( þ B) 6¼ 0 and the following conditions hold: 0 t ¼ 1 t 2 B t ;, t n, ð1þ ¼ 1 n 2 B nþ1 ; : ð2þ To simplify notations, in the above case for transition from (n þ 1)-dimensional filiform Leibniz algebra L() to(n þ 1)-dimensional filiform Leibniz algebra L( 0 ) we write 0 ¼ ð 1, B ; ÞÞ: where and 1, B ; 1 ¼ 1, B ; 1, 2, B ; 1,..., n 1, B ; t ðx, y; zþ ¼x t tþ2 ð y; zþ for 1 t n 2, n 1 ðx, y; zþ ¼x n 2 nþ1 ð y; zþ: Here are main properties of the operator used in this article: 1 0. (1, 0, 1;)) is the identity operator. 2 0 :ð 1 2, B 2 2 ; ð 1 1, B 1 1 ; ÞÞ ¼ ð 1 1 2, 1B 2 þ 2 B 1 þb 1 B ; Þ: 0 If 0 ¼ ð 1, B B ; Þ then ¼ ð, þb ; 0 Þ: From here on, n is a positive integer. We assume that n, since there are complete classifications of complex nilpotent Leibniz algebras of dimension of at most four []. In our study, we proceed from the viewpoint of []. Later on, if no confusion is possible, we write xy for [x, y] as well.,. Classification In this section we classify algebras from F Leib nþ1 for n ¼,, 6. For the purpose of simplification, we establish the following notations and conventions: D ¼, D ¼ þ 2 2, D ¼, D 6 ¼ 6 þ 1, i ¼ i, and D 0 ¼ 0, D0 ¼ 0 þ 202, D0 ¼ 0 0, D 0 6 ¼ 0 6 þ 10, 0 i ¼ 0 0 i, for i ¼,, 6: ðþ

8 Linear and Multilinear lgebra 211 Observe that D i ¼ i (i ¼,, 6) as ¼ 0. fter these notations, the algebra L(,,..., n, ) from F Leib nþ1 becomes L(D, D,..., D n, n )..1. Dimension ccording to the above notations, one can rewrite Theorem.1 for F Leib as follows. THEOREM.1 Two algebras L(D) and L(D 0 ) from F Leib, where D ¼ (D, D, ) and D 0 ¼ (D 0, D 0, 0 ), are isomorphic, if and only if there exist complex numbers and B, such that ( þ B) 6¼ 0 and the following conditions hold: D 0 ¼ 1 1 þ B D, D 0 ¼ þ B D, 0 ¼ 1 2 : ðþ In order to describe orbits in F Leib under the action of the adapted base change, we split it into the following subsets: U 1 ¼fLðDÞ2F Leib : D 6¼ 0, D 6¼ 0g, U 2 ¼fLðDÞ2F Leib : D 6¼ 0, D ¼ 0, 6¼ 0g, U ¼fLðDÞ2F Leib : D 6¼ 0, D ¼ 0, ¼ 0g, U ¼fLðDÞ2F Leib : D ¼ 0, D 6¼ 0, 6¼ 0g, U ¼fLðDÞ2F Leib : D ¼ 0, D 6¼ 0, ¼ 0g, U 6 ¼fLðDÞ2F Leib : D ¼ 0, D ¼ 0, 6¼ 0g, U 7 ¼fLðDÞ2F Leib : D ¼ 0, D ¼ 0, ¼ 0g: It is obvious that {U i }, i ¼ 1, 2,..., 7, is a partition of F Leib. The following proposition shows that U 1 is a union of infinitely many orbits and these orbits can be parametrized by C. PROPOSITION.1 (i) Two algebras L(D) and L(D 0 ) from U 1 are isomorphic if and only if D 2 2 ¼ D0 0 D : ðþ (ii) Orbits in U 1 can be parametrized as L(1, 1, ), 2 C. The next proposition is a description of subsets U i, i ¼ 2,...,7. D 0

9 212 I.S. Rakhimov and S.K. Said Husain PROPOSITION.2 The subsets U 2, U, U, U, U 6 and U 7 are single orbits with the representatives L(1, 0, 1), L(1, 0, 0), L(0, 1, 1), L(0, 1, 0), L(0, 0, 1) and L(0, 0, 0), respectively. We summarize the above observations under the following classification theorem. THEOREM.2 Let L be an element of F Leib. Then, it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras: ð1þ Lð0, 0, 0Þ ¼L s : e 0e 0 ¼ e 2, e i e 0 ¼ e iþ1,1i. (2) L(0, 0, 1): L s, e 0e 1 ¼ e : () L(0, 1, 0): L s, e 0e 1 ¼ e, e 1 e 1 ¼ e : () L(0, 1, 1): L s, e 0e 1 ¼ 2e, e 1 e 1 ¼ e : () L(1, 0, 0): L s, e 0e 1 ¼ e 2e, e 1 e 1 ¼ e 2e, e 2 e 1 ¼ e : (6) L(1, 0, 1): L s, e 0e 1 ¼ e e, e 1 e 1 ¼ e 2e, e 2 e 1 ¼ e : (7) L(1, 1, ): L s, e 0e 1 ¼ e þð 1Þe, e 1 e 1 ¼ e e, e 2 e 1 ¼ e, 2 C:.2. Dimension 6 This section concerns F Leib 6. ccording to the notations (), we rephrase Theorem.1 as follows. THEOREM. Two algebras L(D) and L(D 0 ) from F Leib 6, where D ¼ðD 0, D0, D0, 0 Þ and D 0 ¼ðD 0, D0, D0, 0 Þ, are isomorphic, if and only if there exist complex numbers and B, such that ( þ B) 6¼ 0 and the following conditions hold: D 0 ¼ 1 1 þ B D, D 0 ¼ þ B D, D 0 þ D0 D0 ¼ 1 1 þ B ðd þ D D Þ, 0 ¼ 1 : To classify F Leib 6, we represent it as a disjoint union of the following subsets: U 1 ¼fLðDÞ2F Leib 6 : D 6¼ 0, D 6¼ 0g, U 2 ¼fLðDÞ2F Leib 6 : D 6¼ 0, D ¼ 0, D 6¼ 0, 6¼ 0g, U ¼fLðDÞ2F Leib 6 : D ¼ 0, D 6¼ 0, D 6¼ 0g, U 6 ¼fLðDÞ2F Leib 6 : D 6¼ 0, D ¼ 0, D 6¼ 0, ¼ 0g, U ¼fLðDÞ2F Leib 6 : D 6¼ 0, D ¼ 0, D ¼ 0, 6¼ 0g, U 6 ¼fLðDÞ2F Leib 6 : D 6¼ 0, D ¼ 0, D ¼ 0, ¼ 0g, U 7 ¼fLðDÞ2F Leib 6 : D ¼ 0, D 6¼ 0, D ¼ 0, 6¼ 0g, U 8 ¼fLðDÞ2F Leib 6 : D ¼ 0, D 6¼ 0, D ¼ 0, ¼ 0g, U 9 ¼fLðDÞ2F Leib 6 : D ¼ 0, D ¼ 0, D 6¼ 0, 6¼ 0g, U 10 ¼fLðDÞ2F Leib 6 : D ¼ 0, D ¼ 0, D 6¼ 0, ¼ 0g, U 11 ¼fLðDÞ2F Leib 6 : D ¼ 0, D ¼ 0, D ¼ 0, 6¼ 0g, U 12 ¼fLðDÞ2F Leib 6 : D ¼ 0, D ¼ 0, D ¼ 0, ¼ 0g: ð6þ

10 Since each of these sets is a G ad -stable, the isomorphism problem, for each of them, can be attacked separately. Here each of the subsets U 1, U 2 and U turns out to be a union of infinitely many orbits and they can be described as follows. PROPOSITION. Linear and Multilinear lgebra 21 (i) Two algebras L(D) and L(D 0 ) from U 1 are isomorphic if and only if and D ðd þ D D Þ ¼ D0 ðd0 þ D0 D0 Þ, ð7þ D 2 D 02 D D ¼ D0 0 D 0 : ð8þ (ii) Orbits in U 1 can be parametrized as L(1, 1, 1, 2 ), 1, 2 2 C. PROPOSITION. (i) Two algebras L(D) and L(D 0 ) from U 2 are isomorphic if and only if D D 2 ¼ D0 D : ð9þ (ii) Orbits in U 2 can be parametrized as L(1, 0,, ), 2 C*. PROPOSITION. (i) Two algebras L(D) and L(D 0 ) from U are isomorphic if and only if D D ¼ D0 0 D 0 : ð10þ (ii) Orbits in U can be parameterized as L(0, 1, 1, ), 2 C. However, each of the sets U, U,..., U 12 is a single orbit, here is a description of them. PROPOSITION.6 The subsets U, U, U 6, U 7, U 8, U 9, U 10, U 11 and U 12 are single orbits with the representatives L(1, 0, 1, 0), L(1, 0, 0, 1), L(1, 0, 0, 0), L(0, 1, 0, 1), L(0, 1, 0, 0), L(0, 0, 1, 1), L(0, 0, 1, 0), L(0, 0, 0, 1) and L(0, 0, 0, 0), respectively. The result of all the above observations can be spelled out as follows. THEOREM. Let L be an element of F Leib 6. Then, it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras: ð1þ Lð0, 0, 0, 0Þ ¼L s 6 : e 0e 0 ¼ e 2, e i e 0 ¼ e iþ1,1i. (2) L(0, 0, 0, 1): L s 6, e 0e 1 ¼ e : () L(0, 0, 1, 0): L s 6, e 0e 1 ¼ e, e 1 e 1 ¼ e : () L(0, 0, 1, 1): L s 6, e 0e 1 ¼ 2e, e 1 e 1 ¼ e : () L(0, 1, 0, 0): L s 6, e 0e 1 ¼ e, e 1 e 1 ¼ e, e 2 e 1 ¼ e : (6) L(0, 1, 0, 1): L s 6, e 0e 1 ¼ e þ e, e 1 e 1 ¼ e, e 2 e 1 ¼ e :

11 21 I.S. Rakhimov and S.K. Said Husain (7) L(1, 0, 0, 1): L s 6, e 0e 1 ¼ e 2e þ 6e, e 1 e 1 ¼ e 2e þ e, e 2 e 1 ¼ e 2e, e e 1 ¼ e : (8) L(1, 0, 1, 0): L s 6, e 0e 1 ¼ e 2e þ 6e, e 1 e 1 ¼ e 2e þ 6e, e 2 e 1 ¼ e 2e, e e 1 ¼ e : (9) L(0, 1, 1, ): L s 6, e 0e 1 ¼ e þðþ1þe, e 1 e 1 ¼ e þ e, e 2 e 1 ¼ e, 2 C: (10) L(1, 0,, ): L s 6, e 0e 1 ¼ e 2e þð2þþe, e 1 e 1 ¼ e 2e þðþþe, e 2 e 1 ¼ e 2e, e e 1 ¼ e, 2 C. (11) L(1, 1, 1, 2 ): L s 6, e 0e 1 ¼ e e þð 1 þ 2 þ Þe, e 1 e 1 ¼ e e þð 1 þ Þe, e 2 e 1 ¼ e e, e e 1 ¼ e, 1, 2 2 C. Note.1 The orbit U 6 with the representative L(1, 0, 0, 0) can be included in the parametric family of orbits with the representatives L(1, 0,, ) at ¼ 0... Dimension 7 This section deals with F Leib 7. For F Leib 7, the isomorphism criteria (Theorem.1) can be rewritten as follows. THEOREM. Two algebras L(D) and L(D 0 ) from FLeib 7, where D ¼ (D, D, D, D 6, 6 ) and D 0 ¼ðD 0, D0, D0, D0 6, 0 6Þ, are isomorphic, if and only if there exist complex numbers and B, such that ( þ B) 6¼ 0 and the following conditions hold: D 0 ¼ 1 1 þ B D, D 0 ¼ þ B D, D 0 þ D0 D0 ¼ 1 1 þ B ðd þ D D Þ, D 0 6 þ 6D0 D0 þ 2 9D0 D0 þ 2 D0 ¼ 1 1 þ B ðd 6 þ 6D D þ 9D 2 D þ D 2 Þ, 0 6 ¼ 1 6: ð11þ Under the appropriate constraints to D, D,..., D 6 and 6, the set F Leib 7 can be written as a union of the following subsets: U 1 ¼fLðDÞ2FLeib 7 : D 6¼ 0, D 6¼ 0g, U 2 ¼fLðDÞ2FLeib 7 : D 6¼ 0, D ¼ 0, D 6¼ 0, D 6 þ 6D D 6¼ 0g, U ¼fLðDÞ2FLeib 7 : D 6¼ 0, D ¼ 0, D 6¼ 0, D 6 þ 6D D ¼ 0g, U ¼fLðDÞ2FLeib 7 : D 6¼ 0, D ¼ 0, D ¼ 0, D 6 6¼ 0, 6 6¼ 0g, U ¼fLðDÞ2FLeib 7 : D ¼ 0, D 6¼ 0, D 6¼ 0g, U 6 ¼fLðDÞ2FLeib 7 : D ¼ 0, D 6¼ 0, D ¼ 0, D 6 þ D 2 6¼ 0, U 7 ¼fLðDÞ2FLeib 7 : D ¼ 0, D ¼ 0, D 6¼ 0, D 6 6¼ 0g, U 8 ¼fLðDÞ2FLeib 7 : D 6¼ 0, D ¼ 0, D ¼ 0, D 6 6¼ 0, 6 ¼ 0g, U 9 ¼fLðDÞ2FLeib 7 : D 6¼ 0, D ¼ 0, D ¼ 0, D 6 ¼ 0, 6 6¼ 0g, U 10 ¼fLðDÞ2FLeib 7 : D 6¼ 0, D ¼ 0, D ¼ 0, D 6 ¼ 0, 6 ¼ 0g, U 11 ¼fLðDÞ2FLeib 7 : D ¼ 0, D 6¼ 0, D ¼ 0, D 6 þ D 2 ¼ 0, 6 6¼ 0g, U 12 ¼fLðDÞ2FLeib 7 : D ¼ 0, D 6¼ 0, D ¼ 0, D 6 þ D 2 ¼ 0, 6 ¼ 0g,

12 Linear and Multilinear lgebra 21 U 1 ¼fLðDÞ2F Leib 7 : D ¼ 0, D ¼ 0, D 6¼ 0, D 6 ¼ 0, 6 6¼ 0g, U 1 ¼fLðDÞ2F Leib 7 : D ¼ 0, D ¼ 0, D 6¼ 0, D 6 ¼ 0, 6 ¼ 0g, U 1 ¼fLðDÞ2F Leib 7 : D ¼ 0, D ¼ 0, D ¼ 0, D 6 6¼ 0, 6 6¼ 0g, U 16 ¼fLðDÞ2F Leib 7 : D ¼ 0, D ¼ 0, D ¼ 0, D 6 6¼ 0, 6 ¼ 0g, U 17 ¼fLðDÞ2F Leib 7 : D ¼ 0, D ¼ 0, D ¼ 0, D 6 ¼ 0, 6 6¼ 0g, U 18 ¼fLðDÞ2F Leib 7 : D ¼ 0, D ¼ 0, D ¼ 0, D 6 ¼ 0, 6 ¼ 0g: It is clear that these subsets are disjoint. The following propositions show that each of the subsets U 1 U 7 is a union of infinitely many orbits. The second part of each proposition gives a parametrization of the orbits. PROPOSITION.7 (i) Two algebras L(D) and L(D 0 ) from U 1 are isomorphic if and only if D ðd þ D D Þ ¼ D0 ðd0 þ D0 D0 Þ, ð12þ D 2 D 2 ðd 6 þ 6D D þ 9D 2 D þ D 2 Þ D D 02 2 ¼ D0 ðd0 6 þ 6D0 D0 þ 2 9D0 D0 þ 2 D0 Þ D 0, ð1þ D 6 ¼ D0 D D : ð1þ (ii) Orbits in U 1 can be parametrized as L(1, 1, 1, 2, ), 1, 2, 2 C. Proof (i) ): Let L(D) and L(D 0 ) be isomorphic. Then, due to Theorem (.), there are complex numbers and B: ( þ B) 6¼ 0, such that the action of the adapted group G ad can be expressed by the system as (11). Now, if we substitute the value of D 0, D0, D0 þ D0 D0, D0 6 þ 6D0 D0 þ 9D 02 D0 þ D02 and 0 6 to the expressions D0 ðd0 þd0 D0 Þ D 0 2, D0 2 ðd0 6 þ6d0 D0 þ9d0 2 D0 þd0 2 Þ D 0 and D0 0 D 0 6 we will get the required equalities. (: Let the equalities (12), (1) and (1) hold. We put! 0 ¼ D, B 0 ¼ D D D D D 2 1 ð1þ and 0 0 ¼ D0 D 0, B 0 0 ¼ D0 D 0 Then, D 0 ¼ ð 1 0, B 0 0, DÞ and D 0 0 ¼ ð Section ), where D 0 ¼ L 1, 1, D ðd þ D D Þ D 2! D 0 D 02 1 : ð16þ, B0 0, D 0 Þ (see the convention in 0 0, D2 ðd 6 þ 6D D þ 9D 2 D þ D 2 Þ D,! D 6 D

13 216 I.S. Rakhimov and S.K. Said Husain and D 0 0 ¼ L 1, 1, D0 ðd0 þ D0 D0 Þ 2, D0 D 0 2 ðd0 6 þ 6D0 D0 þ 2 9D0 D0 þ 2 D0 Þ D 0!, 0 : Then, the equalities (12), (1) and (1) imply that D 0 ¼ D 0 0. Now, we make use of the properties of and find the complex numbers and B such that ( þ B) 6¼ 0: D 0 D 0 6 ¼ 0 0, B ¼ B B ð0 0 þ B0 0 Þ : ð17þ Thus we get ¼ D0 D D D 0, B ¼ D0 D D D 0! D 02 D D 2 1 D0 : ð18þ For the given and B, we get the corresponding system of Equation (11): 1 1 þ B D ¼ D 0, 1 1 þ B þ B D ¼ D 0, 1 1 þ B ðd þ D D Þ¼D 0 þ D0 D0, ðd 6 þ 6D D þ 9D 2 D þ D 2 Þ¼D0 6 þ 6D0 D0 þ 9D0 2 D0 þ D0 2, 1 6 ¼ 0 6, meaning that L(D) and L(D 0 ) are isomorphic. (ii) It is easy to see that, for any 1, 2, 2 C, there exists an algebra L(D, D, D, D 6, 6 ) from U 1, such that 1 ¼ D ðd þd D Þ, D 2 2 ¼ D2 ðd 6þ6D D þ9d 2 D þd 2 Þ 6 D and ¼ D D : g PROPOSITION.8 (i) Two algebras L(D) and L(D 0 ) from U 2 are isomorphic if and only if D D ðd 6 þ 6D D Þ 2 ¼ D 0 D 0 ðd0 6 þ, ð19þ 6D0 D0 Þ2 D D 0 6 ¼ D 6 þ 6D D D 0 6 þ 0 6D0 D0 6 : ð20þ (ii) Orbits in U 2 can be represented as L(1, 0, 1, 1, 2 ), 1 2 C *, 2 2 C.

14 Linear and Multilinear lgebra 217 PROPOSITION.9 (i) Two algebras L(D) and L(D 0 ) from U are isomorphic if and only if D ¼ D0 D D : ð21þ (ii) Orbits in U can be parametrized as L(1, 0, 1, 6, ), 2 C. PROPOSITION.10 (i) Two algebras L(D) and L(D 0 ) from U are isomorphic if and only if D 6 D 6 ¼ D0 6 D (ii) Orbits in U can be parametrized as L(1, 0, 0,, ), 2 C *. PROPOSITION.11 : ð22þ (i) Two algebras L(D) and L(D 0 ) from U are isomorphic if and only if D ðd 6 þ D 2 Þ ¼ D0 ðd0 6 þ D02 Þ, ð2þ D 2 D 02 D 6 ¼ D0 0 6 D : ð2þ (ii) Orbits in U can be parametrized as L(0, 1, 1, 1, 2 ), 1, 2 2 C. PROPOSITION.12 (i) Two algebras L(D) and L(D 0 ) from U 6 are isomorphic if and only if D D 6 þ D 2! 2 6 ¼ D 0! 2 D 0 D 0 6 þ : ð2þ D0 (ii) Orbits in U 6 can be parametrized as L(0, 1, 0, 0, ), 2 C. PROPOSITION.1 (i) Two algebras L(D) and L(D 0 ) from U 7 are isomorphic if and only if D 6 ¼ D0 0 6 D : ð26þ 6 (ii) Orbit in U 7 can be parametrized as L(0, 0, 1, 1, ), 2 C. PROPOSITION.1 The subsets U 8, U 9, U 10, U 11, U 12, U 1, U 1, U 1, U 16, U 17 and U 18 are single orbits with the representatives L(1, 0, 0, 1, 0), L(1, 0, 0, 0, 1), L(1, 0, 0, 0, 0), L(0, 1, 0,, 1), L(0, 1, 0,, 0), L(0, 0, 1, 0, 1), L(0, 0, 1, 0, 0), L(0, 0, 0, 1, 1), L(0, 0, 0, 1, 0), L(0, 0, 0, 0, 1) and L(0, 0, 0, 0, 0), respectively. D 0 6

15 218 I.S. Rakhimov and S.K. Said Husain Proof The appropriate base changes leading to the representatives are indicated below: U 8 : For L(D, D, D, D 6, 6 ) 2 U 8, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð1, 0, 0, 1, 0Þ, qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi D where ¼ 6 D 1D and B ¼ 6 D 1D ð 6 1 1D D 1Þ. U 9 : For L(D, D, D, D 6, 6 ) 2 U 9, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð1, 0, 0, 0, 1Þ, qffiffiffiffi qffiffiffiffi qffiffiffiffi where ¼ 6 1 and B ¼ 6 1ð D 1Þ: U 10 : For L(D, D, D, D 6, 6 ) 2 U 10, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð1, 0, 0, 0, 0Þ, where is any nonzero complex number and B ¼ ð D 1Þ. U 11 : For L(D, D, D, D 6, 6 ) 2 U 11, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 1, 0,, 1Þ, qffiffiffiffi qffiffiffiffi qffiffiffiffi where ¼ 6 and B ¼ 6 ð 6 1 D 1Þ. U 12 : For L(D, D, D, D 6, 6 ) 2 U 12, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 1, 0,, 0Þ, where is any nonzero complex number and B ¼ ð 2 D 1Þ. U 1 : For L(D, D, D, D 6, 6 ) 2 U 1, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 0, 1, 0, 1Þ, where ¼ p ffiffiffiffiffiffi ffiffiffiffiffiffi p ffiffiffiffi 6 and B ¼ p 6ð 6 D 1Þ. U 1 : For L(D, D, D, D 6, 6 ) 2 U 1, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 0, 1, 0, 0Þ, where is any nonzero complex number and B ¼ ð D 1Þ. U 1 : For L(D, D, D, D 6, 6 ) 2 U 1, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 0, 0, 1, 1Þ, where ¼ p ffiffiffiffiffiffiffiffiffiffi 6 and B ¼ p ffiffiffiffiffiffiffiffiffiffi 6 ð 6 D 6 1Þ.

16 Linear and Multilinear lgebra 219 U 16 : For L(D, D, D, D 6, 6 ) 2 U 16, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 0, 0, 1, 0Þ, where is any nonzero complex number and B ¼ ð D 6 1Þ. U 17 : For L(D, D, D, D 6, 6 ) 2 U 17, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 0, 0, 0, 1Þ, where ¼ p ffiffiffiffiffiffi ffiffiffiffiffiffi 6 and B is any complex number except for p 6. U 18 : For L(D, D, D, D 6, 6 ) 2 U 18, 1, B : LðD, D, D, D 6, 6 Þ ¼ Lð0, 0, 0, 0, 0Þ, where is any nonzero complex number and B is any complex number except for. g THEOREM.6 Let L be an element of F Leib 7. Then, it is isomorphic to one of the following pairwise non-isomorphic Leibniz algebras: ð1þ Lð0, 0, 0, 0, 0Þ ¼L s 7 : e 0e 0 ¼ e 2, e i e 0 ¼ e iþ1,1i. (2) L(0, 0, 0, 0, 1): L s 7, e 0e 1 ¼ e 6 : () L(0, 0, 0, 1, 0): L s 7, e 0e 1 ¼ e 6, e 1 e 1 ¼ e 6 : () L(0, 0, 0, 1, 1): L s 7, e 0e 1 ¼ 2e 6, e 1 e 1 ¼ e 6 : () L(0, 0, 1, 0, 0): L s 7, e 0e 1 ¼ e, e 1 e 1 ¼ e, e 2 e 1 ¼ e 6 : (6) L(0, 0, 1, 0, 1): L s 7, e 0e 1 ¼ e þ e 6, e 1 e 1 ¼ e, e 2 e 1 ¼ e 6 : (7) L(0, 1, 0,, 1): L s 7, e 0e 1 ¼ e 2e 6, e 1 e 1 ¼ e e 6, e 2 e 1 ¼ e, e e 1 ¼ e 6 : (8) L(0, 1, 0,, 0): L s 7, e 0e 1 ¼ e e 6, e 1 e 1 ¼ e e 6, e 2 e 1 ¼ e, e e 1 ¼ e 6 : (9) L(1, 0, 0, 1, 0): L s 7, e 0e 1 ¼ e 2e 6, e 1 e 1 ¼ e 1e 6, e 2 e 1 ¼ e, e e 1 ¼ e,e e 1 ¼ e 6 : (10) L(0, 0, 1, 1, ): L s 7, e 0e 1 ¼ e þðþ1þe 6, e 1 e 1 ¼ e þ e 6, e 2 e 1 ¼ e 6, 2 C: (11) L(0, 1, 0, 0, ): L s 7, e 0e 1 ¼ e þ e 6, e 1 e 1 ¼ e, e 2 e 1 ¼ e, e e 1 ¼ e 6, 2 C: (12) L(0, 1, 1, 1, 2 ): L s 7, e 0e 1 ¼ e þ e þð 1 þ 2 Þe 6, e 1 e 1 ¼ e þ e þ 1 e 6, e 2 e 1 ¼ e þ e 6, e e 1 ¼ e 6, 1, 2 2 C. (1) L(1, 0, 0,, ): L s 7, e 0e 1 ¼ e þð2 1Þe 6, e 1 e 1 ¼ e þð 1Þe 6, e 2 e 1 ¼ e, e e 1 ¼ e, e e 1 ¼ e 6, 2 C. (1) L(1, 0, 1, 6, ): L s 7, e 0e 1 ¼ e þ 6e þð 20Þe 6, e 1 e 1 ¼ e þ e 20e 6, e 2 e 1 ¼ e þ 6e 6, e e 1 ¼ e, e e 1 ¼ e 6, 2 C. (1) L(1, 0, 1, 1, 2 ): L s 7, e 0e 1 ¼ e þð 1 þ Þe þð 2 1 1Þe 6, e 1 e 1 ¼ e þ ð 1 þ Þe ð 1 þ 1Þe 6, e 2 e 1 ¼ e þ ( 1 þ )e 6, e e 1 ¼ e, e e 1 ¼ e 6, 1, 2 2 C. (16) L(1, 1, 1, 2, ): L s 7, e 0e 1 ¼ e e þð 1 þ Þe þð þ 2 1Þe 6, e 1 e 1 ¼ e e þ ( 1 þ )e þ ( 2 1)e 6, e 2 e 1 ¼ e e þ ( 1 þ )e 6, e e 1 ¼ e e, e e 1 ¼ e 6, 1, 2, 2 C. Note.2 The orbits U 9 and U 10 with the representatives L(1, 0, 0, 0, 1) and L(1, 0, 0, 0, 0) can be included in the parametric family of orbits with representatives L(1, 0, 1, 1, 2 ) and L(1, 0, 0,, ) with the values of parameters 1 ¼ 0, 2 ¼ 1 and ¼ 0, respectively.

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