Solvable Leibniz algebras with NFn⊕ F1m nilradical

Open Access. © 2017 Camacho et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivs 4.0 License. Open Math. 2017; 15: 1371–1388 Open Mathematics Research Article L.M. Camacho, B.A. Omirov, K.K. Masutova, and I.M. Rikhsiboev* Solvable Leibniz algebras with NFnLF1 m nilradical https://doi.org/10.1515/math-2017-0115 Received June 5, 2017; accepted October 4, 2017. Abstract: All finite-dimensional solvable Leibniz algebras L, having NDNFnLF1 mas the nilradical and the dimension of Lequal to nCmC3(the maximal dimension) are described. NFnand F1 mare the null-filiform and naturally graded filiform Leibniz algebras of dimensions nand m; respectively. Moreover, we show that these algebras are rigid. Keywords: Lie algebra, Leibniz algebra, Natural gradation, Null-filiform algebra, Filiform algebra, Solvability, Nilpotence, Nilradical MSC: 17A32, 17A65, 17B30 1 Introduction Leibniz algebras over Kwere first introduced by A. Bloh [1] and called D-algebras. The term Leibniz algebra was introduced in the study of a non-antisymmetric analogue of Lie algebras by Loday [2], being so the class of Leibniz algebras an extension of the one of Lie algebras. In recent years it has been common theme to extend various results from Lie algebras to Leibniz algebras [3,4]. Many results of the theory of Lie algebras have been extended to Leibniz algebras. For instance, the classical results on Cartan subalgebras [5], variations of Engel’s theorem for Leibniz algebras have been proven by different authors [6,7] and Barnes has proven Levi’s theorem for Leibniz algebras [8]. In an effort to classify Lie algebras, many authors place various restrictions on the nilradical. The first work which was devoted to description of such Lie algebras is the paper [9]. Later, Mubarakjanov proposed the description of solvable Lie algebras with a given structure of nilradical by means of outer derivations [10]. In the papers [11-14], the authors apply the Mubarakjanov’s method to classify the solvable Lie algebras with different kinds of nilradicals. Some results of Lie algebra theory generalized to Leibniz algebras in [3] allow us to apply the Mubarakjanov’s method for Leibniz algebras. In this sense, we can see the papers [15-18]. It is important to study solvable Leibniz algebras because thanks to the Levi’s theorem for Leibniz algebras, a Leibniz algebra is a semidirect sum of the solvable radical and a semisimple Lie algebra. As the semisimple part can be described by simple Lie ideals, the main problem is to understand the solvable radical. L.M. Camacho: Dpto. Matemática Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012 Sevilla, Spain, E-mail: [email protected] B.A. Omirov: Institute of Mathematics. National University of Uzbekistan, 29, Do’rmon yo’li street., 100125, Tashkent, Uzbekistan, E-mail: [email protected] K.K. Masutova: Institute of Mathematics. National University of Uzbekistan, 29, Do’rmon yo’li street., 100125, Tashkent, Uzbekistan, E-mail: [email protected] *Corresponding Author: I.M. Rikhsiboev: Universiti Kuala Lumpur, Malaysian Institute of Industrial Technology, Bandar Seri Alam, 81750 Johor Bahru, Malaysia, E-mail: [email protected] Unauthenticated Download Date | 3/20/20 9:15 AM 1372 L.M. Camacho et al. The first aim of the present paper is to classify solvable Leibniz algebras with nilradical NDNFn˚F1 m where NFnand F1 mare the null-filiform and naturally graded filiform Leibniz algebras of dimensions nand m; respectively. To obtain this classification, we use the results obtained in [16-18]. The arrangement of this work is as follows. In Section 2 we recall some essential notions and properties of Leibniz algebras. We start Section 3 by establishing the maximal dimension of a solvable Leibniz algebra whose nilradical is NDNFn˚F1 m; thereafter, we present the classification of solvable Leibniz algebras that can be decomposed as a direct sum of their nilradical and a complementary vector space of maximal dimension. Finally, in Section 4 we study the rigidity of the unique solvable Leibniz algebra obtained in the previous section. Throughout the paper, we consider finite-dimensional vector spaces and algebras over a field of characteristic zero. Moreover, in the multiplication table of an algebra omitted products are assumed to be zero and if it is not noticed we shall consider non-nilpotent solvable algebras. 2 Preliminaries Let us recite some necessary definitions and preliminary results. ALeibniz algebra over a field Fis a vector space Lequipped with a bilinear map, called bracket, Œ;WLL!L; satisfying the Leibniz identity x; Œy; zDŒx; y; zŒx; z; y for all x; y; z 2L. The set Annr.L/ D fx2LjŒy; x D0; y 2Lgis called the right annihilator of the Leibniz algebra L. Note that Annr.L/ is an ideal of Land for any x; y 2Lthe elements Œx; x,Œx; y CŒy; x 2Annr.L/. A linear map dWL!Lof a Leibniz algebra .L; Œ;/ is said to be a derivation if for all x; y 2Lthe following condition holds: d.Œx; y/ DŒd.x/; y CŒx; d.y/: For a given element xof a Leibniz algebra Lthe operator of right multiplication RxWL!L, defined as Rx.y/ D Œy; x for y2L, is a derivation. This kind of derivations are called inner derivations. Any Leibniz algebra Lis associated with the algebra of right multiplications R.L/ D fRxjx2Lg, which is endowed with a structure of Lie algebra by means of the bracket ŒRx; RyDRxRyRyRx:Thanks to the Leibniz identity the equality ŒRx; RyDRŒy;x holds true. In addition, the algebra R.L/ is antisymmetric isomorphic to the quotient algebra L=Annr.L/. Definition 2.1 ([10]).Let d1; d2; : : : ; dnbe derivations of a Leibniz algebra L: The derivations d1; d2; : : : ; dn are said to be nil-independent if ˛1d1C˛2d2C    C ˛ndn is not nilpotent for any scalars ˛1; ˛2; : : : ; ˛n2F: In other words, if for any ˛1; ˛2; : : : ; ˛n2Fthere exists a natural number ksuch that .˛1d1C˛2d2C    C ˛ndn/kD0; then ˛1D˛2D    D ˛nD0: 2.1 Solvable Leibniz algebras For a Leibniz algebra Lwe consider the following lower central and derived series: L1DL LkC1DŒLk; L1; k 1I LŒ1 DL; LŒsC1 DŒLŒs; LŒs; s 1: Unauthenticated Download Date | 3/20/20 9:15 AM Solvable Leibniz algebras with NFnLF1 mnilradical 1373 A Leibniz algebra Lis said to be nilpotent (respectively, solvable), if there exists n2N(m2N) such that LnD0 (respectively, LŒm D0). It should be noted that the sum of any two nilpotent ideals is nilpotent. The maximal nilpotent ideal of a Leibniz algebra is said to be a nilradical of the algebra. Obviously, the index of nilpotency of an n-dimensional nilpotent Leibniz algebra is not greater than nC1. The following theorem describes these algebras, algebras with maximal index of nilpotency. Theorem 2.2 ([4]).Any n-dimensional null-filiform Leibniz algebra admits a basis fe1; e2; :::; engsuch that the table of multiplication in the algebra has the following form: NFnWŒei; e1DeiC1; 1 in1 A Leibniz algebra Lis said to be filiform if dim LiDni, where nDdim Land 2in. Due to [4] and [19] it is known that there are three naturally graded filiform Leibniz algebras. In fact, the third type encloses the class of naturally graded filiform Lie algebras. Theorem 2.3 ([17]).Any complex n-dimensional naturally graded filiform Leibniz algebra is isomorphic to one of the following pairwise non isomorphic algebras: F1 nWŒei; e1DeiC1; 2 in1; F2 nWŒei; e1DeiC1; 1 in2; F3 n.˛/ W(Œei; e1D Œe1; eiDeiC1; 2 in1; Œei; enC1iD ŒenC1i; eiD˛.1/iC1en; 2 in1; where ˛2 f0; 1gfor even nand ˛D0for odd n. The following theorems describe solvable Leibniz algebras of maximal dimension with NF 1 nand F1 nnilradical. Theorem 2.4 ([17]).Let Rbe a solvable Leibniz algebra whose nilradical is NFn. Then there exists a basis fe1; e2; : : : ; en; xgof the algebra Rsuch that the multiplication table of Rwith respect to this basis has the following form: Œei; e1DeiC1; 1 in1; Œei; x Diei; 1 in; Œx; e1D e1: Theorem 2.5 ([18]).An arbitrary .n C2/-dimensional solvable Leibniz algebra with nilradical F1 nis isomorphic to the algebra R.F 1 n/with the multiplication table: Œei; e1DeiC1; 2 in1; Œe1; x De1; Œei; y Dei; 2 in; Œei; x D.i 1/ei; 2 in; Œx; e1D e1: Let Rbe a solvable Leibniz algebra. Then it can be decomposed in the form RDN˚Q; where Nis the nilradical and Qis the complementary vector space. Since the square of a solvable Leibniz algebra is contained into the nilradical [3], we get the nilpotency of the ideal R2and consequently, Q2N: Let us consider the restrictions to Nof the right multiplication operator on an element x2Q(denoted by RxjN). From [17], we know that for any x2Q; the operator RxjNis a non-nilpotent derivation of N. Let fx1; x2; : : : ; xmgbe a basis of Q; then for any scalars f˛1; : : : ; ˛mg 2 Cnf0g;the matrix ˛1Rx1jNC  C˛mRxmjN is non nilpotent, which means that the elements fx1; : : : ; xmgare nil-independent derivations, [10]. Theorem 2.6 ([17]).Let R be a solvable Leibniz algebra and Nbe its nilradical. Then the dimension of the complementary vector space to N is not greater than the maximal number of nil-independent derivations of N. Unauthenticated Download Date | 3/20/20 9:15 AM 1374 L.M. Camacho et al. Moreover, similarly as in Lie algebras, for a solvable Leibniz algebra R; we have dimN dimR 2: A nilpotent Leibniz algebra is called characteristically nilpotent if all its derivations are nilpotent. If the nilradical Nof a Leibniz algebra is characteristically nilpotent then, according to Theorem 2.6, a solvable Leibniz algebra is nilpotent. Therefore, we shall consider solvable Leibniz algebras with non-characteristically nilpotent nilradical. For more details see [17]. 2.2 The second cohomology group of a Leibniz algebra For acquaintance with the definition of cohomology group of Leibniz algebras and its applications to the description of the variety of Leibniz algebras (similar to the Lie algebras case) we refer the reader to the papers [2, 20-24]. Here we just recall that the second cohomology group of a Leibniz algebra Lwith coefficients in a corepresentation Mis the quotient space HL2.L; M / DZL2.L; M /=BL2.L; M /; where the 2-cocycles '2ZL2.L; M / and the 2-coboundaries f2BL2.L; M / are defined as follows .d2'/.a; b; c/ DŒa; '.b; c/ Œ'.a; b/; c CŒ'.a; c/; b C'.a; Œb; c/ '.Œa; b; c/ C'.Œa; c; b/ D0(1) and f .a; b/ DŒd.a/; b CŒa; d.b/ d.Œa; b/ for some linear map d: The linear reductive group GLn.F/acts on the variety of n-dimensional Leibniz algebras, Leibn;as follows: .g /.x; y/ Dg..g1.x/; g1.y///; g 2GLn.F/;  2Leibn: The orbits .O rb.// under this action are the isomorphism classes of algebras. Note that, Leibniz algebras with open orbits are called rigid. Due the work [20], the nullity of the second cohomology group with coefficients itself gives a sufficient condition for the rigidity of the algebras. 3 Main results Let NFnbe an n-dimensional null-filiform Leibniz algebra with a basis fe1; e2; : : : ; engand F1 man m-dimensional filiform Leibniz algebra from the first class with a basis ff1; f2; : : : ; fmg, then we have the following multiplication: NFnWŒei; e1DeiC1; 1 in1IF1 mW8 < : Œf1; f1Df3; Œfi; f1DfiC1; 2 im1: Let us consider the direct sum of these algebras NDNFnLF1 m:The following proposition describes derivations of the algebra N. Proposition 3.1. Any derivation of the algebra NDNFnLF1 mhas the following matrix form: 0 B B B B B B B B B B B B B B B B B B @ ˛1˛2˛3: : : ˛n0 0 0 : : : 0 ˇm 0 2˛1˛2: : : ˛n10 0 0 : : : 0 0 0 0 3˛1: : : ˛n20 0 0 : : : 0 0 : : :: : :: : ::::: : :: : :: : :: : :: : :: : :: : : 0 0 0 : : : n˛n0 0 0 0 0 0 0 0 0 : : : nı1ı2ı3: : : ım1ım 0 0 0 : : : n0 ı1Cı2ı3: : : ım1m 0 0 0 : : : 0 0 0 2ı1Cı2: : : ım2ım1 : : :: : :: : ::::: : :: : :: : :: : :: : :: : :: : : 0 0 0 : : : 0 0 0 0 0 0 .m 1/ı1Cı2 1 C C C C C C C C C C C C C C C C C C A : Unauthenticated Download Date | 3/20/20 9:15 AM Solvable Leibniz algebras with NFnLF1 mnilradical 1375 Proof. The proof is going by straightforward calculation of derivation property. From this proposition it is easy to see that the number of nil-independent outer derivations of the algebra Nis equal to 3. Now we consider solvable Leibniz algebra RDNCQ; where NDNFnLF1 mand the dimension of Qis no more than three. Thus, we study the case dim Q D3; i.e. dim R DnCmC3: Several papers described solvable Leibniz algebras with a given nilradical [15-17]. The most interesting cases are when the complementary space of nilradical has the maximum possible. Namely, they have the second group of cohomology trivial. For this reason, we consider the case dimQ D3. From the work [17], it follows that any solvable Leibniz algebra whose nilradical is NFnhas dimension nC1. It is also known that any solvable Leibniz algebra whose nilradical is F1 mhas dimension either mC1or mC2, [16]. In work [18], it was found a unique .m C2/-dimensional solvable Leibniz algebra with nilradical F1 m. Then in the case of the solvable Leibniz algebras Rwith nilradical NDNFnLF1 mand dim Q D3; there is only one possible case. Taking into account Theorems 2.2 and 2.5, we have the following multiplication of the algebra R: Œei; e1DeiC1; 1 in1; Œfi; f1DfiC1; 2 im1; Œei; x Diei; 1 in; Œf1; y Df1; Œx; e1D e1; Œfi; y D.i 1/fi; 2 im; Œfi; z Dfi; 2 im; Œy; f1D f1; .2/ where fx; y; zgbe a basis of the space Q. In the following theorem, solvable Leibniz algebras with nilradical NDNFnLF1 mand dim Q D3are described. Theorem 3.2. Any .nCmC3/-dimensional solvable Leibniz algebra with nilradical NDNFnLF1 mis isomorphic to the following algebra: LW 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ : Œei; e1DeiC1; 1 in1; Œfi; f1DfiC1; 2 im1; Œei; x Diei; 1 in; Œf1; y Df1; Œx; e1D e1; Œfi; y D.i 1/fi; 2 im; Œfi; z Dfi; 2 im; Œy; f1D f1: Proof. From the above argumentations we have the multiplication (2) and we introduce the following denotations for the algebra R(according to the Mubarakzjanov’s method [10]): Œx; f1D n P iD1 ˛1 ieiC m P iD1 ˇ1 ifi; Œx; y D n P iD1 1 ieiC m P iD1 1 ifi; Œx; f2D n P iD1 ˛2 ieiC m P iD1 ˇ2 ifi; Œy; x D n P iD1 2 ieiC m P iD1 2 ifi; Œy; e1D n P iD1 1 ieiC m P iD1 ı1 ifi; Œx; z D n P iD1 3 ieiC m P iD1 3 ifi; Œz; e1D n P iD1 2 ieiC m P iD1 ı2 ifi; Œz; x D n P iD1 4 ieiC m P iD1 4 ifi: Œfi; x D n P jD1 aij ejC m P jD1 bij fj; 1 im; Œei; y D n P jD1 c1 ij ejC m P jD1 d1 ij fj; 1 in; Œei; z D n P jD1 c2 ij ejC m P jD1 d2 ij fj; 1 in; Unauthenticated Download Date | 3/20/20 9:15 AM 1376 L.M. Camacho et al. From Leibniz identity it follows that Œy; e2DŒy; Œe1; e1 D0and by induction we can easily find that Œy; eiD0; with 2in: Analogously, we have Œz; eiD0; with 2in: We consider Œx; f3DŒx; Œf2; f1 DŒŒx; f2; f1D m P iD3 ˇ2 i1fi:Similarly, and using the induction method, it is possible to show that Œx; fiD m P jDi ˇ2 jiC2fjfor 3im: However, from the equality Œx; Œf3; y DŒŒx; f3; y; it follows ˇ2 iD0; with 2im1; i.e. Œx; fiD0; 3 im; Œx; f2D n P iD1 ˛2 ieiCˇ2 1f1Cˇ2 2f2Cˇ2 mfm: Taking the following change x0Dx1 1f1 m P iD2 1 i11 ifi;we have Œx0; y D n P iD1 1 iei: Next, making y0Dy n P iD1 1 i2 iei;we obtain Œy0; x D m P iD1 2 ifi: Finally, the change of basis z0Dz n P iD1 1 i4 ieiallows to obtain Œz0; x D m P iD1 4 ifi: Let us apply the Leibniz identity on the following triples of elements: fx; x; yg H) 1 iD0; 2 inH) Œx; y D1 1e1I fy; y; xg H) 2 iD0; 2 imH) Œy; x D2 1f1I fz; y; xg H) 4 iD0; 1 imH) Œz; x D0: We observe that Œx; y CŒy; x 2Annr.R/, thus Œe1; 1 1e1C1 1f1D0, i.e. 1 1D0. From the identity 0D Œf2; Œx; y CŒy; x DŒf2; 1 1f1we have 1 1D0: Analogously, Œe1; y CŒy; e1and Œe1; z CŒz; e12Annr.R/, then, Œe1; Œe1; y CŒy; e1 D0and Œf2; Œe1; y CŒy; e1 D0; so we have c1 11 D 1 1; d1 11 D ı1 1and c2 11 D 2 1; d2 11 D ı2 1. Similarly, from Œe1; Œx; z CŒz; x D0and Œf2; Œx; z CŒz; x D0we deduce 3 1D3 1D0: Considering the following equalities: Œx; f2DŒx; Œf2; y D Œx; Œy; f2 D0)ˇ2 1Dˇ2 2Dˇ2 mD0; ˛2 iD0; 1 in; Œx; Œe1; f1 D ŒŒx; f1; e1D0)˛1 iD0; 1 in1; Œy; Œe1; f1 DŒz; Œe1; f1 D0)ı1 iDı2 iD0; 2 im1; we immediately get Œx; f1D˛1 nenC m P iD1 ˇ1 ifi; Œy; e1D n P iD1 1 ieiCı1 1f1Cı1 mfm; Œz; e1D n P iD1 2 ieiCı2 1f1Cı2 mfm: As a result of Œx; f1CŒf1; x 2Annr.R/; we observe that a11 D0and b11 D ˇ1 1: Since, Œf1; Œx; f1 D0and Œf1; Œx; e1 D0it follows that a1;i Db1;j D0; for 1in1; 2 jm1; i.e. Œf1; x Da1;nenˇ1 1f1Cb1;mfm; and from the equalities Œfi; Œx; e1 D0; for 2im; it follows that ai;j D0; for 2imand 1jn1, that is Œfi; x Dai;nenC m X jD1 bij fj; 2 im: Unauthenticated Download Date | 3/20/20 9:15 AM Solvable Leibniz algebras with NFnLF1 mnilradical 1377 We know that Œx; fiCŒfi; x 2Annr.R/ for 2im, then 0DŒf2; Œx; fiCŒfi; x Dbi1f3, that is bi1 D0 for 2im: Now, we proceed by looking at the product of certain elements of Annr.R/: Summarizing the following identities Œe1; Œy; eiCŒei; y DŒf2; Œy; eiCŒei; y DŒe1; Œz; eiCŒei; z DŒf2; Œz; eiCŒei; z D0; we have the following Leibniz brackets of the basic elements: .3/ 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : Œei; e1DeiC1; 1 in1; Œfi; f1DfiC1; 2 im1; Œei; x Diei; 1 in; Œf1; y Df1; Œx; e1D e1; Œfi; y D.i 1/fi; 2 im; Œfi; z Dfi; 2 im; Œy; f1D f1; Œx; f1D˛1 nenC m P iD1 ˇ1 ifi; Œf1; x Da1;nenˇ1 1f1Cb1;mfm; Œy; e1D n P iD1 1 ieiCı1 1f1Cı1 mfm; Œfi; x Dai;nenC m P jD2 bij fj; 2 im; Œz; e1D n P iD1 2 ieiCı2 1f1Cı2 mfm; Œe1; y D 1 1e1C n P jD2 c1 1j ejı1 1f1C m P jD2 d1 1j fj; Œx; z D n P iD2 3 ieiC m P iD2 3 ifi; Œei; y D n P jD2 c1 ij ejC m P jD2 d1 ij fj; 2 in; Œe1; z D 2 1e1C n P jD2 c2 1j ejı2 1f1C m P jD2 d2 1j fj; Œei; z D n P jD2 c2 ij ejC m P jD2 d2 ij fj; 2 in: The Leibniz identity on the following triples imposes futher constraints on .3/. Leibniz identity Constraint ffm; x; f1g ) bm;j D0; 2 jm1 ffm1; x; f1g ) am;n D0; bm1;m1Dˇ1 1Cbm;m; bm1;i D0; 2 im2 As a result of the above constraints we observe that Œfm; x Dbmmfm; Œfm1; x Dam1;nenC.ˇ1 1Cbm;m/fm1Cbm1;mfm: By the induction on decrease i .2 im/ and using the Leibniz identity for the elements ffi; x; f1gwe get Œfi; x D..m i/ˇ1 1Cbmm/fiC m P jDiC1 bijCm;mfj; 3 im; Œf2; x Da2;nenC..m 2/ˇ1 1Cbmm/f2C m P iD3 bmiC2;mfi: Considering the Leibniz identity on the triple fe1; y; e1gwe get Œe2; y D 21 1e2C n P iD3 c1 1;i1ei: Also, using the equalities Œei; y DŒŒei1; e1; y DŒŒei1; y; e1Œei1; Œy; e1; for 3inand by induction method on i; we obtain that Œei; y D i1 1eiC n X jDiC1 c1 1;j iC1ej; 2 in: Analogously, we can get Œei; z D i2 1eiC n P jDiC1 c2 1;j iC1ej;for 2in: Unauthenticated Download Date | 3/20/20 9:15 AM 1378 L.M. Camacho et al. The Leibniz identity on the triples ff2; x; yg;ff1; x; ygand fx; f1; yg;(in this order) gives Œf1; x Da1;nenˇ1 1f1; Œf2; x Da2;nenC..m 2/ˇ1 1Cbmm/f2; Œfi; x D..m i/ˇ1 1Cbmm/fi; 3 im; Œx; f1D˛1 nenCˇ1 1f1Cˇ1 2f2: Now, applying the Leibniz identity on the triples fx; f1; zg;fy; x; f1g;ffi; x; zg;ffi; x; yg;with 1im; fei; z; xg;fei; y; xg;with 1in; (in this order) it follows ˛1 n2 1Dˇ1 2D3 iD0; 2 im1; a1n D0; c1 1i Dc2 1i D0; 2 in1; a2n.n2 1C1/ D0: From the above results we can simplify the family .3/ as follows: .4/ W 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : Œei; e1DeiC1; 1 in1; Œfi; f1DfiC1; 2 im1; Œei; x Diei; 1 in; Œf1; y Df1; Œx; e1D e1; Œfi; y D.i 1/fi; 2 im; Œfi; z Dfi; 2 im; Œy; f1D f1; Œx; f1D˛1 nenCˇ1 1f1; Œf1; x D ˇ1 1f1; Œf2; x Da2nenC..m 2/ˇ1 1Cbmm/f2; Œy; e1D n P iD1 1 ieiCı1 1f1Cı1 mfm; Œfi; x D..m i/ˇ1 1Cbmm/fi; 3 im; Œz; e1D n P iD1 2 ieiCı2 1f1Cı2 mfm; Œe1; y D 1 1e1Cc1 1nenı1 1f1C m P jD2 d1 1j fj; Œx; z D n P iD2 3 ieiC3 mfm; Œei; y D i1 1ei; 2 in Œe1; z D 2 1e1Cc2 1nenı2 1f1C m P jD2 d2 1j fj; Œei; z D i2 1ei; 2 in Finally, considering the Leibniz identity on the following triples we obtain: Leibniz identity Constraint fx; z; e1g H) ı2 1.ˇ1 1C1/ D0; 3 jD0; 2 jn2; 3 n1Dı2 1˛1 nc2 1n; d2 1i D0; 2 im; fx; y; e1g H) c1 1n Dı1 1˛1 n; ı1 1.ˇ1 1C1/ D0; d1 1i D0; 2 im; fx; z; yg H) 3 mD0; fx; z; xg H) 3 nD0; c2 1n Dı2 1˛1 n; fe1; y; xg H) c1 1n D0; fe1; z; xg H) c2 1n D0; fy; e1; xg H) 1 iD0; 2 in; ı1 m.1 bmm/D0; fz; e1; xg H) 2 iD0; 2 in; ı2 m.1 bmm/D0; Unauthenticated Download Date | 3/20/20 9:15 AM Solvable Leibniz algebras with NFnLF1 mnilradical 1379 Leibniz identity Constraint fy; e1; yg H) ı1 m.1 1Cm1/ D0; fz; e1; zg H) ı2 m.2 1C1/ D0; ff2; e1; yg H) ı1 1D0; ff2; e1; zg H) ı2 1D0; fy; e1; zg H) ı1 m.2 1C1/ D0; fz; e1; yg H) ı2 m.1 1Cm1/ D0; fx; f1; xg H) ˛1 n.ˇ1 1Cn/ D0; fx; f1; yg H) ˛1 n.n1 1C1/ D0; fx; f1; zg H) ˛1 n2 1D0: Summarizing the above results and renaming the following parameters a2n Da; ˛1 nD˛; ˇ1 1Dˇ; bmm D b; i 1Di; ıi mDıiwe can simplify the family .4/ as follows: .5/ W 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : Œei; e1DeiC1; 1 in1; Œfi; f1DfiC1; 2 im1; Œx; e1D e1; Œx; f1D˛enCˇf1; Œy; e1D1e1Cı1fm; Œy; f1D f1; Œz; e1D2e1Cı2fm; Œei; x Diei; 1 in; Œei; y D i1ei; 1 in; Œf1; x D ˇf1; Œf1; y Df1; Œf2; x DaenC..m 2/ˇ Cb/f2; Œfi; y D.i 1/fi; 2 im; Œfi; x D..m i/ˇ Cb/fi; 3 im; Œei; z D i2ei; 1 in; Œfi; z Dfi; 2 im with the relations: ık.b 1/ Dık.1Cm1/ Dık.2C1/ D0; k D1; 2; a.n1C1/ Da.n2C1/ D0; ˛.n1C1/ D˛.ˇ Cn/ D˛2D0: Now, by performing a change of basis in the family .5/ x0DxCˇy ..m 1/ˇ Cb/z we have: Œei; e1DeiC1; 1 in1; Œfi; f1DfiC1; 2 im1; Œx; e1D e1Cıfm; Œx; f1D˛en; Œy; e1D1e1Cı1fm; Œy; f1D f1; Œz; e1D2e1Cı2fm; Œei; x Diei1in; Œei; y D i1ei; 1 in; Œf2; x Daen; Œf1; y Df1; Œfi; y D.i 1/fi; 2 im; Œei; z D i2ei; 1 in; Œfi; z Dfi; 2 im Since Rxis not nilpotent, ¤0: Thus we can assume 0D1(making x0D1 x). Unauthenticated Download Date | 3/20/20 9:15 AM 1386 L.M. Camacho et al. .7/ W 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : 2 .nC2/.n1/ n P iD1 ai i;1xanC1 nC1;1yC.anC1 nC1;1 anC2 nC2;1/z; '.e2; x/ Da1 1;1e12a1 nCmC1;1e2C.a2 nCmC1;1 a3 1;1/e3C n P sD4 .as1 1;nCmC1.s 2/as 1;1/esC2 m P sD1 anCs 1;1 fsC C2anCmC1 1;1 xC2anCmC2 1;1 yC2anCmC3 1;1 z; '.ei; x/ D.i 1/.a1 i1;1 anCmC1 i2;1 /e1C i2 P sD2 .i s/s P tD1 at isCt1;1 anCmC1 is1;1esC C.nCi/.niC1/ .nC2/.n1/ i1 P tD1 at t;1 .iC1/.i2/ .nC2/.n1/ n P tDi at t;1ei1ia1 nCmC1;1eiC C.a2 nCmC1;1  i1 P tD1 atC2 t;1 /eiC1C n P sDiC2asiC1 1;nCmC1.s i/ i1 P tD1 asiCtC1 t;1 esCi m P sD1 anCs i1;1fsC CianCmC1 i1;1 xCianCmC2 i1;1 yCianCmC3 i1;1 z; 3 in; '.f1; x/ D n1 P iD1 iaiC1 nC1;1eiCnan nCmC2;nC1enanC1 nCmC1;nC1f1; '.f2; x/ D n1 P iD1 iaiC1 nC2;1eiCan nC2;nCmC1enCanC2 nC2;nCmC1f2CanC3 nC2;nCmC1f3; '.fi; x/ D n1 P sD1 sasC1 nCi;1esnan nCi1;nC1enC.anC2 nC2;nCmC1.i 2/anC1 nCmC1;nC1/fiCanC3 nC2;nCmC1fiC1; 3im; '.x; x/ D n1 P iD2 .iaiC1 nCmC1;1 aiC1 1;nCmC1/eiCan nCmC1;nCmC1en; '.y; x/ D n1 P iD1 iaiC1 nCmC2;1eiCan nCmC2;nCmC1enanC3 nC2;nCmC1f1; '.z; x/ D n1 P iD1 iaiC1 nCmC3;1eiCan nCmC3;nCmC1en; '.e1; y/ D a1 nCmC2;1e1Ca2 nCmC2;1e2CanC3 nC2;1f1C m P iD2 .i 1/anCi nCmC1;1fi; '.ei; y/ D ia1 nCmC2;1eiCa2 nCmC2;1eiC1anC1 i1;1f1 m P sD2 .s 1/anCs i1;1fs; 2 in; '.f1; y/ D  n1 P iD1 asC1 nC1;1esan nCmC2;nC1enanC1 nCmC2;nC1f1C m1 P sD3 .s 2/anCsC1 nC1;nC1fsCanCm nC1;nCmC2fmC Ca1 nC1;1xanCmC2 nCmC2;nC1y1 m1m P tD2 anCt nCt;nC1m.m1/ 2anCmC2 nCmC2;nC1z; '.f2; y/ D  n1 P sD1 asC1 nC2;1es1 nan nC2;nCmC1enC m P sD2 anCs nC2;nCmC2fsCa1 nC2;1xanC1 nC1;nC2yC C.anC1 nC1;nC2anC2 nC2;nC2/z; '.f3; y/ D 2 n1 P sD1 asC1 nC3;1esC2an nC2;nC1enC.anC1 nC2;nC1CanC1 nC1;nC2/f1C C1 m1.m 2/anC2 nC2;nC1 m P tD3 anCt nCt;nC1C.m1/.m2/ 2anCmC2 nCmC2;nC1f2C C.anC2 nC2;nCmC2anC1 nCmC2;nC1/f3C m P sD4anCs1 nC2;nCmC2.s 3/anCs nC2;nC1fsC C2a1 nC3;1xC2anCmC2 nC2;nC1yC2anCmC3 nC2;nC1z; '.fi; y/ D .i 1/ n1 P sD1 asC1 nCi;1esC.i 1/an nCi1;nC1enC.i 2/.anC1 nCi1;nC1anCmC2 nCi2;nC1/f1C C i2 P sD2 .i s/s1 P tD1 anCtC1 nCiCts;nC1fsC CmiC1 m1 i1 P tD2 anCt nCt;nC1i2 m1 m P tDi anCt nCt;nC1C.i2/.miC1/ 2anCmC2 nCmC2;nC1fi1C C.anC2 nC2;nCmC2.i 2/anC1 nCmC2;nC1/fiC m P sDiC1anCsiC2 nC2;nCmC2.s i/ i2 P tD1 anCstC1 nCit;nC1fsC C.i 1/a1 nCi;1xC.i 1/anCmC2 nCi1;nC1yC.i 1/anCmC3 nCi1;nC1z; 4 im; '.x; y/ D a2 nCmC2;1e1CanC3 nC2;nCmC1f1C m1 P iD2 .i 1/anCiC1 nCmC1;nC1fiCanCm nCmC1;nCmC2fm; '.y; y/ D m2 P iD2 .i 1/.anCiC1 nCmC2;nC1anCiC2 nC1;nC1/fiC..m 2/anCm nCmC2;nC1anCm nC1;nCmC2/fm1C CanCm nCmC2;nCmC2fm; Unauthenticated Download Date | 3/20/20 9:15 AM Solvable Leibniz algebras with NFnLF1 mnilradical 1387 .8/ W 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : '.z; y/ DanC1 nCmC3;nCmC2f1C m1 P iD2 .i 1/anCiC1 nCmC3;nC1fiCanCm nCmC3;nCmC2fm; '.e1; z/ D a1 nCmC3;1e1Ca2 nCmC3;1e2C m P iD2 anCi nCmC1;1fi; '.ei; z/ D ia1 nCmC3;1eiCa2 nCmC3;1eiC1 m P sD2 anCs i1;1fs; 2 in; '.f1; z/ D anC1 nCmC3;nC1f1C m1 P sD2 anCsC1 nC1;nC1fsC1 m2anCm nC1;nCmC2fm; '.f2; z/ D  n1 P sD1 asC1 nC2;1es1 nan nC2;nCmC1enanC3 nC2;nC2f1CanC2 nC2;nCmC3f2C CanC1 nCmC3;nCmC2f3Ca1 nC1;1xanC1 nC1;nC2yC.anC1 nC1;nC2anC2 nC2;nC2/z; '.f3; z/ D  n1 P sD1 asC1 nC3;1esCan nC2;nC1enC.anC1 nC2;nC1CanC1 nC1;nC2/f1C C.anC2 nC2;nCmC3anC1 nCmC3;nC1/f3CanC1 nCmC3;nCmC2f4C Ca1 nC3;1xCanCmC2 nC2;nC1yCanCmC3 nC2;nC1z; '.fi; z/ D  n1 P sD1 asC1 nCi;1esCan nCi1;nC1enC.anC1 nCi1;nC1anCmC2 nCi2;nC1/f1C C.anC2 nC2;nCmC3.i 2/anC1 nCmC3;nC1/fiCanC1 nCmC3;nCmC2fiC1C Ca1 nCi;1xCanCmC2 nCi1;nC1yCanCmC3 nCi1;nC1z; 4 im; '.x; z/ D a2 nCmC3;1e1C m1 P iD2 anCiC1 nCmC1;nC1fiC1 m1anCm nCmC1;nCmC2fm; '.y; z/ D anC1 nCmC3;nCmC2f1C m2 P iD2 .anCiC1 nCmC2;nC1anCiC2 nC1;nC1/fiC CanCm nCmC2;nC11 m2anCm nC1;nCmC2fm1C1 m1anCm nCmC2;nCmC2fm; '.z; z/ D m1 P iD2 anCiC1 nCmC3;nC1fiC1 m1anCm nCmC3;nCmC2fm: Taking into account the expressions .6/; .7/ and .8/, we derive that the dimension of ZL2.L; L/ is equal to .m CnC3/25: Based on Proposition 4.2, we have the following corollary. Corollary 4.3. dim HL2.L; L/ D0. Thus, according to the results of the paper [20], we derive the following theorem. Theorem 4.4. The algebra Lis rigid. Acknowledgement: The authors were supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P (European FEDER support included) and Ministry of Education and Science of the Republic of Kazakhstan, grant No. 0828/GF4. The last author was supported by Fundamental Research Grant FRGS/1/2016/STG06/UNIKL/02/1 from Malaysia. References [1] Bloh A., On a generalization of the concept of Lie algebras, Dokl. Akad. Nauk SSSR, 1965, 165, 471–473. [2] Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 1993, (2) 39 (3-4), 269– 293. [3] Ayupov S.A., Omirov B.A., On Leibniz algebras. Algebra and operators theory. Proceedings of the Colloquium in Tashkent 1997. Kluwer Academic Publishers, 1998, 1-12. 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