Solvable Leibniz algebras with naturally graded non-Lie p-filiform nilradicals whose maximal complemented space of its nilradical

SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS AND MAXIMAL COMPLEMENTED SPACE OF ITS NILRADICAL J. Q. ADASHEV1, L.M. CAMACHO2, B. A. OMIROV1,3 1Institute of Mathematics, Uzbekistan Academy of Sciences, 100170, Tashkent, Uzbekistan, [email protected] 2Dpto. Matemática Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012 Sevilla. (Spain) E-mail address: lc[email protected] 3National University of Uzbekistan, 4, University str., 100174, Tashkent, Uzbekistan, omir[email protected] Abstract. The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded p-filiform nonLie Leibniz algebra (n−p≥4) and the complemented space to nilradical has maximal dimension, are described up to isomorphism. Moreover, among obtained algebras we indicate the rigid and complete algebras. 1. Introduction During the last decades the theory of Leibniz algebras has been actively investigated and many results of the Lie Theory have been transferred to Leibniz algebras. Levi’s decomposition asserts that every finite-dimensional Lie algebra is a semidirect sum of a semisimple Lie subalgebra and solvable radical [16], while semisimple Lie algebras over the field of complex numbers have been classified by E. Cartan [11] and over the field of real numbers by F. Gantmacher [12]. Thus, the problem description of finite-dimensional Lie algebras is reduced to the study of solvable Lie algebras. Till present the classification of solvable Lie algebras is known for dimensions not greater than six [13], [22]. Also there are several works devoted to the classification of solvable Lie algebras in an arbitrary finite-dimensions [2–4], [20,23,24]. In fact, there are solvable Lie algebras constructed using the method explained in [21]. For finite-dimensional Leibniz algebras over a field of zero characteristic, there is an analogue of Levi’s decomposition: any Leibniz algebra is decomposed into a semidirect sum of a semisimple Lie algebra and its solvable radical [6]. Therefore, similar to Lie case, the main problem of the study of Leibniz algebras reduced solvable ones. In the paper [10], the method that describes solvable Lie algebras by means of its radical is adapted for Leibniz case. Since the description of finite-dimensional solvable Leibniz algebras is a boundless problem (even for solvable Lie algebras), new approaches are developing. Relevant tools of geometric approaches are properties of Zariski topology and the natural action of linear reductive group on varieties of algebras in a such way that orbits under the action consists of isomorphic algebras. It is a well-known result of algebraic geometry that any algebraic variety (evidently, algebras defined via identities form an algebraic variety) is a union of a finite number of irreducible components. The most important algebras are those whose orbits under the action are open sets in sense of Zariski topology (such algebra are called rigid algebras). The algebras of a variety with open orbits are important since the closures of orbits of such algebras form irreducible components of the variety. At the same time there exists an 2010 Mathematics Subject Classification. 17A32, 17A36, 17B30, 17B56. Key words and phrases. Leibniz algebra, natural gradation, p-filiform algebra, solvability, nilradical, derivation, the second group of cohomology. 1 2 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS irreducible component which is not the closure of orbit of any algebra. This fact does not detract the importance of algebras with open orbits. This was a motivation for many works focused to discovering of algebras with open orbits and to description of sufficient properties of such algebras [7,14,15]. The aim of this article is to describe solvable Leibniz algebras with naturally graded non-Lie pfiliform nilradicals and with maximal dimension of complemented space of its nilradical. Namely, in arbitrary finite dimension, we got three types of such algebras (R(µ1, k), R(µ2, k)and R(µ3, k + 2)) and we established that the algebra R(µ3, k + 2) is complete and cohomologically rigid. Throughout the paper we shall consider finite-dimensional vector spaces and complex algebras. Moreover, in the multiplication table the omitted products are assumed to be zero and we shall consider non-nilpotent solvable algebras (unless stated otherwise). 2. Preliminaries We recall the necessary background in order to make the comprehensive paper. Definition 2.1. [18] A Leibniz algebra Lis a vector space over Fequipped with a bilinear map (multiplication) satisfying the Leibniz identity x, [y, z]=[x, y], z−[x, z], y for all x, y, z ∈L. We refer readers to works [18] and [19] for examples of Leibniz algebras. Further we will use the following notation L(x, y, z) = [x, [y, z]] −[[x, y], z] + [[x, z], y]. It is obvious that the identity L(x, y, z) = 0 determines the Leibniz algebras. For a given Leibniz algebra Lwe can define the following two-sided ideals Annr(L) = {x∈L|[y, x] = 0,for all y∈L}, Center(L) = {x∈L|[x, y] = [y, x] = 0,for all y∈L} called the right annihilator and the center of L, respectively. From the Leibniz identity we conclude that the following elements [x, x],[x, y] + [y, x]in Annr(L)for any x, y ∈L. A linear map d:L→Lof a Leibniz algebra Lis said to be a derivation if for all x, y ∈L, the following condition holds: d([x, y]) = [d(x), y] + [x, d(y)].(2.1) The set of all derivations of L(denoted by Der(L)) forms a Lie algebra with respect to the commutator. Note that the operator of right multiplication on elements x∈L(further denoted by Rx) is a derivation, which is called inner derivation. Definition 2.2. A Leibniz algebra Lis called complete if Center(L) = 0 and all derivations of Lare inner. Analogously to Lie algebras, we define the following sequences: L1=L, Lk+1 = [Lk, L], k ≥1, L[1] =L, L[s+1] = [L[s], L[s]], s ≥1, so-called the lower central and the derived series of L, respectively. Definition 2.3. A Leibniz algebra Lis nilpotent (respectively, solvable), if there exists n∈N(m∈N) such that Ln= 0 (respectively, L[m]= 0). The maximal nilpotent ideal of a Leibniz algebra is said to be the nilradical of the algebra. An analogue of Mubarakzjanov’s methods has been applied for solvable Leibniz algebras which shows the importance of the consideration of non-characteristically nilpotent Leibniz algebra [10]. Consider a solvable Leibniz algebra R=N⊕Qwith the nilradical Nand complementary vector space Qof Nwith a basis {x1,...,xm}. It is known that for an element x∈Qthe operator Rx|N is a non-nilpotent derivation of N. Moreover, for any scalars {α1,...,αm} ∈ C\ {0}, the operator α1Rx1|N+··· +αmRxm|Nis non-nilpotent, which means that the elements {x1,...,xm}are nilindependent. Therefore, the dimension of complementary vector space to Nis no greater than the maximal number of nil-independent derivations of N( [10, Theorem 3.2]). SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 3 For a nilpotent Leibniz algebra Land x∈L\L2we consider the decreasing sequence C(x) = (n1, n2,...,nk)as the dimensions of the Jordan blocks of the operator Rx. On the set of such sequences we consider lexicographic order. Definition 2.4. The sequence C(L) = max x∈L\L2C(x)is called the characteristic sequence of the Leibniz algebra L. Similar to the Lie algebras, we have the following definition. Definition 2.5. A Leibniz algebra Lis called p-filiform if C(L) = (n−p, 1,...,1 |{z } p ), where p≥0. Note that above definition, when p > 0agrees with the definition of p-filiform Lie algebras [8]. Since in the case of Lie algebras there is no singly-generated algebra, the notion of 0-filiform algebra for Lie algebras has no sense, while for the Leibniz algebras case in each dimension there exists up to isomorphism a unique null-filiform algebra [5]. Definition 2.6. Given an n-dimensional p-filiform Leibniz algebra L, put Li=Li/Li+1,1≤i≤n−p, and gr L=L1⊕L2⊕ · · · ⊕ Ln−p. Then [Li, Lj]⊆Li+jand we obtain the graded algebra gr L. If gr L and Lare isomorphic, gr L∼ =L, we say that Lis naturally graded. In this paper, we consider naturally graded p-filiform non-Lie Leibniz algebras. Their classification is given in the next theorem. Theorem 2.7. [9] An arbitrary n-dimensional naturally graded non-split non-Lie p-filiform Leibniz algebra (n−p≥4) is isomorphic to one of the following non-isomorphic algebras: p= 2kis even µ1:([ei, e1] = ei+1,1≤i≤n−2k−1, [e1, fj] = fk+j,1≤j≤k, µ2:           [ei, e1] = ei+1,1≤i≤n−2k−1, [e1, f1] = e2+fk+1, [ei, f1] = ei+1,2≤i≤n−2k−1, [e1, fj] = fk+j,2≤j≤k, p= 2k+ 1 is odd µ3:     [ei, e1] = ei+1,1≤i≤n−2k−2, [e1, fj] = fk+1+j,1≤j≤k, [ei, fk+1] = ei+1,1≤i≤n−2k−2, where {e1, e2,...,en−p, f1, f2,...,fp}is a basis of the algebra. In order to simplify our next calculations, the following change of basis in µ3: e′ 1=fk+1, e′ 2=e1−fk+1, e′ i+1 =ei,2≤i≤n−2k−1, f′ j=fj, f′ k+j=fk+1+j,1≤j≤k, allows to obtain a more convenient form of µ3: µ3:([ei, e1] = ei+1,2≤i≤n−2k−1, [e2, fj] = fk+j,1≤j≤k. 2.1. Cohomology Leibniz algebras. Since in the last section of this paper we study the cohomological rigidity of obtained algebras, we need some concepts of the second cohomology group of Leibniz algebras. For more details, we refer to [18], [19] and references therein. The second cohomology group of a Leibniz algebra Lwith coefficient itself is the quotient space HL2(L, L) := ZL2(L, L)/BL2(L, L), where the elements ψ∈BL2(L, L)and ϕ∈ZL2(L, L)are defined by: ψ(x, y) = [d(x), y] + [x, d(y)] −d([x, y]),for some linear map d∈Hom(L, L), [x, ϕ(y, z)] −[ϕ(x, y), z] + [ϕ(x, z), y] + ϕ(x, [y, z]) −ϕ([x, y], z) + ϕ([x, z], y) = 0,(2.2) respectively. It is obvious that a Leibniz 2-cocycle ϕof a Leibniz algebra Lis determined by the identity Φ(ϕ)(x, y, z) = 0,where Φ(ϕ)(x, y, z) = [x, ϕ(y, z)] −[ϕ(x, y), z] + [ϕ(x, z), y] + ϕ(x, [y, z]) −ϕ([x, y], z) + ϕ([x, z], y). 4 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS Definition 2.8. A Leibniz algebra Lis called cohomologically rigid if HL2(L, L) = 0. Due to results of the paper [5], we have that a Leibniz algebra is rigid if the second cohomology group with coefficients in itself is trivial. 3. Solvable Leibniz algebras with abelian nilradical and maximal dimension of complemented space Q. In this section we recall some results of the paper [1], which will be used below. We denote by akthe k-dimensional abelian algebras and by R(ak, s)the solvable Leibniz algebra with akas nilradical and sas the dimension of complemented space to ak. Theorem 3.1. [1] The maximal possible dimension of algebras of the family R(ak, s)is equal to 2k, that is, s=k. Moreover, an arbitrary algebra of the family R(ak, k)is decomposed into a direct sum of copies of two-dimensional non-trivial solvable Leibniz algebras. Consider the solvable Leibniz algebras L(γi)with nilradical akunder the condition that the complemented space to the nilradical have maximal dimension. Then there exists a basis {f1, f2,...,fk, x1, x2,...,xk}of L(γi)such that the multiplication table has the form: L(γi) : [fi, xi] = fi,[xi, fi] = γifi,1≤i≤k, where γi∈ {−1,0}. The algebra L(γi)is a rigid algebra for any γi∈ {−1,0},1≤i≤k, [1]. Lemma 3.2. Any automorphism ϕof the algebra L(γi)has the following form: ϕ(fi) = αifi, ϕ(xi) = βifi+xi,1≤i≤k, where (1 + γi)βi= 0 for 1≤i≤k. Proof. Let ϕbe an automorphism of L(γi).Since the automorphism of algebra maps nilradical to nilradical we can assume ϕ(fi) = k X j=1 Di,jfj, ϕ(xi) = k X j=1 Fi,j fj+ k X j=1 Hi,jxj,1≤i≤k. From the following equalities [ϕ(fi), ϕ(xj)] = ϕ([fi, xj]) and [ϕ(xi), ϕ(xj)] = ϕ([xi, xj]) with 1≤ i≤k, we derive      Di,mHj,m = 0,1≤i6=j, m ≤k, Di,mHi,m =Di,m,1≤i, m ≤k, Fi,mHj,m +γmFj,mHi,m = 0,1≤i, j, m ≤k. So, for a given value of jwe have a linear system with respect to Hj,1, Hj,2, . . . , Hj,k. Let us prove that for a fixed j, 1≤j≤kthere exists only m0such that Hj,m0= 1 and Hj,m = 0 with 1≤m6=m0≤k. Let us suppose that Hj,m0=Hj,m1= 1, then we get Di,m0=Di,m1= 0 for 1≤i6=j≤k. On the other hand, det(Di,m)k i,m=1 = 0,that is, we arrive at contradiction. Without loss of generality, we can assume that Hj,j = 1 and Hj,i = 0 with 1≤j6=i≤k. Then, we obtain the following restrictions: (Dj,j 6= 0, Dj,m = 0,1≤m6=j≤k, Fi,j = 0,(1 + γj)Fj,j = 0,1≤i6=j≤k, which imply ϕ(fi) = Di,ifi, ϕ(xi) = Fi,ifi+xi,1≤i≤k, where (1 + γi)Fi,i = 0 and γi∈ {−1,0}for 1≤i≤k.  4. solvable leibniz algebras with n-dimensional naturally graded p-filiform non-Lie Leibniz algebra and maximal dimension of Q. In this section we give a description of solvable Leibniz algebras whose nilradical is a naturally graded p-filiform Leibniz algebra and the dimension of Qis maximal. Firstly, we recall the derivations of the algebras µi, i = 1,2,3given in [1]. SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 5 4.1. Derivations of algebras µi, i = 1,2,3. Proposition 4.1. Any derivation of the algebra µ1has the following matrix form: D=A B C D, with D =D1D2 0a1E+D1, where A= n−2k X i=1 ia1ei,i + n−2k−1 X i=1 n−2k X j=i+1 aj−i+1ei,j, B = 2k X i=1 bie1,i + k X i=1 bie2,k+i, C = k X i=1 ciei,n−2k, A∈Mn−2k,n−2k, B ∈Mn−2k,2k, C ∈M2k,n−2k, D1, D2,E∈Mk,k and matrix units ei,j. Proposition 4.2. Any derivation of the algebra µ2has the following matrix form: D=A B C D, with D =D1D2 0D3, where A= n−2k X i=1 (ia1+ (i−1)b1)ei,i + n−2k−1 X i=1 n−2k X j=i+1 aj−i+1ei,j, B = 2k X i=1 bie1,i + k X i=1 bie2,k+i, C= k X i=1 ciei,n−2k, D1= k X i=1 k X j=2 di,jei,j + (a1+b1)e1,1, D3=D1+a1E− k X j=1 bje1,j, with A∈Mn−2k,n−2k, B ∈Mn−2k,2k, C ∈M2k,n−2k, D1, D2, D3,E∈Mk,k and matrix units ei,j . Proposition 4.3. Any derivation of the algebra µ3has the following matrix form: D=A B C D, with D =D1D2 0a2E+D1, where A=a1e1,1+ n−2k X i=2 ((i−2)a1+a2)ei,i +βe1,n−2k+ n−2k−1 X i=2 n−2k X j=i+1 aj−i+2ei,j, B= 2k X i=1 b1,ie1,i + k X i=1 b2,ie2,k+i+ k X i=1 b1,ie3,k+i, C = k X i=1 ciei,n−2k, with A∈Mn−2k,n−2k, B ∈Mn−2k,2k, C ∈M2k,n−2k, D1, D2,E∈Mk,k and matrix units ei,j . The theorem bellow describes the maximal dimensions of the complemented space to µi, i = 1,2,3. Theorem 4.4. Let Rbe a solvable Leibniz algebra whose nilradical is µi, i = 1,2,3. Then the dimension of complemented space to nilradical verifies that: dim Q(µi)≤k+ 2 i 3. Proof. According to Propositions 4.1 and 4.2, we have the following expresions for R(µ1, s)and R(µ2, s), respectively:        [e1, x] = n−2k P i=1 aiei+ 2k P i=1 bifi, [e2, x] = 2a1e2+ n−2k P i=3 ai−1ei+ k P i=1 bifk+i,        [e1, x] = n−2k P i=1 aiei+ 2k P i=1 bifi, [e2, x] = (2a1+b1)e2+ n−2k P i=3 ai−1ei+ k P i=1 bifk+i, Let us introduce the following notations: [x, e1] = n−2k X i=1 βiei+ 2k X i=1 βn−2k+ifi,[x, f1] = n−2k X i=1 γiei+ 2k X i=1 ϕifi. The equalities L(x, f1, e1) = L(e1, x, e1) = 0 imply a1= 0. Note that {f1, f2,...,fk}form the algebra akand the space of derivations of the algebra akcoincided with Mk,k. 6 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS It is easy to see that Rx|ak◦ Ry|ak=Ry|ak◦ Rx|ak for any x, y ∈Q. This implies that all operators Rxi|ak,1≤i≤scould be simultaneously transformed to their Jordan forms by a basis transformation. Therefore, the matrix operator Rx|ak(in our case Rx|ak=D1) has the following form: D1=          d1,1d1,20... 0 0 0d2,2d2,3... 0 0 0 0 d3,3... 0 0 . . .. . .. . .. . .. . .. . . 0 0 0 . . . dk−1,k−1dk−1,k 0 0 0 ... 0dk,k          , where di,i+1 ∈ {0,1}for 1≤i≤k−1. Now we are going to investigate the nilpotency of matrix D.Due to Propositions 4.1-4.3 the nilpotency of Ddepends on the matrices Aand D1. Let us consider the matrix Das follows D=A B C D =A1+A2B C K1+K2, where A1, K1are diagonal matrices and A2, K2are nilpotent such that A1=     diag{0,0,0,...,0},for µ1, diag{0, b1,2b1,...,(n−2k−1)b1},for µ2, diag{a1, a2, a1+a2,...,(n−2k−2)a1+a2},for µ3, K1=     diag{d1,1, d2,2,...,dk,k, d1,1, d2,2,...,dk,k},for µ1, diag{b1, d2,2,...,dk,k,0, d2,2,...,dk,k},for µ2, diag{d1,1, d2,2,...,dk,k, a2+d1,1, a2+d2,2,...,a2+dk,k},for µ3. It is easy to see that CB = 0 and the matrices A1A2, A2 2, BC, K1K2, K2 2are nilpotent. Moreover, matrices C(A1+A2),(K1+K2)Chave the type of Cand matrices (A1+A2)B, B(K1+K2) have the type of B. According to the above arguments we have the following recurrence formula: Dt= At 1+e A2e B e C Kt 1+e K2!, t ≥1, where e A2,e K2−nilpotent matrices and matrices e B, e Chave the types of Band C, respectively. To sum up, we conclude that the matrix Dis nilpotent if and only if A1and K1are nilpotents. Therefore, we obtain the following conclusions: •For µ1, the nilpotency of Ddepends on di,i,1≤i≤k, that is Dnilpotent if only if di,i = 0,1≤i≤k. •For µ2, the nilpotency of Ddepends on b1and di,i,2≤i≤k, that is Dnilpotent if only if b1=di,i = 0,2≤i≤k. •For µ3,the nilpotency of Ddepends on a1, a2and di,i,1≤i≤k, that is Dnilpotent if only if a1=a2=di,i = 0,1≤i≤k. Applying the result in [10, Theorem 3.2], the stated inequalities follow.  The following results will be used in the description of solvable Leibniz algebras whose nilradicals are µi, i = 1,2,3and with maximal dimensional complemented space of nilradicals. Proposition 4.5. Let Rbe a solvable Leibniz algebra whose nilradical is a naturally graded p-filiform non-Lie Leibniz algebra. Then {e1, f1,...,fk} ∩ Annr(R) = 0 and {e2,...,en−2k, fk+1,...,f2k} ⊆ Annr(R), with a26= 0 for the algebra R(µ3, s). SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 7 Proof. Using Theorem 2.7 and the properties of the right annihilator (that is, [x, x],[x, y] + [y, x]∈ Annr(R)) the assertion easily follows for R(µ1, s)and R(µ2, s). Consider the algebra R(µ3, s).It is easy to see that e1, f1,...,fk/∈Annr(R)and e3,...,en−2k, fk+1,...,f2k∈Annr(R). Let us suppose a26= 0. Then, from the derivation of µ3we get [e2, x] = n−2k X i=2 aiei+ k X i=1 b2,ifk+i,[x, e2] = n−2k X i=1 αiei+ 2k X i=1 βifi. The equality L(x, e2, e1) = 0 implies αi= 0 for 2≤i≤n−2k−1.Since [e2, x] + [x, e2]∈Annr(R) and a26= 0, we have e2∈Annr(R)which complete the proof.  Lemma 4.6. Let Rbe a solvable Leibniz algebra whose nilradical is a naturally graded p-filiform Leibniz algebra. Then the maximal solvable Leibniz subalgebra with nilradical ak=< f1,...,fk>of R is isomorphic to L(γi)with γi=−1for 1≤i≤k. Proof. Clearly, ak={f1, f2,...,fk}forms an abelian subalgebra of R. By Theorem 3.1 the maximal solvable Leibniz algebra with nilradical akis isomorphic to L(γi).Since [fi, xi] + [xi, fi]∈Annr(R) with 1≤i≤kand fi/∈Annr(R)with 1≤i≤k, the proof of lemma is complete.  In the following theorem we present the description of algebras of the family R(µ1, k). Theorem 4.7. An arbitrary algebra of the family R(µ1, k)admits a basis such that the non-vanishing Leibniz brackets become: R(µ1, k)(ai,j, ϕi,j , δi,j) :                        [ei, xj] = n−2k P t=i+1 at−i+1,jet,1≤i≤n−2k, 1≤j≤k, [fi, xi] = fi,1≤i≤k, [fk+i, xi] = fk+i,1≤i≤k, [xi, fi] = −fi,1≤i≤k, [xi, fj] = ϕi,jfk+j,1≤i6=j≤k, [xi, xj] = δi,jen−2k,1≤i, j ≤k. Proof. According to Propositions 4.1, 4.5, Theorem 4.4 and Lemma 4.6 we have the following brackets for R(µ1, k):                                                                          [e1, xi] = n−2k P t=2 at,iet+ 2k P t=1 bt,ift,1≤i≤k, [e2, xi] = n−2k P i=3 at−1,iet+ k P t=1 bt,ifk+t,1≤i≤k, [ej, xi] = n−2k P t=j+1 at−j+1,iet,3≤j≤n−2k, [fi, xi] = ci,ien−2k+fi+ 2k P t=k+1 dt i,ift,1≤i≤k, [fi, xj] = ci,jen−2k+ 2k P t=k+1 dt i,j ft,1≤i6=j≤k, [fk+i, xi] = fk+i,1≤i≤k, [xi, e1] = n−2k P t=2 βt,iet− k P t=1 bt,ift+ k P t=1 βn−k+t,ifk+t,1≤i≤k, [xi, fi] = n−2k P t=2 γt i,iet−fi+ 2k P t=k+1 ϕt i,ift,1≤i≤k, [xi, fj] = n−2k P t=2 γt i,jet+ 2k P t=k+1 ϕt i,j ft,1≤i6=j≤k, [xi, xj] = n−2k P t=1 δt i,jet+ 2k P t=k+1 θt i,jft,1≤i, j ≤k. 8 SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE P-FILIFORM NILRADICALS By taking the change of basis e′ 1=e1− k P t=1 bt,tft, e′ 2=e2− k P t=1 bt,tfk+t, f′ i=fi−γi,ien−2k− 2k P t=k+1 ϕt i,ift, x′ i=xi− k P t=1 θk+t i,t fk+t,1≤i≤k, we can assume bi,i =γn−2k i,i =ϕk+t i,i =θk+t i,t = 0,1≤i, t ≤k. Applying the Leibniz identity, the following relations are obtained (L(xi, fj, e1) = 0,⇒γt i,j = 0,1≤i, j ≤k, 2≤t≤n−2k−1, L(e1, xi, xj) = 0,⇒bi,j =bk+i,j =δ1 i,i =δ1 i,j = 0,1≤i6=j≤k. Putting e′ 1=e1− k P t=1 bk+t,tfk+t, x′ i=xi− n−2k P t=2 βt,iet−1,we conclude that βt,i =bk+i,i = 0 with 1≤i≤k, 2≤t≤n−2k. Considering the Leibniz identity, we obtain the following restrictions on structure constants:                L(xi, fi, xi) = 0,⇒ci,i =dt i,i = 0,1≤i≤k, k + 1 ≤t≤2k, L(xi, fi, xj) = 0,⇒ci,j =dt i,j = 0,1≤i6=j≤k, k + 1 ≤t≤2k, L(xi, fj, xj) = 0,⇒γn−2k i,j =ϕk+t i,j = 0,1≤i6=j6=t≤k, L(xi, e1, xj) = 0,⇒δt i,j =βn−k+i,j = 0,1≤i, j ≤k, 2≤t≤n−2k−1, L(xi, xj, xs) = 0,⇒θk+s i,j = 0,1≤i, j 6=s≤k.  Below the necessary and sufficient conditions of the existence of an isomorphism between two algebras of the family R(µ1, k)(ai,j, ϕi,j, δi,j)are established. Proposition 4.8. Two algebras R(µ1, k)′(a′ i,j, ϕ′ i,j, δ′ i,j)and R(µ1, k)(ai,j, ϕi,j, δi,j)are isomorphic if and only if there exists A∈C∗such that a′ i,j =ai,j Ai−1,2≤i≤n−2k+ 1,1≤j≤k, ϕ′ i,j =ϕi,j A,1≤i6=j≤k, δ′ i,j =δi,j An−2k,1≤i, j ≤k. Proof. Taking into account Lemma 3.2 we consider the general change of generator basis elements of an algebra from R(µ1, k)(ai,j , ϕi,j, δi,j): e′ 1= n−2k P i=1 Aiei+ 2k P i=1 Bifi, f′ i= n−2k P j=1 Ci,jej+Di,ifi+ 2k P j=k+1 Di,jfj, x′ i= n−2k P j=1 Ei,jej+Fi,ifi+ 2k P j=k+1 Fi,j fj+xi,1≤i≤k. From the following brackets in R(µ1, k)′(a′ i,j, ϕ′ i,j , δ′ i,j) : [e′ i, e′ 1] = e′ i+1,1≤i≤n−2k−1,[f′ i, e′ 1] = 0,[e′ 1, f′ i] = f′ k+i,1≤i≤k, we derive e′ 2=A1 n−2k P i=2 Ai−1ei+A1 k P i=1 Bifk+i, e′ i=Ai−1 1 n−2k P j=i Aj−i+1ej,3≤i≤n−2k, f′ k+i=A1Di,ifk+i, Ci,j = 0,1≤i≤k, 1≤j≤n−2k−1. The following vanishing parameters Bi=Ei,j =Ci,n−2k=Di,k+t= 0 with 1≤i, t ≤kand 1≤j≤n−2k−1have been obtained from the products: [x′ i, e′ 1] = 0,[x′ i, f′ i] = f′ i,1≤i≤k. Therefore, we obtain e′ 1= n−2k P i=1 Aiei+ 2k P i=k+1 Bifi, e′ i=Ai−1 1 n−2k P j=i Aj−i+1ej,2≤i≤n−2k, f′ i=Di,ifi, f′ k+i=A1Di,ifk+i, x′ i=Ei,n−2ken−2k+Fi,ifi+ 2k P j=k+1 Fi,jfj+xi,1≤i≤k. SOLVABLE LEIBNIZ ALGEBRAS WITH NATURALLY GRADED NON-LIE p-FILIFORM NILRADICALS 9 Let us consider the products [e′ 1, x′ j] = n−2k X i=2 a′ i,je′ i,[x′ i, f′ j] = ϕ′ i,j f′ k+j,[x′ i, x′ j] = δ′ i,je′ n−2k,1≤i, j ≤k. From which we get the following restrictions:          a′ i,j =ai,j Ai−1 1 ,2≤i≤n−2k, 1≤j≤k, ϕ′ i,j =ϕi,j A1,1≤i6=j≤k, δ′ i,j =δi,j An−2k 1 ,1≤i, j ≤k, where A1Fj,j +Bk+j=Fi,k+i=Fi,k+j+ϕi,j Fj,j = 0,1≤i6=j≤k.  Below we describe solvable Leibniz algebras R(µ2, k). Theorem 4.9. An arbitrary algebra of the family R(µ2, k)admits a basis such that its multiplication table has the following form: R(µ2, k)(bi, βi, ϕi,j, θi,j) :                                              [e1, x1] = f1+b1fk+1, [e2, x1] = e2+fk+1, [ej, x1] = (j−1)ej,3≤j≤n−2k, [x1, e1] = −f1+β1fk+1, [e1, xi] = bifk+1,2≤i≤k, [fi, xi] = fi,1≤i≤k, [fk+i, xi] = fk+i,2≤i≤k, [xi, e1] = βifk+1,2≤i≤k, [xi, fi] = −fi,1≤i≤k, [xi, fj] = ϕi,j fk+j,1≤i≤k, 2≤j≤k, i 6=j, [xi, xj] = θi,jfk+1,1≤i, j ≤k. Proof. The description of R(µ2, k)follows from Proposition 4.2, 4.5, Theorem 4.4 and Lemma 4.6. In fact, firstly, we consider derivations of µ2and since the parameters b1, d2, d3,...,dkare in the diagonal, we have only knil-independent derivations which correspond to the values of (b1, d2, d3,...dk) : (1,0,0, . . . , 0),(0,1,0,...,0), ..., (0,0,0,...,1). Later, assuming these derivations as Rx1,Rx2, ..., Rxk(respectively) we complete the proof by applying similar arguments as used in the proof of Theorem 4.7.  In the next proposition necessary and sufficient conditions of the existence of an isomorphism between two algebras of the family R(µ2, k)(bi, βi, ϕi,j, θi,j)are established. Proposition 4.10. Two algebras R(µ2, k)′(b′ i, β′ i, ϕ′ i,j, θ′ i,j)and R(µ2, k)(bi, βi, ϕi,j , θi,j)are isomorphic if and only if there exists A∈C∗such that b′ i=bi A,1≤i≤k, β′ i=βi A,1≤i≤k, ϕ′ 1,i =ϕ1,i A,2≤i≤k, ϕ′ i,j =ϕi,j A,2≤i6=j≤k, θ′ i,j =θi,j A2,1≤i, j ≤k. Proof. Analogously to the proof of Proposition 4.8.  To complete the description of solvable Leibniz algebras with the nilradicals µi, i = 1,2,3and maximal complemented space to nilradical, we give the following theorem.