Solvable Leibniz algebras with NFn⊕ Fm1 $\begin{array}{} F_{m}^{1} \end{array} $ nilradical

Abstract All finite-dimensional solvable Leibniz algebras L, having N = NFn⊕ Fm1 $\begin{array}{} F_{m}^{1} \end{array} $ as the nilradical and the dimension of L equal to n+m+3 (the maximal dimension) are described. NFn and Fm1 $\begin{array}{} F_{m}^{1} \end{array} $ are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m, respectively. Moreover, we show that these algebras are rigid.


Introduction
Leibniz algebras over K were 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][12][13][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][16][17][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.
The first aim of the present paper is to classify solvable Leibniz algebras with nilradical N D NF n˚F 1 m where NF n and F 1 m are the null-filiform and naturally graded filiform Leibniz algebras of dimensions n and m; respectively. To obtain this classification, we use the results obtained in [16][17][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 N D NF n˚F 1 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.

Preliminaries
Let us recite some necessary definitions and preliminary results.
A Leibniz algebra over a field F is a vector space L equipped with a bilinear map, called bracket, OE ; W L L ! L; satisfying the Leibniz identity x; OEy; z D OEx; y; z OEx; z; y for all x; y; z 2 L. The set Ann r .L/ D fx 2 L j OEy; x D 0; y 2 Lg is called the right annihilator of the Leibniz algebra L. Note that Ann r .L/ is an ideal of L and for any x; y 2 L the elements OEx; x, OEx; y C OEy; x 2 Ann r .L/.
A linear map d W L ! L of a Leibniz algebra .L; OE ; / is said to be a derivation if for all x; y 2 L the following condition holds: d.OEx; y/ D OEd.x/; y C OEx; d.y/: For a given element x of a Leibniz algebra L the operator of right multiplication R x W L ! L, defined as R x .y/ D OEy; x for y 2 L, is a derivation. This kind of derivations are called inner derivations. Any Leibniz algebra L is associated with the algebra of right multiplications R.L/ D fR x j x 2 Lg, which is endowed with a structure of Lie algebra by means of the bracket OER x ; R y D R x R y R y R x : Thanks to the Leibniz identity the equality OER x ; R y D R OEy;x holds true. In addition, the algebra R.L/ is antisymmetric isomorphic to the quotient algebra L=Ann r .L/.

Solvable Leibniz algebras
For a Leibniz algebra L we consider the following lower central and derived series:

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A Leibniz algebra L is said to be nilpotent (respectively, solvable), if there exists n 2 N (m 2 N) such that L n D 0 (respectively, L OEm D 0). 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 n C 1. The following theorem describes these algebras, algebras with maximal index of nilpotency. 4]). Any n-dimensional null-filiform Leibniz algebra admits a basis fe 1 ; e 2 ; :::; e n g such that the table of multiplication in the algebra has the following form: 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. 17]). Any complex n-dimensional naturally graded filiform Leibniz algebra is isomorphic to one of the following pairwise non isomorphic algebras: where˛2 f0; 1g for even n and˛D 0 for odd n.
The following theorems describe solvable Leibniz algebras of maximal dimension with NF 1 n and F 1 n nilradical.
Theorem 2.4 ( [17]). Let R be a solvable Leibniz algebra whose nilradical is NF n . Then there exists a basis fe 1 ; e 2 ; : : : ; e n ; xg of the algebra R such that the multiplication table of R with respect to this basis has the following form: OEe i ; e 1 D e i C1 ; 1 Ä i Ä n 1; OEx; e 1 D e 1 : 18]). An arbitrary .n C 2/-dimensional solvable Leibniz algebra with nilradical F 1 n is isomorphic to the algebra R.F 1 n / with the multiplication table: OEx; e 1 D e 1 : Let R be a solvable Leibniz algebra. Then it can be decomposed in the form R D N˚Q; where N is the nilradical and Q is 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 R 2 and consequently, Q 2 Â N: Let us consider the restrictions to N of the right multiplication operator on an element x 2 Q (denoted by R x j N ). From [17], we know that for any x 2 Q; the operator R x j N is a non-nilpotent derivation of N . Let fx 1 ; x 2 ; : : : ; x m g be a basis of Q; then for any scalars f˛1; : : : ;˛mg 2 Cnf0g; the matrix˛1R x 1 j N C C˛mR x m j N is non nilpotent, which means that the elements fx 1 ; : : : ; x m g are nil-independent derivations, [10].
Theorem 2.6 ( [17]). Let R be a solvable Leibniz algebra and N be 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.
Moreover, similarly as in Lie algebras, for a solvable Leibniz algebra R; we have d i mN d imR 2 : A nilpotent Leibniz algebra is called characteristically nilpotent if all its derivations are nilpotent. If the nilradical N of 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].

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][21][22][23][24]. Here we just recall that the second cohomology group of a Leibniz algebra L with coefficients in a corepresentation M is the quotient space The linear reductive group GL n .F/ acts on the variety of n-dimensional Leibniz algebras, Lei b n ; as follows: .g /.x; y/ D g. .g 1 .x/; g 1 .y///; g 2 GL n .F/; 2 Lei b n : 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.

Main results
Let NF n be an n-dimensional null-filiform Leibniz algebra with a basis fe 1 ; e 2 ; : : : ; e n g and F 1 m an m-dimensional filiform Leibniz algebra from the first class with a basis ff 1 ; f 2 ; : : : ; f m g, then we have the following multiplication: : Let us consider the direct sum of these algebras N D NF n L F 1 m : The following proposition describes derivations of the algebra N .
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 N is equal to 3. Now we consider solvable Leibniz algebra R D N C Q; where N D NF n L F 1 m and the dimension of Q is no more than three. Thus, we study the case d i m Q D 3; i.e. d i m R D n C m C 3: Several papers described solvable Leibniz algebras with a given nilradical [15][16][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 d i mQ D 3.
From the work [17], it follows that any solvable Leibniz algebra whose nilradical is NF n has dimension n C 1. It is also known that any solvable Leibniz algebra whose nilradical is F 1 m has dimension either m C 1 or m C 2, [16]. In work [18], it was found a unique .m C 2/-dimensional solvable Leibniz algebra with nilradical F 1 m . Then in the case of the solvable Leibniz algebras R with nilradical N D NF n L F 1 m and d i m Q D 3; there is only one possible case.
Taking into account Theorems 2.2 and 2.5, we have the following multiplication of the algebra R: .2/ where fx; y; zg be a basis of the space Q.
In the following theorem, solvable Leibniz algebras with nilradical N D NF n L F 1 m and d i m Q D 3 are described.
Theorem 3.2. Any .nCmC3/-dimensional solvable Leibniz algebra with nilradical N D NF n L F 1 m is isomorphic to the following algebra: 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]): From Leibniz identity it follows that OEy; e 2 D OEy; OEe 1 ; e 1 D 0 and by induction we can easily find that OEy; e i D 0; with 2 Ä i Ä n: Analogously, we have OEz; e i D 0; with 2 Ä i Ä n: We consider OEx; i 1 f i : Similarly, and using the induction method, However, from the equality OEx; Taking the following change Let us apply the Leibniz identity on the following triples of elements: We observe that OEx; y C OEy; x 2 Ann r .R/, thus OEe 1 ; 1 As a result of OEx; f 1 C OEf 1 ; x 2 Ann r .R/; we observe that a 11 D 0 and b 11 D ˇ1 1 : and from the equalities OEf i ; OEx; e 1 D 0; for 2 Ä i Ä m; it follows that a i;j D 0; for 2 Ä i Ä m and 1 Ä j Ä n 1, that is Now, we proceed by looking at the product of certain elements of Ann r .R/: Summarizing the following identities OEe 1 ; OEy; e i C OEe i ; y D OEf 2 ; OEy; e i C OEe i ; y D OEe 1 ; OEz; e i C OEe i ; z D OEf 2 ; OEz; e i C OEe i ; z D 0; we have the following Leibniz brackets of the basic elements: The Leibniz identity on the following triples imposes futher constraints on .3/.

Leibniz identity Constraint
As a result of the above constraints we observe that By the induction on decrease i .2 Ä i Ä m/ and using the Leibniz identity for the elements ff i ; x; f 1 g we get Considering the Leibniz identity on the triple fe 1 ; y; e 1 g we get OEe 2 ; y D 2 1 1 e 2 C n P i D3 c 1 1;i 1 e i : Also, using the equalities OEe i ; y D OEOEe i 1 ; e 1 ; y D OEOEe i 1 ; y; e 1 OEe i 1 ; OEy; e 1 ; for 3 Ä i Ä n and by induction method on i; we obtain that Analogously, we can get OEe i ; z D i 2 1 e i C n P j DiC1 c 2 1;j i C1 e j ; for 2 Ä i Ä n: The Leibniz identity on the triples ff 2 ; x; yg; ff 1 ; x; yg and fx; f 1 ; yg; (in this order) gives OEf 1 ; x D a 1;n e n ˇ1 1 f 1 ; Now, applying the Leibniz identity on the triples fx; f 1 ; zg; fy; x; f 1 g; ff i ; x; zg; ff i ; x; yg; with 1 Ä i Ä m; fe i ; z; xg; fe i ; y; xg; with 1 Ä i Ä n; (in this order) it follows From the above results we can simplify the family .3/ as follows: Finally, considering the Leibniz identity on the following triples we obtain:
The Leibniz identity on the following triples imposes further constraints on the parameters. We can distinguish the following cases: -Case a ¤ 0: The restrictions imply that 1 D 2 D 1 n : By performing a change of basis

Leibniz identity Constraint
we obtain the algebra L: -Case a D 0: Making the change y 0 D y C 1 x; z 0 D z C 2 x; we obtain the algebra L:

Rigidity of the algebra L
In order to describe the second group of cohomology of the algebra L we need the description of its derivations. The general form of the derivations of L is given in the following proposition.
Theorem 4.4. The algebra L is rigid.