Solvable Leibniz Algebras with Abelian Nilradicals
Abstract.
We extend the classification of solvable Lie algebras with abelian nilradicals to classify solvable Leibniz algebras which are one dimensional extensions of abelian nilradicals.
1. Introduction
Leibniz algebras were defined by Loday in 1993 [12, 13]. Recently, there has been a trend to show how various results from Lie algebras extend to Leibniz algebras [1, 3, 16]. In particular, there has been interest in extending classifications of certain classes of Lie algebras to classifications of corresponding Leibniz algebras [2, 4, 5, 8, 9, 11].
Some of the classifications of Lie algebras come from placing certain restrictions on the nilradical [7, 15, 17, 18]. Several authors have been able to extend these results to Leibniz algebras or show similar results for Leibniz algebras with certain restrictions on the nilradical [5, 8, 9, 10, 11]. In praticular, Ndogmo and Winternitz [15] study solvable Lie algebras with abelian nilradicals. The goal of this paper is to utilize the results of [15] to develop similar results in the Leibniz setting.
We construct a general classification theorem for solvable Leibniz algebras with abelian nilradicals over . Furthermore, we discuss the case of 1dimensional extensions; then provide an explicit classification of all 1dimensional extensions of 1, 2, and 3dimensional abelian nilradicals. In [6], Cañete and Khudoyberdiyev classify all nonnilpotent 4dimensional Leibniz algebras over . Our classification recovers their result for 4dimensional algebras with 3dimensional abelian nilradicals. We also develop some results on 2dimensional extensions of abelian nilradicals, specifically we show that all 4dimensional solvable Leibniz algebras with 2dimensional abelian nilradicals are in fact Lie algebras.
2. Preliminaries
A Leibniz algebra, , is a vector space over a field (which we will take to be or ) with a bilinear operation (which we will call multiplication) defined by which satisfies the Jacobi identity
(1) 
for all . In other words , leftmultiplication by , is a derivation. Some authors choose to impose this property on , rightmultiplication by , instead. Such an algebra is called a “right” Leibniz algebra, but we will consider only “left” Leibniz algebras (which satisfy (1)). is a Lie algebra if additionally .
The derived series of a Leibniz (Lie) algebra is defined by , for . is called solvable if for some . The lowercentral series of is defined by , for . is called nilpotent if for some . It should be noted that if is nilpotent, then must be solvable.
The nilradical of is defined to be the (unique) maximal nilpotent ideal of , denoted by . It is a classical result that if is solvable, then . From [14], we have that
An abelian Lie algebra is the dimensional Lie algebra with basis
, and multiplication
(2) 
The leftannihilator or leftnormalizer of a Leibniz algebra is the ideal . Note that the elements and are in , for all , because of (1).
An element in a Leibniz algebra is nilpotent if both for some . In other words, for all in
A set of matrices is called linearly nilindependent if no nonzero linear combination of them is nilpotent. In other words, if
then implies that for all . A set of elements of a Leibniz algebra is called linearly nilindependent if no nonzero linear combination of them is a nilpotent element of .
3. Classification
Let be the dimensional abelian (Lie) algebra over the field ( or ) with basis and products given by (2). By appending the basis of with linearly nilindependent elements , we construct an dimensional solvable Leibniz algebra, where . In doing so, we create a Leibniz algebra whose nilradical is . Henceforth, we shall only consider such that are indecomposable. As in [15], we have the following constraints on , , and :
(3) 
Let be a finitedimensional solvable nonnilpotent Leibniz algebra over the field , of characteristic zero, subject to (3). Let have an dimensional abelian nilradical, namely . For , we may choose the basis , where and . In general, the bracket relations for elements in are defined by , . Letting , then the left and right bracket relations of elements are defined, for , by
(4a)  
(4b) 
since . Note that we are employing Einstein notation in (4b), summing over all .
The classification of these Leibniz algebras is equivalent to the classification of the matrices and and the constants . The Jacobi identities for the triples , , for and yield, respectively,
Unlike the Lie case, these give nontrivial relations for or when . By the linear nilindependence of the , we have the following relations on the matrices and
(5a)  
(5b)  
(5c) 
The Jacobi identity for the triple for gives us
(6) 
summed over . Again, we do not require to be distinct, which in particular gives nontrivial relations for . Equivalently, since , we have and, again, we are summing over index .
Lemma 3.1.
Proof.
Recall that, for Leibniz algebra , , . Hence, , . In particular, , which implies that
where here we are summing over all . Additionally, this is true for all and all . Thus, , which implies that . Therefore, . What is more, if , then it immediately follows that , as well. ∎
It is advisable to note that Lemma 3.1 does not imply that or need be nontrivial elements of .
In an effort to simplify matrices and and constants , we employ several transformations which leave bracket relations (2) invariant. Namely,

redefine the elements of :
(7) 
redefine the elements of the extension:
(8) 
redefine the extension by linear combinations of :
(9) where .
The matrix used in (7) is simply required to be invertible to preserve (2). So, we may choose to be the appropriate permutation matrix which transforms into Jordan canonical form. Therefore, the classification of Leibniz algebras with abelian nilradical will rely primarily on classes defined by the Jordan canonical form of the matrix . Unfortunately, it cannot be guaranteed that the same will also transform or, or , , into Jordan canonical form. These matrices, however, will be determined by other constraints. For example, note that if has no zero eigenvalues, then exists, and so by (5c), for all .
Observe that, in (9), for the special case of a 1dimensional extension of , the matrix will be a scalar.
Note that (8) leaves and the matrices and invariant, but it will have an effect on . That is,
(10) 
In the special case where , we see that (10) appears as , implying that we cannot use (8) to manipulate when , for . This is of particular value in the case, as it implies that we may not change when .
4. 1Dimensional Extensions of
We will explore the nonLie Leibniz algebras whose abelian nilradical is extended by a single semisimple element, . In this case we have a specialized version of the previous result Lemma 3.1, as well as a general classification theorem for nonLie Leibniz algebras . Throughout this section, we will assume that is transformed by (7) into Jordan canonical form, which places the eigenvalues of on its diagonal. Note that at least one of these eigenvalues must be nonzero, or else would act nilpotently on from the right. Once this transformation is complete, we will reorder the basis elements of such that any Jordan blocks associated with zero eigenvalues are moved to the bottom right corner of . We do this reordering in such a way so as to preserve each Jordan block. Lastly, we perform transformation (9). Since , is any nonzero scalar of our choosing. Let , where is the first eigenvalue of , associated with the nilradical basis element . This ensures that . Putting the result back in Jordan form preserves the 1’s off the main diagonal.
Lemma 4.1.
Proof.
Statement (1) follows directly from Lemma 3.1, in the case of a 1dimensional extension.
For statement (2), recall that we may choose the appropriate permutation matrix which transforms into Jordan canonical form by conjugation. Once this transformation is applied, consists of Jordan blocks with each nonzero column of either having a single nonzero entry or two nonzero entries. Suppose that column has a single nonzero entry in row . Then, by (6), we know that .
Suppose now that column has two nonzero entries. Then, these entries must be and , since they are components of , an Jordan block associated with the eigenvalue . (Hence or .) So, there is a smallest integer such that the column of has a single nonzero entry, namely , the first nonzero entry of , and hence . Now consider the column of , which must have two nonzero entries and . By (6), it must be the case that . Since and , then . An iteration of this process will yield that , so as needed. ∎
Lemma 4.2.
Let be a solvable Leibniz algebra that is a 1dimensional extension by of an dimensional abelian nilradical, . If the matrix , which defines the rightaction of on , is nonsingular, then is a Lie algebra.
Proof.
Theorem 4.3.
Let be an dimensional solvable nonnilpotent Leibniz algebra over the field , of characteristic zero with . Let have the dimensional abelian nilradical, namely , and let be subject to (3). We may choose the basis of to be , where and . The bracket relations for elements in are defined by , . Letting , then the left and right bracket relations of x are defined by
(11a)  
(11b) 
where
We will specifically describe in detail in the cases of .
4.1. Leibniz algebras of nonLie type
For , the products are , , and , with . By Lemma 4.2, we see that if , then is a Lie algebra. Suppose, then, that , implying that , else . Thus, using (9) with , we obtain the algebra described in Table 1. Note that since , there is no restriction placed on . These algebras are classified in section 1 of Table 1.
4.2. Leibniz algebras of nonLie type
For , the only two Jordan canonical forms are
(12a)  
(12b) 
where and are eigenvalues of the matrix, , that we are transforming, where and are not necessarily distinct. By Lemma 4.2, must be singular. This leaves us with three possibilities for :
4.2.1. Case 1: .
4.2.2. Case 2:
.
4.2.3. Case 3:
4.3. Leibniz algebras of nonLie type
Solvable Leibniz algebras of nonLie type are 1dimensional extensions of a 3dimensional abelian nilradical. Similar to the previous section, we begin classification by first considering the possible Jordan canonical forms for matrices. Again, we arrange the eigenvalues so that the nonzero eigenvalues, if there are any, come before zero eigenvalues. The possible Jordan canonical forms that matrix may take are as follows:
(14a)  
(14b)  
(14c) 
where , , and are not necessarily distinct.
As before, by Lemma 4.2, must be singular. This means that we can force at least one to be zero. If , then once we perform transformation (9) with , the leading eigenvalue becomes 1. Putting the result back in Jordan form, the following possibilities for remain:
(14)  
(15) 
where .
4.3.1. Case 1: .
As before, since we can use (8) to put in Jordan form. If is nilpotent (one of the matrices in (14)) then is nilpotent, which contradicts . Thus we can assume , and we can use (9) to make . This means that can be written in the form of one of the matrices in (14a), (14b), or (14c) with . The resulting algebras are classified in Table 1.
4.3.2. Case 2:
4.3.3. Case 3:
4.3.4. Case 4:
where and are not necessarily distinct. If or is nonzero, then by (10) an appropriate choice of in (8) will make the corresponding be zero. Additionally, if in the second case, choosing makes . If in the first case (), then our only constraint on is that , , and are not all zero (or else is Lie). However in this case, we are free to interchange and (hence and ), so we can assume . The resulting algebras are classified in Table 1.
4.3.5. Case 5:
4.3.6. Case 6:
4.3.7. Case 7:
5. 2Dimensional Extensions of
The case of , which are dimensional Leibniz algebras with an dimensional abelian nilradical, seems significantly more complex when . This is also true in the Lie case, as seen in [15]. So, we narrow our focus to the case of 2dimensional extensions of 2dimensional abelian nilradicals.
The Leibniz algebra has basis , satisfying relations (2), (4a), and (4b). As before, we may select so that is transformed into Jordan canonical form. Therefore, will be of the form (12a) or (12b). Once is determined, the structure of can likewise be determined based on other restrictions, namely (5a), (5b), and (5c). In particular, (5c) guarantees that if is nonsingular, then and .
5.1. not diagonal
We first consider of the form (12b). In this case, since , is nonsingular. By (5c), this implies that and . So by (5a), we have . This implies that must be of the following form:
By our requirement for nilindipendence, it must be true that and . However, if and , then and will not be nilindependent. So, there are no Leibniz algebras defined by of the form (12b).
5.2. diagonal
We now consider of the form (12a). Note here that, since at least one of or must be nonzero, we will specify that . Due to the freedom of , we further divide such algebras into three cases.

and


5.2.1. Case 1:
Suppose that and . Then, is invertible and we have that and , by (5c). Then, by (5a) and since , is a diagonal matrix. That is, the products of and on are defined, respectively, by the matrices
where we are denoting by and by for ease of notation.
Since and are nilindependent, using (9) we can take a linear combination of so that
Applying Lemma 3.1, we find that for all , . Therefore, the solvable Leibniz algebra with products of defined by the above assumptions on , is also isomorphic to a Lie algebra.
5.2.2. Case 2:
Suppose . Then is invertible, so by (5c) and . Since is a scalar multiple of the identity matrix, we can use (7) to put in Jordan form and will be left invariant. Then will be of form (12a) or (12b), but if , then and will not be nilindependent, a contradiction. Hence is a diagonal matrix with , so either or plus a multiple of will have and , so we obtain an algebra from Case 1 by interchanging and . Thus we find no new Leibniz algebras in this case.
5.2.3. Case 3:
Suppose . Then (5b) implies that and are diagonal. Applying (5c) with , we find that is uppertriangular. Since and are nilindependent this implies that . Applying (5c) with other choices of and we find that and (and is diagonal). Again, either or plus a multiple of will have and , so we obtain an algebra from Case 1 and find no new Leibniz algebras in this case.
5.3. General result for solvable Leibniz algebras
By the allowable transformations (7) and (9), with the arguments given in 5.1 and 5.2, we have a general result on all solvable Leibniz algebras of type .
Theorem 5.1.
All solvable Leibniz algebras , 2dimensional extensions of a 2dimensional abelian nilradical, are also Lie algebras.
Acknowledgements. This work was completed as part of a undergraduate independent research project at Spring Hill College. The authors would like to extend gratitutde to the support given by the Department of Mathematics at Spring Hill College.
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No.  restrictions  

(1)  
(2.1)  
(2.2)  
(2.3)  
(2.4)  
(2.5)  , not both zero  
(3.1)  
(3.2)  
(3.3)  
(3.4)  
(3.5)  
(3.6)  
(3.7)  or  
(3.8)  
(3.9)  
(3.10)  
(3.11)  
(3.12)  
(3.13)  
(3.14) 