On Some Structural Components of Nilsolitons

In this paper, we study nilpotent Lie algebras that admit nilsoliton metric with simple pre-Einstein derivation. Given a Lie algebra η, we would like to compute as much of its structure as possible. (e structural components we consider in this study are the structure constants, the index, and the rank of the nilsoliton derivations. For this purpose, we prove necessary or sufficient conditions for an algebra to admit such metrics. Particularly, we prove theorems for the computation of the Jacobi identity for a given algebra so that we can solve the system of the equation(s) and find the structure constants of the nilsoliton.


Introduction
On any Lie group, it is possible to define several different Riemannian metrics. Considering any Riemannian metrics, Einstein metrics are the most preferable metrics, as the Ricci tensor complies the Einstein metric: Ric � cg, for some constant c ∈ R. But, it is not possible to define Einstein metrics on nonabelian nilpotent Lie algebras; therefore, we consider the following weaker condition on a left invariant metric g on a nilpotent Lie group G: for some β ∈ R and D ∈ Der(η), where Ric g denotes the Ricci operator of (η, g), η is the Lie algebra of G, and Der(η) denotes the Lie algebra of derivations of η. Equation (1) is called the nilsoliton condition, D is called the nilsoliton derivation, and β is called the nilsoliton constant. Nilsoliton metric Lie algebras are unique up to isometry and scaling. is is one of the reasons that makes nilsolitons an important topic. On the other hand, there is a one to one correspondence between Einstein nilradicals with nilsolitons. In [1], eorem 2.11 states that a nilpotent Lie algebra η is an Einstein nilradical if and only if η admits a nilsoliton metric. erefore, it indicates that classification of nilsoliton metrics on a nilpotent Lie algebra is equivalent to the same of Einstein nilradicals. Additionally, if δ is Einstein solvmanifold, then the metric restricted to the submanifold can completely be determined by the nilsoliton metric Lie algebra η � [δ, δ]. On the contrary, any nilsoliton can uniquely extend to an Einstein solvmanifold. erefore, the study of solvmanifolds is actually the study of nilsolitons. See [1,2], for a survey on nilsoliton metric Lie algebras.
ere are three methods to represent a Lie algebra and its related structures: representing a Lie algebra as a linear Lie algebra, i.e., subalgebra of gl(n), using table of its structure constants or using generators and relations [3]. In this paper, we use the table of structure constants related to a Lie algebra. e main reason is that this representation helps one to create and classify Lie algebras by the computer software programs as in [4][5][6]. We vary the Lie algebra structure by finding structure constants. Namely, we determine a Lie algebra η with a fixed basis X i : 1 ≤ i ≤ n explicitly by given multiplication table, consisting of structure constants α k ij which are defined by the relations In this work, the nonzero structure constants are encoded by using the index set Λ � (i, j, k)|α k ij ≠ 0, i < j < k , ignoring repetitions due to skew-symmetry. While indexing the structure constants, we use triples (i, j, k) ∈ Λ such that i < j < k, and if (i, j, k), (i, j, m) ∈ Λ, then k � m and (i, j 1 , k), (i, j 2 , k) ∈ Λ, then j 1 � j 2 . For this purpose, we fix a basis X 1 , . . . , X n for a nilpotent Lie algebra η with [Xi, Xj] � α k ij X k ≠ 0 such that for every i, j, # k: α k ij ≠ 0 ≤ 1, and for every i, k, # j: α k ij ≠ 0 ≤ 1. Such basis X j is called nice and defined by Nikolayevsky in [7]. In our paper, we call this basis as "Nikolayevsky basis." Using this special basis, we do not need to use sum symbol in equation (2).
In this paper, we prove theorems regarding to some structural components of nilsolitons for possible algorithmic approach in classifications. In particular, we present some new concepts and theorems regarding to create Jacobi identity equations for a given nilsoliton metric Lie algebra. We also present methods for the computations of the index and the rank of a nilsoliton. We provide several examples to illustrate the newly proposed concepts and methods.
is paper consists of four sections. In the second section, we present preliminary background. In the third section, we prove necessary theorems that help us to calculate structural elements of nilsolitons. In the last section, we present concluding statements.

Preliminaries
Let (η μ , < , > ) be a metric algebra, where μ ∈ Λ 2 η ⊗ η * . Let B � X i n i�1 be a 〈, 〉-orthonormal basis of η μ (we always assume that basis is ordered). e nil-Ricci endomorphism Ric μ is defined as for X, Y ∈ η. When η is a nilpotent Lie algebra, the nil-Ricci endomorphism is the Ricci endomorphism. If all elements of the basis are eigenvectors for the nil-Ricci endomorphism Ric μ , we call the orthonormal basis a Ricci eigenvector basis. Let Der(η) denote the derivation algebra of η. A maximal abelian subalgebra of Der(η) comprised of semisimple elements which is called a maximal torus. e dimension of a maximal torus is called the rank of η.
In the following, we define root vectors and root matrix, Gram matrix, Nikolayevsky basis.
is a finite set which indexes the set of nonzero structure constants corresponding to a Lie algebra η, ignoring repetitions due to skew-symmetry. For 1 ≤ i, j, k ≤ n, we define 1 × n row vector y k ij to be ϵ is the standard orthonormal basis for R n . We call the vectors in y k ij |(i, j, k) ∈ Λ root vectors for Λ. Let y 1 , y 2 , . . . , y m (where m � |Λ|) be an enumeration of the root vectors in dictionary order. We define root matrix Y Λ for Λ to be the m × n matrix whose rows are the root vectors y 1 , y 2 , . . . , y m . e Gram matrix U Λ for Λ is the m × m matrix defined by U Λ � Y Λ Y T Λ ; the (i, j) entry of U Λ is the inner product of the i th and j th root vectors (unless otherwise stated, the matrix U means the Gram matrix corresponding to the index set Λ. erefore, from now on, we do not use U Λ ). From eorem 5, in [8], we know that U is a symmetric matrix where its all diagonal entries are 3 and its off-diagonal entries are in the set − 2, − 1, 0, 1, 2 { }. Nikolayevsky showed that every Lie algebra admitting a derivation with all the eigenvalues of multiplicity one has a nice basis [7]. We use this type of basis in our study. is way our Gram matrices corresponding to metric nilpotent Lie algebras does not have a 2 and − 2 as an entree (Lemma 2 in [4]). Now, suppose that |Λ| � m and [1] m represents a col- Theorem 1 (Theorem 1 in [8]). Let η be a nonabelian metric algebra with Ricci eigenvector basis B. Let U and [α 2 ] be the Gram matrix and the structure vector for η with respect to B. en, η satisfies the nilsoliton condition with nilsoliton The above theorem indicates a Lie algebra η which admits a nilsoliton metric iff there exists a solution v ∈ R m of the linear system Uv � [1] m where all entries are positive real numbers.

The Structural Elements
In this section, we introduce notions and prove theorems for the computation structural elements of the nilsoliton metric Lie algebras.

e Jacobi Identity(s)
. Now, we present the theorems which help to create possible Jacobi identity equations.
Theorem 2 (see [8]). Let η be an n-dimensional vector space; B � X i n i�1 be a basis for η. Suppose that a set of nonzero structure constants α k i,j relative to B, indexed by Λ, defines a skew symmetric product on η. Assume that if (i, j, k) ∈ Λ, then i < j < k. en, η is a Lie algebra if and only if whenever there exists m so that the inner product of root vectors 〈y l ij , y m lk 〉 � − 1 for triples (i, j, l) and (l, k, m) or (k, l, m) in Λ, the equation holds. Furthermore, a term of form α l i,j α m l,k is nonzero if and only if 〈y l i,j , y m l,k 〉 � − 1.
Lemma 1 (Lemma 2.8 in [4]). Let η be an n-dimensional nonabelian nilpotent Lie algebra. Suppose that η admits a derivation D having distinct real positive eigenvalues. Let B be a basis consisting of eigenvectors for the derivation D, and let Λ index the nonzero structure constants with respect to B. Let Y be the m × n root matrix for Λ. If rank(Y) � m, then the following hold: (1) |Λ| ≤ n − 1.
□ e counter example provided in Remark 1 illustrates none of the inner products of root vectors being − 1 does not imply that the Gram matrix of the nilsoliton is nonsingular. Additionally, if the cardinality of the index set |Λ| ≤ n, − 1 does not imply that the Gram matrix is nonsingular.

Constructing the System of Jacobi Identities.
In order an algebra to be a Lie algebra, one needs to satisfy the Jacobi identity. Using our index set Λ, the corresponding Jacobi identity turns into equation (4). Also, there has to be at least two product couples in the Jacobi identity. Otherwise, if there is one product couple in Jacobi identity, it leads to α s i,j α m s,k � 0; therefore, it contradicts to the fact that (i, j, s), (s, k, m) ∈ Λ. On the other hand, the Jacobi identity is created by each vector triples from the given fixed Nikolayevsky basis. For example, the Jacobi identity for Since we use the Nikolayevsky basis, there is a unique k which appears in equation (9). erefore, each Jacobi identity is created with fixed X i , X j , X k , and m ∈ AoM. Also, because there has to be at least one product couple in the Jacobi identity, then there exist 2 or three product couples in the Jacobi identity. In equation (9), the product couple α s ij α m sk corresponds to the index triples in Λ such that (i, j, s), (s, k, m) ∈ Λ, or (i, j, s), (k, s, m) ∈ Λ. In the following definition, we define the set of all product couples P m related to the Jacobi identity for a given subset X i , X j , X k of the fixed basis. As one can see, there exist at most 3 product couples in the same Jacobi identity.

Definition 2.
If there are t product couples in the Jacobi identity for the same m such that (i, j, s), (s, k, m) ∈ Λ, or (i, j, s), (k, s, m) ∈ Λ, then the Jacobi identity which was created by the basis vectors X i , X j , X k . Now, we define the set of all nonzero product couples for a given index triple basis vectors X i , X j , X k as follows: It is called set of product couples for #m and for in equation (4)

(11)
Here, AoM is the matrix of #m that appears in equation (4).

Remark 2.
Let η be an algebra that is indexed by Λ, then the Jacobi identity for a given X i , X j , X k with each m ∈ AoM is given by e proof of Remark 2 follows from the definition of Jacobi identity for a given algebra.

Theorem 3. Let η be an algebra that is indexed by Λ, and U be the Gram matrix related to Λ. Let v p belong to the solution space to the linear system Uv � [1], then the Jacobi identity for each m ∈ AoM is given by
���� Proof. Let η be an algebra that is indexed by Λ. By Remark 3, the Jacobi identity for each m ∈ AoM and each basis vector triple X i , X j , X k is given by Here, p s � α s ij α m s,k . By eorem 1, the solution vectors of the linear system Uv � [1] are the squares of the structure vectors α s ij for all (i, j, s) ∈ Λ. erefore, for any v that us, for each p s ∈ P m , where a, b ∈ i 1 , i 2 , i 3 , j 1 , j 3 , and v a � (α s ij ) 2 and v b � (α m sk ) 2 . Solving the equations for the structure constants, we have for all (i, j, s) ∈ Λ. erefore, equation (12) turns into which finishes the proof.

□
In some cases, there can be more than two square root product couples in equation (13), i.e., there is more than one m ∈ AoM. In that case, we need to consider all the cases of the signs between the product couples α s i,j α m s,k , α s j,k α m s,i , and α s k,i α m s,j . e following lemma deals with this matter. Proof. Suppose that p i ∈ P m for m in equation (4) where P m is the set of all product couples, as in Definition 2. erefore, the Jacobi identity turns into p 1 ∓p 2 ∓p 3 � 0. Without loss of generality, we assume that p 1 > 0. For each p s ∈ P m , where 2 ≤ s ≤ n, there are two possible sign choices +, − { }. erefore, we have 2 t− 1 possible sign choices. Since all the product couples are nonzero, then they can not all be +. erefore, we drop the case (+, +, . . . , +). us, there are 2 t− 1 − 1 possible sign choices for the set P m .
For m � 8, we have two product couples of the triples (i, j, l) and (l, k, m) or (k, l, m) in Λ.
ey are (2, 4, 7) (1,7,8) and (1,4,5) (2,5,8). So, the corresponding Jacobi identity is It can easily be seen that there is a unique sign choice for the above equations. e common solution is as follows: 3.2. e Rank and the Index of a Nilsoliton. e following proposition and its following corollary help us to compute rank of a nilsoliton metric Lie algebra.
Proposition 1 (see Proposition 4.7 in [9]). Let η be a nonabelian Lie algebra that admits a simple derivation D. Let B � Xi { } n i�1 be an eigenvector basis with index set Λ, and let Y be the root matrix associated to Λ. en, the rank of the nilsoliton metric Lie algebra η equals to the nullity of the root matrix Y. Corollary 1. Let η be a nonabelian n− dimensional Lie algebra that admits a simple derivation. Let B be an eigenvector basis with index set Λ, and let Y be the root matrix associated to Λ. en, rank(η) � n + nullity YY t − |Λ|. (26) Proof. Suppose that η is an n− dimensional nonabelian Lie algebra, admitting a simple derivation. Let B be an eigenvector basis with index set Λ where |Λ| � m, and let Y be the root matrix associated to Λ. By Proposition 1, the root matrix Y is an m × n matrix, whose nullity is rank(η). erefore, from rank-nullity theorem, rank(η) � n − Rank(Y). We also know that rank(Y) � rank(YY T ); therefore, we have On the other hand, YY T is an m × m matrix. erefore, Rank(YY T ) � m − Nullity(YY T ). en, we have rank(η) � n − m − Nullity YY T � n + Nullity YY T − |Λ|.
(28) □ Definition 4. Let X ∈ η, ad X , denote the adjoint representation and η * denote the dual of the Lie algebra η. en, the skew symmetric bilinear form Ψ f , where f ∈ η * , is defined by (29) e index of a Lie algebra η is the integer Proposition 2 (see Proposition 4 in [10]). e index of an n-dimensional Lie algebra η is the integer, where R(η) is the quotient field of symmetric algebra S(η).
Remark 3. above proposition tells us that the index of a Lie algebra is the nullity of the matrix E η � ([X i , X j ]). Now, we present an example regarding to this notion with the use of the index set Λ.

Proposition 3.
For an n− dimensional nilsoliton represented by an index set Λ with a Nikolayevsky basis, the matrix E η has the following properties: (1) e matrix is an n × n matrix with zero diagonal entries (2) e rank of E η is even (3) e n th row and column of E η are zero matrices (4) If |Λ| � K, then E η has 2K nonzero entries Proof. Suppose that the Nikolayevsky basis is represented by erefore, E η is a skew symmetric matrix, which implies the first and second properties. On the other hand, since η is Mathematical Problems in Engineering nilpotent Lie algebra with Nikolayevsky basis, then Λ consists of the elements of form (i, j, k) such that i < j < k. erefore, n cannot appear in the first or the second component of the triples in Λ. us, [X i , X n ] � [X n , X i ] � 0 for all i ∈ 1, 2, . . . , n { }, which implies that (E η ) in � (E η ) ni � 0, i.e., the last row and the last column are zero matrices. e entries of E η are defined by the index set Λ. If |Λ| � K, then there exists K non-zero entries on the upper triangular part of E η . Since the matrix is skew symmetric, then there exists other K nonzero entries in the lower triangular part of the matrix. So, in total, there exists 2K nonzero entries in E η . □ Corollary 2. Suppose that η is an n− dimensional nilsoliton. en, index(η) is even, if n is even, and index is odd if n is odd.
Proof. e index of the nilsoliton is index(η) � Nullity(E η ) � n − rank(E η ). Since E η is a skew symmetric matrix, its rank is always even, which finishes the proof.

Conclusion
In this paper, we prove theorems for the computations of structural elements of an n-dimensional nilsoliton η. We prove theorems regarding to the Jacobi identity or identities that have to be satisfied, rank, and index of the nilsoliton. In the future, we plan to use these theorems to create a computer algorithm for the classifications of nilsolitons for a given dimension. e theorems appearing in this study will allow us to pare down the number of cases to consider in our procedure.

Data Availability
No data were used in this study.

Conflicts of Interest
e author declares no conflicts of interest.