In this paper, we study nilpotent Lie algebras that admit nilsoliton metric with simple pre-Einstein derivation. Given a Lie algebra
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:
Nilsoliton metric Lie algebras are unique up to isometry and scaling. This 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 [
There 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
In this work, the nonzero structure constants are encoded by using the index set
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.
This 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.
Let
Let
In the following, we define root vectors and root matrix, Gram matrix, Nikolayevsky basis.
Suppose that
The Gram matrix
Nikolayevsky showed that every Lie algebra admitting a derivation with all the eigenvalues of multiplicity one has a nice basis [
Now, suppose that
(Theorem 1 in [
The above theorem indicates a Lie algebra
In this section, we introduce notions and prove theorems for the computation structural elements of the nilsoliton metric Lie algebras.
Now, we present the theorems which help to create possible Jacobi identity equations.
(see [
(Lemma 2.8 in [ If
The reverse of Lemma
Suppose that
Its corresponding Gram matrix is as follows:
It is a singular matrix with
The counter example provided in Remark
In order an algebra to be a Lie algebra, one needs to satisfy the Jacobi identity. Using our index set
Since we use the Nikolayevsky basis, there is a unique
If there are
It is called set of product couples for
Here, AoM is the matrix of
Let
The proof of Remark
Let
Let
Here,
In some cases, there can be more than two square root product couples in equation (
If there are
Suppose that
As we know that there is at most three, at least two product couples in the Jacobi identity, the there is at most three different sign choices for a Jacobi identity.
We define the matrix of all sign choices
The equations are created by
Therefore, we have
Suppose that
The solution of the equation
For
It can easily be seen that there is a unique sign choice for the above equations. The common solution is as follows:
The following proposition and its following corollary help us to compute rank of a nilsoliton metric Lie algebra.
(see Proposition 4.7 in [
Let
Suppose that
On the other hand,
Let
The index of a Lie algebra
(see Proposition 4 in [
above proposition tells us that the index of a Lie algebra is the nullity of the matrix
Now, we present an example regarding to this notion with the use of the index set
Suppose that
Since there is no
Its rank is 2; therefore,
For an The matrix is an The rank of The If
Suppose that the Nikolayevsky basis is represented by
Therefore,
The entries of
Suppose that
The index of the nilsoliton is
In this paper, we prove theorems for the computations of structural elements of an
No data were used in this study.
The author declares no conflicts of interest.