Classification of Ordered Type Soliton Metric Lie Algebras by a Computational Approach

Abstract

We classify some soliton nilpotent Lie algebras and possible candidates in dimensions 8 and 9 up toisomorphy. We focus on 1<2<⋯<𝑛 type of derivations, where 𝑛 is the dimension of the Lie algebras. We present algorithms to generate possible algebraic structures.

1. Introduction

In this paper, we compute and classify 𝑛-dimensional (𝑛=8,9) nilsoliton metric Lie algebras with eigenvalue type 1<2<⋯<𝑛, which will be called “ordered type of Lie algebra” throughout this paper. We use MATLAB to achieve this goal. In the literature, six-dimensional nilpotent Lie algebras have been classified by algorithmic approaches [1]. In dimension seven and lower, nilsoliton metric Lie algebras have been classified [2–9]. Summary and details of some other classifications can be found in [10]. In our paper, we focus on dimensions eight and nine. We note that we have found that our algorithm gives consistent results with the literature in lower dimensions. We use a computational procedure that is similar to the one that we have used in our previous paper [4].

In our previous paper, we classified all the soliton and nonsoliton metric Lie algebras where the corresponding Gram matrix is invertible and of dimensions 7 and 8 up to isomorphism. If corresponding Gram matrix is invertible, then the soliton metric condition 𝑈𝑣=[1] has a unique solution. So in this case, it is easy to check if the algebra is soliton or not. But in noninvertible case, there is more than one solution. Therefore it is hard to guess if one of the solutions provides the soliton condition without solving Jacobi identity which is nonlinear. On the other hand, it may be easy if we can eliminate some algebras which admit a derivation 𝐷 that does not have ordered eigenvalues without solving the following soliton metric condition 𝑈𝑣=[1]. For this, we prove that if the nilpotent Lie algebra admits a soliton metric with corresponding Gram matrix of 𝜂 being noninvertible, all the solutions of 𝑈𝑣=[1] have a unique derivation. This theorem allows us to omit several cases that come from nonordered eigenvalues without considering Jacobi identity.

This paper is organized as follows. In Section 2, we provide some preliminaries that we use for our classifications. In Section 3, we give specific Jacobi identity conditions for Lie algebras up to dimension nine. This allows us to decide whether the Lie algebra has a soliton metric or not. In Section 4, we give details of our classifications with specific examples and provide algorithmic procedures. Section 5 contains our concluding remarks.

2. Preliminaries

Let (𝜂_𝜇,𝑄) be a metric algebra, where 𝜇∈Λ²𝜂⊗𝜂^∗. Let 𝐵={𝑋_𝑖}^𝑛_𝑖=1 be a 𝑄-orthonormal basis of 𝜂_𝜇 (we always assume that bases are ordered). The nil-Ricci endomorphism Ric_𝜇 is defined as ⟨Ric_𝜇𝑋,𝑌⟩=ric_𝜇(𝑋,𝑌), where ric_𝜇()1𝑋,𝑌=−2^𝑛_𝑖=1𝑋,𝑋_𝑖,𝑌,𝑋_𝑖+14^𝑛_𝑖=1𝑋_𝑖,𝑋_𝑗,𝑋𝑋_𝑖,𝑋_𝑗,𝑌 (1) for 𝑋,𝑌∈𝜂 (we often write an inner product 𝑄(⋅,⋅) as ⟨⋅,⋅⟩). 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.

Now we define some combinatorial objects associated to a set of integer triples Λ⊂{(𝑖,𝑗,𝑘)∣1≤𝑖,𝑗,𝑘≤𝑛}. For 1≤𝑖,𝑗,𝑘≤𝑛, define 1×𝑛 row vector 𝑦^𝑘_𝑖𝑗 to be 𝜖^𝑇_𝑖+𝜖^𝑇_𝑗−𝜖^𝑡_𝑘, where {𝜖}^𝑛_𝑖=1 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 𝑦₁,𝑦₂,…,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 𝑦₁,𝑦₂,…,𝑦_𝑚. The Gram matrix 𝑈_Λ for Λ is the 𝑚×𝑚 matrix defined by 𝑈_Λ=𝑌_Λ𝑌^𝑇_Λ; the (𝑖,𝑗) entry of 𝑈_Λ is the inner product of the 𝑖th and 𝑗th root vectors. It is easy to see that 𝑈 is a symmetric matrix. It has the same rank as the root matrix; that is, Rank(𝑈_Λ)=Rank(𝑌_Λ). Diagonal elements of 𝑈 are all three, and the off-diagonal entries of 𝑈 are in the set {−2,−1,0,1,2}. For more information, see [11]. Let 𝐷 have distinct real positive eigenvalues, and let Λ index the structure constants for 𝜂 with respect to eigenvector basis 𝐵. If (𝑖₁,𝑗₁,𝑘₁)∈Λ and (𝑖₂,𝑗₂,𝑘₂)∈Λ, then ⟨𝑦^𝑘1_𝑖1,𝑗1,𝑦^𝑘2_𝑖2,𝑗2⟩≠2. Thus 𝑈 does not contain two as an entry [4].

Lemma 1. Let (𝜂,𝑄) be an 𝑛-dimensional inner product space, and let 𝜇 be an element of Λ²𝜂^∗⊗𝜂. Suppose that 𝜂_𝜇 admits a symmetric derivation 𝐷 having 𝑛 distinct eigenvalues 0<𝜆₁<𝜆₂<⋯<𝜆_𝑛 with corresponding orthonormal eigenvectors 𝑋₁,𝑋₂,…,𝑋_𝑛. Let 𝛼^𝑘_𝑖𝑗 denote the structure constants for 𝜂 with respect to the ordered basis 𝐵={𝑋_𝑖}^𝑛_𝑖=1. Let 1≤𝑖<𝑗≤𝑛. Then(1)if there is some 𝑘∈{1,2,…,𝑛} such that 𝜆_𝑘=𝜆_𝑖+𝜆_𝑗, then [𝑋_𝑖,𝑋_𝑗] is a scalar multiple of 𝑋_𝑘; otherwise 𝑋_𝑖 and 𝑋_𝑗 commute; (2)𝛼^𝑘_𝑖𝑗≠0 if and only if 𝑋_𝑘∈[𝜂_𝜇,𝜂_𝜇].

Theorem 2 (see [11]). Let 𝜂 be a vector space, and let 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. Then the algebra 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 _𝑠<𝑚𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗=0 (2) holds. Furthermore, a term of form 𝛼^𝑙_𝑖,𝑗𝛼^𝑚_𝑙,𝑘 is nonzero if and only if ⟨𝑦^𝑙_𝑖,𝑗,𝑦^𝑚_𝑙,𝑘⟩=−1

Theorem 3 (see [11]). Let (𝜂_𝜇,𝑄) be a metric algebra and 𝐵={𝑋_𝑖}^𝑛_𝑖=1 a Ricci eigenvector basis for 𝜂_𝜇. Let 𝑌 be the root matrix for 𝜂_𝜇. Then the eigenvalues of the nil-Ricci endomorphism are given by Ric^𝐵_𝜇1=−2𝑌^𝑇𝑣, (3) where 𝑣=[𝛼²].

Theorem 4 (see [4, 11]). Let (𝜂,𝑄) be a nonabelian metric algebra with Ricci eigenvector basis 𝐵. The following are equivalent. (1)(𝜂_𝜇,𝑄) satisfies the nilsoliton condition with nilsoliton constant 𝛽. (2)The eigenvalue vector 𝑉_𝐷 for 𝐷=Ric−𝛽Id with respect to 𝐵 lies in the kernel of the root matrix for (𝜂_𝜇,𝑄) with respect to 𝐵. (3)For noncommuting eigenvectors 𝑋 and 𝑌 for the nil-Ricci endomorphism with eigenvalues 𝜅_𝑋 and 𝜅_𝑌, the bracket [𝑋,𝑌] is an eigenvector for the nil-Ricci endomorphism with eigenvalue 𝜅_𝑋+𝜅_𝑌−𝛽. (4)𝛽=𝑦^𝑘_𝑖𝑗Ric for all (𝑖,𝑗,𝑘) in Λ(𝜂_𝜇,𝐵).

Theorem 5 (see [4]). Let 𝜂 be an 𝑛-dimensional nonabelian nilpotent Lie algebra which 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 𝑈 be the 𝑚×𝑚 Gram matrix. If 𝑈 is invertible, then the following hold: (i)|Λ|≤𝑛−1; (ii)if (𝑖₁,𝑗₁,𝑘₁)∈Λ and (𝑖₂,𝑗₂,𝑘₂)∈Λ, then ⟨𝑦^𝑘2_𝑖1,𝑗1,𝑦^𝑘2_𝑖2,𝑗2⟩≠−1.

3. Theory

This section provides some theorems and their proofs that allow us to consider fewer cases for our algoritm. The following theorem gives a pruning method while Gram matrix is noninvertible.

Theorem 6. Let 𝜂 be an 𝑛-dimensional nilsoliton metric Lie algebra, and 𝑈 the corresponding Gram matrix which is noninvertible. Then Ker(𝑌^𝑇)=Ker(𝑈). Furthermore all of the solutions of 𝑈𝑣=[1] correspond to a unique derivation.

Proof . Since rank of a matrix is equal to the rank of its Gram matrix, then 𝑝=Rank(𝑌)=Rank(𝑈). Let 𝑈∶ℝ^𝑝→ℝ^𝑝 and 𝑌^𝑇∶ℝ^𝑝→ℝ^𝑛 denote the linear functions (with respect to the standard basis) that correspond to the Gram matrix 𝑈 and the transpose of the root matrix 𝑌, respectively. Since Rank(𝑈)=Rank(𝑌)=Rank(𝑌^𝑇) and by rank-nullity theorem, we have Ker(𝑈)=Ker𝑌^𝑇. (4) Let 𝑣 be a particular solution and 𝑣₀ the last column of reduced row echelon matrix [𝑈,[1]]. Then 𝑣₀ is also a solution of 𝑈𝑣₀=[1]. Therefore 𝑈(𝑣−𝑣₀)=0; that is, (𝑣−𝑣₀)∈Ker(𝑈). Using (4), then (𝑣−𝑣₀)∈Ker(𝑌^𝑇). For the solution 𝑣₀, suppose that we denote 𝐷₀ for the Nikolayevsky derivation, Ric₀ for the Ricci tensor, and 𝛽₀ for the soliton constant. Then using (3), we have Ric−Ric₀=12𝑌^𝑡𝑣−𝑣₀=0. (5) Then Ric=Ric₀. Using Theorem 4, we have 𝛽=𝛽₀, which implies that 𝐷=𝐷₀.

Lemma 7. If nilsoliton metric Lie algebra 𝜂 has ordered type of derivations 1<2<⋯<𝑛, then its index set Λ consists of triples (𝑖,𝑗,𝑖+𝑗).

Proof. If 𝑉_𝐷=(𝜆₁,𝜆₂,…,𝜆_𝑛)^𝑇 is the eigenvalue vector of 𝐷 with eigenvector basis 𝐵={𝑋_𝑖}^𝑛_𝑖=1 for 𝜂, then by Theorem 4, 𝑉_𝐷 lies in the kernel of 𝑌. Thus for each element (𝑖,𝑗,𝑘)∈Λ, 𝜆_𝑖+𝜆_𝑗−𝜆_𝑘=0; that is, 𝜆_𝑖+𝜆_𝑗=𝜆_𝑘. By Lemma 1, [𝑋_𝑖,𝑋_𝑗]=𝜆𝑋_𝑘 for some 𝜆∈ℝ. Since 𝜆_𝑖=𝑖 for all 𝑖∈{1,2,…,𝑛}, 𝑘=𝑖+𝑗 and [𝑋_𝑖,𝑋_𝑗]=𝜆𝑋_𝑖+𝑗. Hence, the index set for ordered type of derivations is of form (𝑖,𝑗,𝑖+𝑗).

The next corollary describes the index triples (𝑖,𝑗,𝑘) and the Jacobi identity for algebras with ordered type of derivations.

Corollary 8. The algebra 𝜂 is a Lie algebra if and only if for all pairs of form (𝑖,𝑗,𝑙) and (𝑙,𝑘,𝑚) or (𝑖,𝑗,𝑙) and (𝑘,𝑙,𝑚) in Λ_𝐵 with 𝑘∉{𝑖,𝑗} and for all 𝑚≥max{𝑖+3,𝑗+2,5}, the following equation holds: _{3≤𝑠<𝑚,𝑠∉𝑖,𝑗,𝑘}𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗=0. (6)
If in addition 𝜆_𝑖=𝑖 for 𝑖=1,…,𝑛, then the algebra 𝜂 is a Lie algebra if and only if for all pairs of form (𝑖,𝑗,𝑖+𝑗) and (𝑖+𝑗,𝑘,𝑖+𝑗+𝑘) or (𝑖,𝑗,𝑖+𝑗) and (𝑘,𝑖+𝑗,𝑖+𝑗+𝑘) in Λ_𝐵 with 𝑘∉{𝑖,𝑗,𝑖+𝑗} and for all 𝑚=𝑖+𝑗+𝑘≥max{2𝑖+2,𝑗+2,6}, the equation _{4≤𝑠<𝑚,𝑠∉𝑖,𝑗}𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗=0 (7) holds.

Proof. By Theorem 7 of [11], the algebra 𝜂_𝜇 defined by 𝜇 is a Lie algebra if and only if whenever there exists 𝑚 so that ⟨𝑦^𝑙_𝑖𝑗,𝑦^𝑚_𝑙𝑘⟩=−1 for triples (𝑖,𝑗,𝑙) and (𝑙,𝑘,𝑚) or (𝑘,𝑙,𝑚) in Λ_𝐵, (2) _𝑠<𝑚𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗=0 (8) holds. Furthermore, if 𝑖, 𝑗, and 𝑘 are distinct, the product 𝛼^𝑙_𝑖𝑗𝛼^𝑚_𝑙𝑘 is nonzero if and only if ⟨𝑦^𝑙_𝑖𝑗,𝑦^𝑚_𝑙𝑘⟩=−1.
Suppose that ⟨𝑦^𝑙_𝑖𝑗,𝑦^𝑚_𝑙𝑘⟩=−1 for (𝑖,𝑗,𝑙)∈Λ_𝐵 and (𝑙,𝑘,𝑚) or (𝑘,𝑙,𝑚) in Λ_𝐵. By definition of Λ_𝐵, we have 𝑖<𝑗. By Lemma 1, 𝑗<𝑙, 𝑙<𝑚, and 𝑘<𝑚. Since 𝑖<𝑗<𝑙<𝑚, we know that 𝑚≥𝑖+3. Similarly, 𝑗<𝑙<𝑚 implies that 𝑗+2≤𝑚. If 𝑖=𝑘 or 𝑗=𝑘, then ⟨𝑦^𝑙_𝑖𝑗,𝑦^𝑚_𝑙𝑘⟩=0, and so 𝑖,𝑗, and 𝑘 must be distinct. Since 𝑖,𝑗,𝑘, and 𝑙 are all distinct and less than 𝑚, we know that 𝑚≥5. Thus an expression of form 𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗 (9) is nonzero only if 𝑚≥max{𝑖+3,𝑗+2,5}, and 𝑘∉{𝑖,𝑗}.
Suppose that 𝜆_𝑖=𝑖, and (𝑖,𝑗,𝑙) and (𝑙,𝑘,𝑚) are in the index set. Then from Lemma 7, 𝑙=𝑖+𝑗 which implies that 𝑚=𝑖+𝑗+𝑘. We know that 1≤𝑖<𝑗<𝑙<𝑚. Then, since 𝑗≥𝑖+1 and 𝑘≥1, we have 2𝑖+2≤𝑖+𝑗+𝑘=𝑚. Since 2𝑖+2≤𝑚, 𝑚=5 implies that 𝑖=1. So there is no possible (𝑖,𝑗,𝑘), where all 𝑖,𝑗,𝑘 are distinct and 𝑖<𝑗<𝑚 with 𝑖+𝑗+𝑘. Thus if 𝑚=5, then ⟨𝑦^𝑙_𝑖𝑗,𝑦^𝑚_𝑙𝑘⟩≠−1. Therefore, if 𝜆_𝑖=𝑖, an expression of form 𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗 (10) is nonzero only if 𝑚≥max{2𝑖+2,𝑗+2,6}, and 𝑘∉{𝑖,𝑗}.
By Lemma 1, 𝑋₁ and 𝑋₂ are in [𝜂_𝜇,𝜂_𝜇]^⟂, and so 𝛼^𝑠_𝑟𝑡≠0 which implies that 𝑠≥3 and 𝑟≠𝑠, 𝑡≠𝑠. Therefore all expressions in (10) with 𝑠<3 or 𝑠∈{𝑖,𝑗,𝑘} are identically zero and may be omitted from the summation for any 𝑚.

The next corollary describes some equations in the structure constants of a nilpotent metric Lie algebra that are equivalent to the Jacobi identity. Each of the terms 𝛼^𝑠_𝑖𝑗𝛼^𝑚_𝑠𝑘 in the following equations corresponds to each of −1 entry in the Gram matrix 𝑈. Therefore, the following equations are useful for noninvertible case since there is no −1 entry in the Gram matrix for the invertible case.

Corollary 9. Let(𝜂_𝜇,⟨⋯,⋯⟩) be an 𝑛-dimensional inner product space where 𝑛≤9, and, 𝜇 be an element of Λ²𝜂^∗⊗𝜂. Suppose that the algebra 𝜂_𝜇 defined by 𝜇 admits a symmetric derivation 𝐷 having 𝑛 eigenvalues 1<2<⋯<𝑛 with corresponding orthonormal eigenvectors 𝑋₁,𝑋₂,…,𝑋_𝑛. 𝛼^𝑘_𝑖,𝑗 denote the structure constants for 𝜂 with respect to the ordered basis 𝐵={𝑋_𝑖}^𝑛_𝑖=1, and let 𝜆_𝐵 index the nonzero structure constants as defined in (2). The algebra 𝜂_𝜇 is a Lie algebra if and only if 𝛼⁴₁₃𝛼⁶₄₂+𝛼⁵₂₃𝛼⁶₅₁𝛼=0,₃₁₂𝛼⁷_3,4−𝛼⁵₁₄𝛼⁷₅₂−𝛼⁶₂₄𝛼⁷₆₁𝛼=0,₃₁₂𝛼⁸₃₅+𝛼⁵₁₄𝛼⁸₅₃+𝛼⁶₁₅𝛼⁸₆₂+𝛼⁷₂₅𝛼⁸₇₁+𝛼⁷₃₄𝛼⁸₇₁𝛼=0,₃₁₂𝛼⁹₃₆+𝛼⁴₁₃𝛼⁹₄₅−𝛼⁵₂₃𝛼⁹₄₅−𝛼⁶₂₄𝛼⁹₃₆−𝛼⁷₃₄𝛼⁹₂₇−𝛼⁸₃₅𝛼⁹₁₈=0 (11)(12)(13)(14) holds.

Proof. Following Lemma 7, the index set consists of elements of form (𝑖,𝑗,𝑖+𝑗). Therefore the number 𝑠 equals to 𝑖+𝑗 in the expression 𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘. From the previous corollary, 𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗 (15) is nonzero only if 𝑚≥6, and 𝑘∉{𝑖,𝑗}. Also 𝑘≠𝑖+𝑗; otherwise 𝛼^𝑚_𝑠,𝑘=0.
If 𝑚=6, 2𝑖+2≤𝑚 implies that 𝑖≤2; that is, possible numbers for “𝑖” are 1 and 2. Possible and not possible (𝑖,𝑗,𝑘) triples, which are being used in _{3≤𝑠<𝑚,𝑠∉𝑖,𝑗,𝑘}𝛼^𝑠_𝑖,𝑗𝛼^𝑚_𝑠,𝑘+𝛼^𝑠_𝑗,𝑘𝛼^𝑚_𝑠,𝑖+𝛼^𝑠_𝑘,𝑖𝛼^𝑚_𝑠,𝑗=0 (16) and where 𝑖+𝑗+𝑘=6, are illustrated in Table 1. The notations in the table are as follows ✓∶=yes; −=no; PT= possible triple.
In the case (a), 𝑖+𝑗=𝑘, and then it is not a possible triple. In the case (b), 𝑖<𝑗, 𝑖,𝑗,𝑘 are distinct, and 𝑘≠𝑖+𝑗. So it is a possible triple. In the case (c), 𝑖=𝑘, and so it is not a possible triple. In the case (d), 𝑖<𝑗, and all 𝑖,𝑗,𝑘 are distinct as 𝑖+𝑗≠𝑘. Thus it is a possible triple. In the case (e), 𝑘 is not a natural number, and so it is not a possible triple. Therefore only possible (𝑖,𝑗,𝑘) triples are (1,3,2) and (2,3,1). Triples (1,3,2) and (2,3,1) correspond to nonzero products 𝛼⁴₁₃𝛼⁶₄₂ and 𝛼⁵₂₃𝛼⁶_5,1, respectively. Using the skew-symmetry, (2) turns into the following equation: 𝛼⁴₁₃𝛼⁶₂₄+𝛼⁵₂₃𝛼⁶₁₅=0, (17) which gives (11). Using the same procedure for 𝑚=7, possible (𝑖,𝑗,𝑘) triples are (1,2,4),(1,4,2) and (2,4,1), which correspond to nonzero 𝛼³₁₂𝛼⁷₃₄, 𝛼⁵₁₄𝛼⁷₅₂, and 𝛼⁶₂₄𝛼⁷₆₁ respectively. Therefore (12) is obtained. Equations (13) and (14) can be obtained by the same way.

Table 1 

Possible (𝑖,𝑗,𝑘) for 𝑚=6.

As an illustration, we show how to use the results of this section in the following example.

Example 10. Let 𝜂 be an 8-dimensional algebra with nonzero structure constants relative to eigenvector basis 𝐵 indexed by ),(),(),(),(),(),(),(.Λ={(1,2,31,3,41,4,51,6,72,3,52,6,83,4,73,5,8)} (18)
Computation shows that the structure vector [𝛼²] is a solution to 𝑈𝑣=[1]_8×1 if and only if it is of form 𝛼²=𝛼³₁₂²𝛼⁴₁₃²𝛼⁵₁₄²𝛼⁷₁₆²𝛼⁵₂₃²𝛼⁸₂₆²𝛼⁷₃₄²𝛼⁸₃₅²=−317101715−817−517171271700100001011101.17+𝑋⋅−1−1+𝑌⋅−1−1−1−1 (19)
Equation (13) from the previous corollary leads −317+𝑋+𝑌⋅𝑌=15917−𝑋−𝑌⋅𝑌⟹𝑋+𝑌=.17 (20) Moreover, using (12), we find that 𝑋=3/17 and 𝑌=6/17, which means that (𝛼⁷₁₆)²=−2/17. Thus 𝜂 is not a Lie algebra.

4. Algorithm and Classifications

In this section, we describe our computational procedure and give the results in dimensions 8 and 9.

4.1. Algorithm

Now we describe the algorithm. The following algorithm can be used for both invertible and noninvertible cases.

Input. The input is the integer 𝑛 which represents the dimension.

Output. The output is two 0−1 matrices Wsoliton and Uninv listing characteristic vectors for index sets Λ of Θ_𝑛. The matrix Wsoliton has as its rows all possible characteristic vectors for canonical index sets Λ for nilpotent Lie algebras of dimension 𝑛 with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix 𝑈 is invertible. The matrix Uninv has as its rows all possible characteristic vectors for canonical index sets Λ for nilpotent Lie algebras of dimension 𝑛 with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix 𝑈 is noninvertible. In the dimensions 8 and 9, there is no example for invertible case. Thus Wsoliton is an empty matrix. Therefore we give the algorithm for the noninvertible case.

Algorithm for the Noninvertible Case. Consider the following.(i)Enter the dimension 𝑛. (ii)Compute the matrix 𝑍_𝑛. (iii)Compute the matrix 𝑊. (iv)Delete all rows of 𝑊 containing abelian factor which is the row that represents direct sums of Lie algebras. (v)Remove all rows of 𝑊 such that the canonical Gram matrix 𝑈 associated to the index set Λ is invertible. (vi)Define eigenvalue vector 𝑣_𝐷=(1,2,3,…,𝑛)^𝑇 in dimension 𝑛. (vii)Remove all rows of 𝑊 if 𝑣(𝑖)=𝑣₀(𝑖)≤0 where 𝑣 is the general solution of 𝑈_Λ𝑣=[1]_𝑚×1 and 𝑣₀ is the vector that we have defined in the proof of Theorem 6. (viii)Remove all the rows of 𝑊 such that the corresponding algebra does not have a derivation of eigenvalue type 1<2<⋯<𝑛. (ix)Remove all the rows of  𝑊 such that the corresponding algebra does not satisfy Jacobi identity condition, which is obtained in Corollary 9.

After this process, we solve nonlinear systems which follow from Jacobi identity. In order to see how the algorithm works, we give the following example for 𝑛=6.

Example 11. Let 𝑛=6. Then Θ₆={(1,2,3),(1,3,4),(1,4,5),(1,5,6),(2,3,5),(2,4,6)}. (21) So, matrix 𝑍₆ is 6×3 of form 𝑍₆=123134145156235246. (22) Since |Θ₆|=6, the matrix 𝑊 is of size 2⁶×6 as follows 𝑊_Θ=000000000001000010⋮⋮⋮⋮⋮⋮111111. (23) The first row of 𝑊_𝜃 represents empty matrix, row two represents the subset {(2,4,6)} of Θ₆, and so forth. Eliminating rows that represent direct sums, we have 33 rows in 𝑊 matrix. Therefore none of the rows of 𝑊 corresponds to Lie algebras that can be written as direct sums. These algebras correspond both to invertible and noninvertible Gram matrices. There is no example for the invertible case. For the noninvertible case, there is one ordered type nilsoliton metric Lie algebra 𝜂. Let 𝐵 be the eigenvector basis for 𝜂, whose nonzero structure constants are indexed by Λ_𝜂={(1,2,3),(1,3,4),(1,4,5),(1,5,6),(2,3,5),(2,4,6)}. (24) Computation shows that the structure vector [𝛼²] is a solution to 𝑈𝑣=[1]_6×1 if and only if it is of form 𝛼²=𝛼³₁₂²𝛼⁴₁₃²𝛼⁵₁₄²𝛼⁶₁₅²𝛼⁵₂₃²𝛼⁶₂₄²=1212550+𝑡1430101.143−1−1 (25) By Corollary 9, 𝜂 satisfies (11). Solving the equation for 𝑡, we find that 𝛼⁴₁₃𝛼⁶₂₄²=𝛼⁵₂₃𝛼⁶₁₅²,(11+𝑡)𝑡=(55−𝑡)²,𝑡=25. (26) After rescaling and solving for structure constants from [𝛼²], we see that letting 𝑋₁,𝑋₂=√22𝑋₃,𝑋₁,𝑋₃=6𝑋₄,𝑋₁,𝑋₄=√22𝑋₅,𝑋₁,𝑋₅=√30𝑋₆,𝑋₂,𝑋₃=√30𝑋₅,𝑋₂,𝑋₄=5𝑋₆ (27) defines a nilsoliton metric Lie algebra, previously found in [3].

4.2. Classifications

Classification results for dimensions 8 and 9 appear in Tables 2 and 3, respectively. We use vector notations to represent Lie algebra structures. For example, the list 0,0,0,0,√45⋅23,√14⋅24,√91⋅25+√91⋅34,√+√136⋅1729⋅26+√14⋅35,√104⋅18+4⋅27 (28) in the first row of Table 3 is meant to encode the metric Lie algebra (𝑛,⟨⋅,⋅⟩) with orthonormal basis 𝐵={𝑋_𝑖}⁸_𝑖=1 and bracket relations 𝑋₂,𝑋₃=√45𝑋₅,𝑋₂,𝑋₄=√14𝑋₆,𝑋₂,𝑋₅=√91𝑋₇,𝑋₃,𝑋₄=√91𝑋₇,𝑋₁,𝑋₇=√136𝑋₈,𝑋₂,𝑋₆=√29𝑋₈,𝑋₃,𝑋₅=√14𝑋₈,𝑋₁,𝑋₈=√104𝑋₉,𝑋₂,𝑋₇=4𝑋₉. (29)

Table 2 

8-dimensional nilsoliton metric Lie algebras.

Table 3 

The 9-dimensional nilsoliton metric Lie algebras of nullities 1 and 2.

4.2.1. Candidates of Nilsoliton Metrics

Table 5 illustrates how many possible candidates of Lie algebras appear in dimension 9 up to the nullity of its Gram matrix. The algebras illustrated in Table 4 are possible candidates of nilsoliton metric Lie algebras with ordered type of derivations in dimension 8. Here, as an example we give potential Lie algebra structures when the nullity of their corresponding Gram matrices are 3, 6 and 8 in Tables 6, 7, and 8 respectively for dimension nine.

Table 4 

The 8-dimensional nilsoliton metric Lie algebra candidates.

Table 5 

Number of 9-dimensional nilsoliton metric Lie algebra candidates.

Table 6 

The 9-dimensional candidates of nullities 6 and 8.

Table 7 

The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.

Table 8 

The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.

5. Conclusion

In this work, we have focused on nilpotent metric Lie algebras of dimensions eight and nine with ordered type of derivations. We have given specific Jacobi identity conditions for Lie algebras which allowed us to simplify the Jacobi identity condition. We have classified nilsoliton metric Lie algebras for the corresponding Gram matrix 𝑈 being invertible and noninvertible. For dimension 8, we have focused on nilsoliton metric Lie algebras with noninvertable Gram matrix which leads to more than one solution for 𝑈𝑣=[1]. We have proved that if the nilpotent Lie algebra admits a soliton metric with corresponding Gram matrix being noninvertible, all the solutions of 𝑈𝑣=[1] correspond to a unique derivation. This theorem has allowed us to omit several cases that come from nonordered eigenvalues without considering Jacobi condition. Moreover, we have classified some nilsoliton metric Lie algebras with derivation types 1<2<⋯<𝑛 and provided some candidates that may be classified. We are currently working on an algorithm that provides a full list of classifications for dimensions eight and nine.

Appendix

See Tables 5, 6, 7, and 8.

Conflict of Interest

Author declares that she has no competing interest.

Acknowledgment

Hülya Kadıoğlu would like to thank the Idaho State University Department of Mathematics, in particular department chair Professor Dr. Robert Fisher Jr. and Professor Dr. Tracy L. Payne for their advices and technical supports during her research.