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

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 $$ denote the structure constants for η with respect to the ordered basis $$. 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)

  $$ if and only if X_k ∈ [η_μ, η_μ].

Theorem 2 (see [11].)Let η be a vector space, and let $$ be a basis for η. Suppose that a set of nonzero structure constants $$ 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 $$ for triples (i, j, l) and (l, k, m) or (k, l, m) in Λ, the equation

Theorem 3 (see [11].)Let (η_μ, Q) be a metric algebra and $$ 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

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 − β Id 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)

  $$ 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 $$.

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 → ℝⁿ 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

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 $$ 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).

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:

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

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 $$ for triples (i, j, l) and (l, k, m) or (k, l, m) in Λ_B, (2)

Suppose that $$ for (i, j, l) ∈ Λ_B and (l, k, m) or (k, l, m) in Λ_B. By definition of Λ_B, we have i < j. By Lemma 1, j < l, l < m, and k < m. Since i < j < l < m, we know that m ≥ i + 3. Similarly, j < l < m implies that j + 2 ≤ m. If i = k or j = k, then $$, 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

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 $$. Therefore, if λ_i = i, an expression of form

By Lemma 1, X₁ and X₂ are in [η_μ, η_μ] ^⊥, and so $$ 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.

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. $$ denote the structure constants for η with respect to the ordered basis $$, 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

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 $$. From the previous corollary,

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

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 $$ and $$, respectively. Using the skew-symmetry, (2) turns into the following equation:

Example 10. Let η be an 8-dimensional algebra with nonzero structure constants relative to eigenvector basis B indexed by

Computation shows that the structure vector [α²] is a solution to Uv = [1] _{8  ×  1} if and only if it is of form

Equation (13) from the previous corollary leads