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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 871930, 9 pages
Classification of Ordered Type Soliton Metric Lie Algebras by a Computational Approach
Department of Mathematics Education, Yıldız Technical University, 34210 Esenler, İstanbul, Turkey
Received 25 February 2013; Accepted 22 April 2013
Academic Editor: Mustafa Bayram
Copyright © 2013 Hülya Kadıoğlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We classify some soliton nilpotent Lie algebras and possible candidates in dimensions 8 and 9 up toisomorphy. We focus on type of derivations, where is the dimension of the Lie algebras. We present algorithms to generate possible algebraic structures.
In this paper, we compute and classify -dimensional nilsoliton metric Lie algebras with eigenvalue type , 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 . 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 . 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 .
In our previous paper, we classified all the soliton and nonsoliton metric Lie algebras where the corresponding Gram matrix is invertible and of dimensions and up to isomorphism. If corresponding Gram matrix is invertible, then the soliton metric condition 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 that does not have ordered eigenvalues without solving the following soliton metric condition . For this, we prove that if the nilpotent Lie algebra admits a soliton metric with corresponding Gram matrix of being noninvertible, all the solutions of 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.
Let be a metric algebra, where . Let be a -orthonormal basis of (we always assume that bases are ordered). The nil-Ricci endomorphism is defined as , where for (we often write an inner product as ). When 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 , we call the orthonormal basis a Ricci eigenvector basis.
Now we define some combinatorial objects associated to a set of integer triples . For , define row vector to be , where is the standard orthonormal basis for . We call the vectors in root vectors for . Let (where ) be an enumeration of the root vectors in dictionary order. We define root matrix for to be the matrix whose rows are the root vectors . The Gram matrix for is the matrix defined by ; the entry of is the inner product of the th and th root vectors. It is easy to see that is a symmetric matrix. It has the same rank as the root matrix; that is, . Diagonal elements of are all three, and the off-diagonal entries of are in the set . For more information, see . Let have distinct real positive eigenvalues, and let index the structure constants for with respect to eigenvector basis . If and , then . Thus does not contain two as an entry .
Lemma 1. Let be an -dimensional inner product space, and let be an element of . Suppose that admits a symmetric derivation having distinct eigenvalues with corresponding orthonormal eigenvectors . Let denote the structure constants for with respect to the ordered basis . Let . Then(1)if there is some such that , then is a scalar multiple of ; otherwise and commute; (2) if and only if .
Theorem 2 (see ). Let be a vector space, and let be a basis for . Suppose that a set of nonzero structure constants relative to , indexed by , defines a skew symmetric product on . Assume that if , then . Then the algebra is a Lie algebra if and only if whenever there exists so that the inner product of root vectors for triples and or in , the equation holds. Furthermore, a term of form is nonzero if and only if
Theorem 3 (see ). Let be a metric algebra and a Ricci eigenvector basis for . Let be the root matrix for . Then the eigenvalues of the nil-Ricci endomorphism are given by where .
Theorem 4 (see [4, 11]). Let be a nonabelian metric algebra with Ricci eigenvector basis . The following are equivalent. (1) satisfies the nilsoliton condition with nilsoliton constant . (2)The eigenvalue vector for with respect to lies in the kernel of the root matrix for with respect to . (3)For noncommuting eigenvectors and for the nil-Ricci endomorphism with eigenvalues and , the bracket is an eigenvector for the nil-Ricci endomorphism with eigenvalue . (4) for all in .
Theorem 5 (see ). Let be an dimensional nonabelian nilpotent Lie algebra which admits a derivation having distinct real positive eigenvalues. Let be a basis consisting of eigenvectors for the derivation , and let index the nonzero structure constants with respect to . Let be the Gram matrix. If is invertible, then the following hold: (i); (ii)if and , then .
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 be an -dimensional nilsoliton metric Lie algebra, and the corresponding Gram matrix which is noninvertible. Then . Furthermore all of the solutions of correspond to a unique derivation.
Proof . Since rank of a matrix is equal to the rank of its Gram matrix, then . Let and denote the linear functions (with respect to the standard basis) that correspond to the Gram matrix and the transpose of the root matrix , respectively. Since and by rank-nullity theorem, we have Let be a particular solution and the last column of reduced row echelon matrix . Then is also a solution of . Therefore ; that is, . Using (4), then . For the solution , suppose that we denote for the Nikolayevsky derivation, for the Ricci tensor, and for the soliton constant. Then using (3), we have Then . Using Theorem 4, we have , which implies that .
Lemma 7. If nilsoliton metric Lie algebra has ordered type of derivations , then its index set consists of triples .
Proof. If is the eigenvalue vector of with eigenvector basis for , then by Theorem 4, lies in the kernel of . Thus for each element , ; that is, . By Lemma 1, for some . Since for all , and . Hence, the index set for ordered type of derivations is of form .
The next corollary describes the index triples and the Jacobi identity for algebras with ordered type of derivations.
Corollary 8. The algebra is a Lie algebra if and only if for all pairs of form and or and in with and for all , the following equation holds:
If in addition for , then the algebra is a Lie algebra if and only if for all pairs of form and or and in with and for all , the equation holds.
Proof. By Theorem 7 of , the algebra defined by is a Lie algebra if and only if whenever there exists so that for triples and or in , (2)
holds. Furthermore, if , , and are distinct, the product is nonzero if and only if .
Suppose that for and or in . By definition of , we have . By Lemma 1, , , and . Since , we know that . Similarly, implies that . If or , then , and so , and must be distinct. Since , and are all distinct and less than , we know that . Thus an expression of form is nonzero only if , and .
Suppose that , and and are in the index set. Then from Lemma 7, which implies that . We know that . Then, since and , we have . Since , implies that . So there is no possible , where all are distinct and with . Thus if , then . Therefore, if , an expression of form is nonzero only if , and .
By Lemma 1, and are in , and so which implies that and , . Therefore all expressions in (10) with or are identically zero and may be omitted from the summation for any .
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 in the following equations corresponds to each of entry in the Gram matrix . Therefore, the following equations are useful for noninvertible case since there is no entry in the Gram matrix for the invertible case.
Corollary 9. Let be an -dimensional inner product space where , and, be an element of . Suppose that the algebra defined by admits a symmetric derivation having eigenvalues with corresponding orthonormal eigenvectors . denote the structure constants for with respect to the ordered basis , and let index the nonzero structure constants as defined in (2). The algebra is a Lie algebra if and only if holds.
Proof. Following Lemma 7, the index set consists of elements of form . Therefore the number equals to in the expression . From the previous corollary,
is nonzero only if , and . Also ; otherwise .
If , implies that ; that is, possible numbers for “” are and . Possible and not possible triples, which are being used in and where , are illustrated in Table 1. The notations in the table are as follows ; ; possible triple.
In the case (a), , and then it is not a possible triple. In the case (b), , are distinct, and . So it is a possible triple. In the case (c), , and so it is not a possible triple. In the case (d), , and all are distinct as . Thus it is a possible triple. In the case (e), is not a natural number, and so it is not a possible triple. Therefore only possible triples are and . Triples and correspond to nonzero products and , respectively. Using the skew-symmetry, (2) turns into the following equation: which gives (11). Using the same procedure for , possible triples are and , which correspond to nonzero , , and respectively. Therefore (12) is obtained. Equations (13) and (14) can be obtained by the same way.
As an illustration, we show how to use the results of this section in the following example.
Example 10. Let be an -dimensional algebra with nonzero structure constants relative to eigenvector basis indexed by
Computation shows that the structure vector is a solution to if and only if it is of form
Equation (13) from the previous corollary leads Moreover, using (12), we find that and , which means that . Thus 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.
Now we describe the algorithm. The following algorithm can be used for both invertible and noninvertible cases.
Input. The input is the integer which represents the dimension.
Output. The output is two matrices Wsoliton and Uninv listing characteristic vectors for index sets of . The matrix Wsoliton has as its rows all possible characteristic vectors for canonical index sets for nilpotent Lie algebras of dimension with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix is invertible. The matrix Uninv has as its rows all possible characteristic vectors for canonical index sets for nilpotent Lie algebras of dimension with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix is noninvertible. In the dimensions and , 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 . (ii)Compute the matrix . (iii)Compute the matrix . (iv)Delete all rows of containing abelian factor which is the row that represents direct sums of Lie algebras. (v)Remove all rows of such that the canonical Gram matrix associated to the index set is invertible. (vi)Define eigenvalue vector in dimension . (vii)Remove all rows of if where is the general solution of and is the vector that we have defined in the proof of Theorem 6. (viii)Remove all the rows of such that the corresponding algebra does not have a derivation of eigenvalue type . (ix)Remove all the rows of 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 .
Example 11. Let . Then So, matrix is of form Since , the matrix is of size as follows The first row of represents empty matrix, row two represents the subset of , and so forth. Eliminating rows that represent direct sums, we have 33 rows in matrix. Therefore none of the rows of 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 . Let be the eigenvector basis for , whose nonzero structure constants are indexed by Computation shows that the structure vector is a solution to if and only if it is of form By Corollary 9, satisfies (11). Solving the equation for , we find that After rescaling and solving for structure constants from , we see that letting defines a nilsoliton metric Lie algebra, previously found in .
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 in the first row of Table 3 is meant to encode the metric Lie algebra with orthonormal basis and bracket relations
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.
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 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 . We have proved that if the nilpotent Lie algebra admits a soliton metric with corresponding Gram matrix being noninvertible, all the solutions of 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 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.
Conflict of Interest
Author declares that she has no competing interest.
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.
- W. A. de Graaf, “Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2,” Journal of Algebra, vol. 309, no. 2, pp. 640–653, 2007.
- J. Lauret, “Finding Einstein solvmanifolds by a variational method,” Mathematische Zeitschrift, vol. 241, no. 1, pp. 83–99, 2002.
- C. Will, “Rank-one Einstein solvmanifolds of dimension 7,” Differential Geometry and its Applications, vol. 19, no. 3, pp. 307–318, 2003.
- H. Kadioglu and T. L. Payne, “Computational methods for nilsoliton metric Lie algebras I,” Journal of Symbolic Computation, vol. 50, pp. 350–373, 2013.
- E. A. Fernández-Culma, “Classification of 7-dimensional Einstein nilradicals,” Transformation Groups, vol. 17, no. 3, pp. 639–656, 2012.
- E. A. Fernández-Culma, “Classification of 7-dimensional Einstein nilradicals II,” Transformation Groups. In press.
- J. Lauret and C. Will, “Einstein solvmanifolds: existence and non-existence questions,” Mathematische Annalen, vol. 350, no. 1, pp. 199–225, 2011.
- Y. Nikolayevsky, “Einstein solvmanifolds with a simple Einstein derivation,” Geometriae Dedicata, vol. 135, pp. 87–102, 2008.
- R. M. Arroyo, “Filiform nilsolitons of dimension 8,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 4, pp. 1025–1043, 2011.
- J. Lauret, “Einstein solvmanifolds and nilsolitons,” in New Developments in Lie Theory and Geometry, vol. 491 of Contemporary Mathematics, pp. 1–35, American Mathematical Society, Providence, RI, USA, 2009.
- T. L. Payne, “The existence of soliton metrics for nilpotent Lie groups,” Geometriae Dedicata, vol. 145, pp. 71–88, 2010.