Classification of the Quasifiliform Nilpotent Lie Algebras of Dimension 9

On the basis of the family of quasifiliform Lie algebra laws of dimension 9 of 16 parameters and 17 constraints, this paper is devoted to identify the invariants that completely classify the algebras over the complex numbers except for isomorphism. It is proved that the nullification of certain parameters or of parameter expressions divides the family into subfamilies such that any couple of them is nonisomorphic and any quasifiliform Lie algebra of dimension 9 is isomorphic to one of them. The iterative and exhaustive computation with Maple provides the classification, which divides the original family into 263 subfamilies, composed of 157 simple algebras, 77 families depending on 1 parameter, 24 families depending on 2 parameters, and 5 families depending on 3 parameters.


Introduction
The interest in classifying nilpotent Lie algebras is broad both within the academic community and the industrial engineering community, since they are applied in classical mechanical problems and current research in scientific disciplines as modern geometry, solid state physics, or particle physics [1][2][3][4][5]. Lie algebras classification consists in determining equivalence relations that subdivide the original set in equivalence classes defined by at least one element in each set, and it is usual to classify the algebras except for isomorphisms. The solvable Lie algebras classification problem comes down in a sense to the nilpotent Lie algebras classification [6] and computer algebra has been indispensable. However, the more the dimension increases, the more and more complex is the determination of exhaustive lists of Lie algebras, so new computation methodologies are a present field of research [7,8] with current symbolic manipulation programs such as Reduce, Mathematica, or Maple [9].
The classification of nilpotent Lie algebras over the complex numbers experimented an important advance based on the works of Ancochéa-Bermúdez and Goze [10] introducing an invariant more potent than the previously existing: the characteristic sequence or Goze's invariant (defined in Section 2.1). Those authors were able, by using the characteristic sequence as an invariant, to classify the nilpotent Lie algebras of dimension 7 [11] and the filiform Lie algebras of dimension 8 [12]. Later, by using that invariant, Gomez and Echarte [13] classify the filiform Lie algebras of dimension 9. Afterward, Castro et al. [14] develop an algorithm for symbolic language for finding the generic families of filiform Lie algebras in any dimension with the restrictions required to the parameters. Subsequent works about quasifiliform Lie algebras classification were centered on specific types of families or subclasses, obtaining results applicable to higher dimensions. For instance, the classifications of naturally graded [15] and graded by derivations [16] quasifiliform Lie algebras. These works extended to other algebras, with a high nilindex, the classification of graded filiform Lie algebras, studied initially by Vergne [17,18], obtained from the gradation related to the filtration produced in a natural way by the descending central sequence.
In this paper we focus on a method of identification of the invariants that completely classify the nilpotent Lie algebras of dimension 9 over the complex numbers except for isomorphisms. With this aim, the dimensions of the subalgebras of its derived series, of its descending central On the whole all the bracket products can be described by where are the algebra structure constants. The laws of every complex quasifiliform Lie algebra (QFLA) of dimension 9 can be described by the following family with 16 parameters and 17 polynomial restriction equations [19] derived from the Jacobi identity: [ 1 , 2 ] = 1 4 + 2 5 + 3 6 + 4 7 + 5 8 , [ 1 , 4 ] = 6 5 + 7 6 + 8 7 + 9 8 , (3d)    Table 1 shows the structure constants corresponding with the 16 parameters. From here forward the Lie Algebra Families will be denoted as ( 1 1 , . . . , 16 ).
Our objective is to study exhaustively the case of dimension 9; therefore the coefficients identification is tackled in an iterative and interactive way by imposing the Jacobi identity. Maple programs have been developed so that all the equations resulting from the application of the abovementioned conditions are obtained, the simplest conditions are applied, and the process is repeated until there are no restrictions of simple application.

Case
The nullity of 2 − 8 constitutes a new classification criterion. Proposition 6. The nilpotent QFLA of dimension 9 with 11 ̸ = 0, 1 = 0, and 2 ̸ = 8 can be classified in ten nonisomorphic subfamilies with from 4 to 13, described in Figure 1, according to the conditions described in Figure 2.

Proposition 7.
The nilpotent QFLA of dimension 9 with 11 ̸ = 0, 1 = 0, and 2 = 8 can be classified in six nonisomorphic subfamilies with from 14 to 19, described in Figure 1, according to the conditions described in Figure 2.

Concluding Remarks
Computational aid has been indispensable in this piece of research. A PC Pentium 4 of 2.4 GHz and the programming language Maple 6 have been used in the process. The library modules developed represent approximately 12,000 lines of code. In some cases, in this massive application of computational resources and looking for the simplification of some laws, procedures that perhaps can be considered of "inverse engineering" have been used in order to find some very complex changes of base, which have allowed us to  10/9 7/9        Classifcation criterion from previous cases Additional result from previous cases eliminate some parameters in the laws involved. In any case, the massive application of changes of base and characteristic vector has allowed us to obtain the complete classification in 263 subfamilies of the QFLA laws of dimension 9. The 263 families have been represented in the paper, consisting of 157 simple algebras, 77 families depending on 1 parameter, 24 families depending on 2 parameters, and 5 families depending on 3 parameters. The classification is complete since any couple of the obtained 263 families is nonisomorphic and any quasifiliform Lie algebra of dimension 9 is isomorphic to one of them. The nonisomorphism of the 263 Lie algebra families has been proved in the 10 propositions of the paper, and the completeness of the classification is proved by the "exhaustive" analysis of all the possible cases, depending on the combination of the values of the 16 parameters ( 1 ⋅ ⋅ ⋅ 16).