Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 173072, 12 pages http://dx.doi.org/10.1155/2014/173072
Research Article Classification of the Quasifiliform Nilpotent Lie Algebras of Dimension 9 Mercedes Pérez,1 Francisco P. Pérez,2 and Emilio Jiménez3 1
Department of Mechanical Engineering, University of La Rioja, 26004 Logro˜no, Spain Department of Applied Mathematics I, University of Sevilla, 41012 Seville, Spain 3 Department of Electrical Engineering, University of La Rioja, 26004 Logro˜no, Spain 2
Correspondence should be addressed to Emilio Jim´enez;
[email protected] Received 3 November 2013; Accepted 2 January 2014; Published 6 March 2014 Academic Editor: Peter G. L. Leach Copyright © 2014 Mercedes P´erez et al. 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. 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.
1. 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– 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 Ancoch´ea-Berm´udez 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
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series, and of its descending central series centralizers are used as class invariants. The exhaustive analysis has been developed with significant computational effort; the total code is 2820 pages in 37 files, summing more than 12000 lines of Maple code, and these programs have provided 3038 pages of results [19]. We strongly recommend the reading of B¨auerle and de Kerf [20], Benjumea et al. [21], and Sendra et al. [22] to become familiar with Lie algebras terminology and symbolic computation with Maple.
[𝑥1 , 𝑥4 ] = 𝛼6 𝑥5 + 𝛼7 𝑥6 + 𝛼8 𝑥7 + 𝛼9 𝑥8 ,
(3d)
[𝑥1 , 𝑥5 ] = 2𝛼6 𝑥6 + (2𝛼7 − 𝛼1 ) 𝑥7 ,
(3e)
[𝑥1 , 𝑥6 ] = 𝛼10 𝑥7 + 𝛼11 𝑥8 ,
(3f)
[𝑥1 , 𝑥8 ] = 𝛼12 𝑥3 + 𝛼13 𝑥4 + 𝛼14 𝑥5 + 𝛼15 𝑥6 + 𝛼16 𝑥7 ,
(3g)
[𝑥2 , 𝑥3 ] = −𝛼6 𝑥5 + (𝛼1 − 𝛼7 ) 𝑥6 + (𝛼2 − 𝛼8 ) 𝑥7 − 𝛼9 𝑥8 , (3h)
2. The Subfamilies of Laws
[𝑥2 , 𝑥4 ] = −𝛼6 𝑥6 + (𝛼1 − 𝛼7 ) 𝑥7 ,
(3i)
2.1. Preliminaries. Let g be a nilpotent Lie algebra; the characteristic sequence of 𝑎𝑑(𝑋) is denoted by 𝑐(𝑋) = (𝑐1 , . . . , 𝑐𝑘 , 1), and for the lexicographic order 𝑐(g) = Max𝑋∈g−[g,g] 𝑐(𝑋) is known as the Goze’s invariant or characteristic sequence [23]. Obviously 𝑐(g) is an invariant for the isomorphisms and, by construction, there is at least one vector 𝑋 ∈ g − [g, g] such that 𝑐(g) = 𝑐(𝑋); all vector verifying this condition is called characteristic vector of the algebra. The abelian algebra of dimension 𝑛 is the only one with Goze’s invariant (1, . . . , 1), in metabelian algebras the characteristic series is (2, . . . 2, 1, . . . 1), in Heisenberg algebras it is (2, 1, . . . , 1), in filiform algebras it is (𝑛 − 1, 1), and in quasifiliform algebras it is (𝑛 − 2, 1, 1). A Lie algebra g is nilpotent if and only if the characteristic polynomial of the matrix 𝑎𝑑(𝑥) is 𝜆9 , for every vector 𝑥 of g. Anyway this condition is often difficult to be applied, so the moment in the process, when the nilpotence condition should be applied or, much better, when the condition should be applied for each vector, has to be chosen carefully. The condition of being quasifiliform can be also interpreted in terms of matrices. Thus the vectors candidate to characteristic vectors, that is, the vectors in g − [g, g], have to satisfy that the respective adjoint matrices do not have nonnull minors of order ⩽7. As in the case of the nilpotence, this condition has to be applied with caution and in several stages. Every quasifiliform Lie algebra of dimension 9 can have an adapted base {𝑥0 , 𝑥1 , . . . , 𝑥8 } such that
[𝑥2 , 𝑥5 ] = (2𝛼6 − 𝛼10 ) 𝑥7 − 𝛼11 𝑥8 ,
(3j)
[𝑥2 , 𝑥8 ] = 𝛼12 𝑥4 + 𝛼13 𝑥5 + 𝛼14 𝑥6 + 𝛼15 𝑥7 ,
(3k)
[𝑥3 , 𝑥4 ] = (−3𝛼6 + 𝛼10 ) 𝑥7 + 𝛼11 𝑥8 ,
(3l)
[𝑥3 , 𝑥8 ] = 𝛼12 𝑥5 + 𝛼13 𝑥6 + 𝛼14 𝑥7 ,
(3m)
[𝑥4 , 𝑥8 ] = 𝛼12 𝑥6 + 𝛼13 𝑥7 ,
(3n)
[𝑥5 , 𝑥8 ] = 𝛼12 𝑥7
(3o)
𝛼5 𝛼12 = 0,
(4a)
𝛼6 𝛼12 = 0,
(4b)
𝛼6 𝛼13 = 0,
(4c)
𝛼9 𝛼12 = 0,
(4d)
𝛼9 𝛼13 = 0,
(4e)
𝛼9 𝛼14 = 0,
(4f)
𝛼10 𝛼12 = 0,
(4g)
𝛼11 𝛼12 = 0,
(4h)
𝛼11 𝛼13 = 0,
(4i)
𝛼11 𝛼14 = 0,
(4j)
𝛼11 𝛼15 = 0,
(4k)
𝛼11 𝛼16 = 0,
(4l)
𝛼11 (3𝛼1 − 𝛼7 ) = 0,
(4m)
𝛼12 (𝛼1 − 𝛼7 ) = 0,
(4n)
𝛼5 𝛼13 − 2𝛼62 − 𝛼9 𝛼15 = 0,
(4o)
[𝑥0 , 𝑥𝑖 ] = 𝑥𝑖+1 ,
1 ⩽ 𝑖 ⩽ 6;
[𝑥0 , 𝑥𝑖 ] = 0,
7 ⩽ 𝑖 ⩽ 8. (1)
On the whole all the bracket products can be described by [𝑥𝑖 , 𝑥𝑗 ] =
𝑛−1
∑ 𝐶𝑖𝑗𝑘 𝑘=0
⋅ 𝑥𝑘 ,
0 ⩽ 𝑖, 𝑗 ⩽ 𝑛 − 1,
(2)
where 𝐶𝑖𝑗𝑘 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: [𝑥0 , 𝑥𝑖 ] = 𝑥𝑖+1 ,
1 ⩽ 𝑖 ⩽ 6,
[𝑥1 , 𝑥2 ] = 𝛼1 𝑥4 + 𝛼2 𝑥5 + 𝛼3 𝑥6 + 𝛼4 𝑥7 + 𝛼5 𝑥8 , [𝑥1 , 𝑥3 ] = 𝛼1 𝑥5 + 𝛼2 𝑥6 + 𝛼3 𝑥7 ,
(3a) (3b) (3c)
subject to
2 (𝛼2 − 𝛼8 ) 𝛼12 + 3 (𝛼1 − 𝛼7 ) 𝛼13 + 2 (𝛼6 − 𝛼10 ) 𝛼14 = 0, (4p) 𝛼5 𝛼14 − 2 (2𝛼1 + 𝛼7 ) 𝛼6 − 𝛼9 𝛼16 + (3𝛼1 − 𝛼7 ) 𝛼10 = 0 (4q) with the application of the Jacobi identity to the 3-tuple (𝑥0 , 𝑥𝑖 , 𝑥𝑗 ), where 𝑥𝑖 , 𝑥𝑗 are base vectors different from
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Table 1: Notation for the QFLA parameters. 4 𝛼1 = 𝐶12
8 𝛼5 = 𝐶12 8 𝛼9 = 𝐶14 4 𝛼12 = 𝐶18
5 𝛼2 = 𝐶12
5 𝛼6 = 𝐶14 7 𝛼10 = 𝐶16 5 𝛼14 = 𝐶18
6 𝛼3 = 𝐶12
6 𝛼7 = 𝐶14 8 𝛼11 = 𝐶16 6 𝛼15 = 𝐶18
7 𝛼4 = 𝐶12
7 𝛼8 = 𝐶14 3 𝛼12 = 𝐶18 7 𝛼16 = 𝐶18
𝑥0 vector. Table 1 shows the structure constants corresponding with the 16 parameters. From here forward the Lie Algebra Families will be denoted as 𝜇(𝛼11 , . . . , 𝛼16 ). 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. The exhaustiveness of the classification is developed by analyzing 𝑎𝑙𝑙 the possible combinations of values of the 16 parameters (𝛼1 , . . . , 𝛼16 ), which is summarized within the cases shown in the following subsections: case A.1 (𝛼11 ≠ 0 and 𝛼1 ≠ 0), A.2 (𝛼11 ≠ 0 and 𝛼1 = 0), B.1.1 (𝛼11 = 0, 𝛼9 ≠ 0, and 𝛼6 ≠ 0), B.1.2 (𝛼11 = 0, 𝛼9 ≠ 0, and 𝛼6 = 0), B.2.1 (𝛼11 = 0, 𝛼9 = 0, and 𝛼5 ≠ 0), and B.2.2 (𝛼11 = 0, 𝛼9 = 0, and 𝛼5 = 0). In all the cases the nonisomorphism is proved in the corresponding propositions. 2.2. General Case Proposition 1. The nilpotent QFLA of dimension 9 and 𝛼11 ≠ 0 are nonisomorphic to the algebras with 𝛼11 = 0. Proof. For the family described by (3a)–(3o) and (4a)– (4q), its descending central series is C1 g = ⟨𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝛼5 𝑥8 , 𝛼9 𝑥8 , 𝛼11 𝑥8 ⟩, C2 g = ⟨𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝛼5 𝑥8 , 𝛼9 𝑥8 , 𝛼11 𝑥8 ⟩, C3 g = ⟨𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝛼9 𝑥8 , 𝛼11 𝑥8 ⟩, C4 g = ⟨𝑥5 , 𝑥6 , 𝑥7 , 𝛼9 𝑥8 , 𝛼11 𝑥8 ⟩, C5 g = ⟨𝑥6 , 𝑥7 , 𝛼11 𝑥8 ⟩, and so forth. Thus Dim[C5 g] = 3 if 𝛼11 ≠ 0 and Dim[C5 g] = 2 if 𝛼11 = 0. Therefore the nullity of 𝛼11 constitutes the first classification criterion.
(5c)
[𝑥1 , 𝑥4 ] = 3𝛼1 𝑥6 + 𝛼8 𝑥7 + 𝛼9 𝑥8 ,
(5d)
[𝑥1 , 𝑥5 ] = 5𝛼1 𝑥7 ,
(5e)
[𝑥1 , 𝑥6 ] = 𝛼10 𝑥7 + 𝛼11 𝑥8 ,
(5f)
[𝑥2 , 𝑥3 ] = −2𝛼1 𝑥6 + (𝛼2 − 𝛼8 ) 𝑥7 − 𝛼9 𝑥8 ,
(5g)
[𝑥2 , 𝑥4 ] = −2𝛼1 𝑥7 ,
(5h)
[𝑥2 , 𝑥5 ] = −𝛼10 𝑥7 − 𝛼11 𝑥8 ,
(5i)
[𝑥3 , 𝑥4 ] = 𝛼10 𝑥7 + 𝛼11 𝑥8
(5j)
without restrictions derived from the Jacobi identity (4a)– (4q). It can be observed that 𝑥7 and 𝑥8 are now central; thus the application of the elementary change of base 𝑦𝑖 = 𝑥𝑖 ,
Proposition 2. The nilpotent QFLA of dimension 9 with 𝛼11 ≠ 0∧𝛼1 ≠ 0 are nonisomorphic to the algebras with 𝛼11 ≠ 0∧ 𝛼1 = 0. Proof. If 𝛼11 ≠ 0, from restrictions (4a)–(4q) it can be deduced that 𝛼6 = 𝛼12 = 𝛼13 = 𝛼14 = 𝛼15 = 𝛼16 = 0 and 𝛼7 = 3𝛼1 . By computing the Jacobi equations, the family of laws is reduced to 1 ⩽ 𝑖 ⩽ 6,
[𝑥1 , 𝑥2 ] = 𝛼1 𝑥4 + 𝛼2 𝑥5 + 𝛼3 𝑥6 + 𝛼4 𝑥7 + 𝛼5 𝑥8 ,
(5a) (5b)
𝑖 ≠ 8,
𝑦8 = 𝛼10 ⋅ 𝑥7 + 𝛼11 ⋅ 𝑥8
(6)
permits us to suppose that 𝛼10 = 0 and 𝛼11 = 1. Then (5f), (5i), and (5j) are simplified and the derived series is D1 g = ⟨𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 ⟩, D2 g = ⟨−2𝛼1 𝑥6 , (𝛼2 − 𝛼8 )𝑥7 , −𝛼9 𝑥8 , −2𝛼1 𝑥7 , 𝑥8 ⟩, and so forth. Thus Dim[D2 g] = 3 if 𝛼1 ≠ 0 and Dim[D2 g] ⩽ 2 if 𝛼1 = 0. Therefore the nullity of 𝛼1 constitutes a new classification criterion. In this subsection (case A), the notation to describe the parameters of the subfamily 𝑖 is reduced to 𝜇𝑖 (𝛼1 , 𝛼2 , 𝛼3 , 𝛼4 , 𝛼5 , 𝛼7 , 𝛼8 , 𝛼9 ) for simplification. Figure 1 shows the classification in 19 subfamilies in the case A. They are classified with the criteria summarized in Figure 2 and detailed in the following cases. 2.3.1. Case A.1. One has 𝛼11 ≠ 0 and 𝛼1 ≠ 0. Proposition 3. Case A.1 permits us to suppose that 𝛼1 = 1. Proof. With the elementary change of base CB, 𝑦0 = 𝑥0 , 𝑦𝑖 =
2.3. Case A. 𝛼11 ≠ 0.
[𝑥0 , 𝑥𝑖 ] = 𝑥𝑖+1 ,
[𝑥1 , 𝑥3 ] = 𝛼1 𝑥5 + 𝛼2 𝑥6 + 𝛼3 𝑥7 ,
𝑥𝑖 , 𝛼1 𝑦8 =
1 ⩽ 𝑖 ⩽ 7,
(7)
𝑥8 , 𝛼12
|CB| = 1/𝛼19 ≠ 0 and then 𝛼1 = 1. The subfamilies of laws with the structure (5a)–(5j) are 𝜇𝑖 (1, 𝛼2 , 𝛼3 , 𝛼4 , 𝛼5 , 3, 𝛼8 , 𝛼9 ) with 𝑖 = 1, 2, 3. Let us denote, from here forward, by 𝛿 the new parameters obtained from the changes of base and 𝜇 the Lie algebra families depending on these new parameters 𝛿 (which in general depend on the 16 parameters 𝛼), in order to differentiate the new representation 𝜇𝑖 (𝛿𝑗 ) from the representation of the families depending in general on the 16 parameters 𝜇𝑖 (𝛼1 , . . . , 𝛼16 ).
4
Journal of Applied Mathematics 𝜇
𝛼1
𝛼2
𝛼3
𝛼4
𝛼5
𝛼7
𝛼8
𝛼9
1
1
0
0
0
0
3
1
2
1
0
0
0
0
3
0
𝜆 1
3
1
0
0
0
0
3
0
0
4
0
𝜆
1
0
0
0
1
0
5
0
1/4
1
𝜆
0
0
1
0
6
0
1
1
𝜆
0
0
0
0
7
0
𝜆
0
1
0
0
1
0
8
0
1
0
1
0
0
4
0
9
0
1
0
0
0
0
4
0
10
0
1
0
1
0
0
0
0
11
0
1
0
0
0
0
0
0
12
0
0
1
0
0
0
1
0
13
0
0
0
1
0
0
1
0
14
0
1
1
0
0
0
1
0
15
0
1
0
0
0
0
1
0
16
0
0
1
1
0
0
0
0
17
0
0
1
0
0
0
0
0
18
0
0
0
1
0
0
0
0
19
0
0
0
0
0
0
0
0
Aa Aa Aa Aa Aa
apply (8) again and the only restriction is 𝑃0 ≠ 0. Thus, with the coefficient identifications, the new parameters are 𝛿2 = 𝛿3 =
(−2𝑃1 + 𝛼2 𝑃0 ) , 𝑃02
(9)
(−2𝑄22 + 4𝑃02 𝑄3 − 7𝑃1 𝛼2 𝑃03 + 7𝑃12 𝑃02 + 𝛼3 𝑃04 ) 𝑃06
,
(10)
𝛿4 = (𝛼4 𝑃06 − 2𝛼22 𝑃05 𝑃1 − 2𝛼2 𝑃04 𝑄3 + 𝛼2 𝑃02 𝑄22 − 𝛼2 𝑃05 𝑃1 𝛼8 + 2𝑄23 − 12𝑃1 𝛼3 𝑃05 − 42𝑃1 𝑃03 𝑄3 − 44𝑃13 𝑃03 + 56𝑃04 𝑃12 𝛼2 + 18𝑃1 𝑄22 𝑃0 − 6𝑃02 𝑄3 𝑄2 + 6𝑃04 𝑄4
(11)
+ 6𝑃2 𝑃03 𝑄2 + 𝑃04 𝑃12 𝛼8 − 6𝑃05 𝑃3 −1
+2𝛼8 𝑃04 𝑄3 − 𝑄22 𝑃02 𝛼8 ) × (𝑃09 ) , 𝛿5 = (2𝑃0 𝑃1 𝑄3 𝑄2 + 2𝑃2 𝛼2 𝑃03 𝑄2 + 6𝑃12 𝛼3 𝑃04 − 2𝑃2 𝑃0 𝑄22 − 2𝑃02 𝑄4 𝑄2 + 6𝑃04 𝑃3 𝑃1
Classification criterion Classifcation criterion from previous cases Additional result Additional result from previous cases Direct result
− 8𝑃12 𝑄22 − 2𝑃0 𝑃1 𝑄22 𝛼2 − 4𝑃1 𝑃2 𝑃02 𝑄2 + 2𝑃2 𝑃03 𝑄3 + 2𝑃3 𝑃03 𝑄2 − 4𝑃1 𝑃03 𝑄4
Figure 1: Case A: 𝛼11 ≠ 0; classification of the nonisomorphic QFLA of dimension 9 (the last 7 values, 0, 1, 0, 0, 0, 0, 0, common to the 19 families have been omitted from the figure for simplicity).
+ 20𝑃02 𝑃12 𝑄3 − 𝑄22 𝑃02 𝛼9 + 2𝑄3 𝑃04 𝛼9 + 𝑃1 𝑄22 𝑃0 𝛼8 − 𝑃1 𝑃05 𝛼4 + 𝑃02 𝑄32 + 2𝑃04 𝑄5 + 𝑃06 𝛼5 + 2𝑃1 𝑃03 𝑄3 𝛼2 − 2𝑃1 𝛼8 𝑃03 𝑄3 + 𝛼2 𝑃04 𝑃12 𝛼8 − 𝛼2 𝑃05 𝑃1 𝛼9 − 2𝑃05 𝑃3 𝛼2
Proposition 4. The nilpotent QFLA of dimension 9 with 𝛼11 ≠ 0 and 𝛼1 ≠ 0 can be classified in three nonisomorphic subfamilies 𝜇𝑖 with 𝑖 from 1 to 3, described in Figure 1, according to the conditions described in Figure 2.
− 2𝑃05 𝑃4 + 15𝑃02 𝑃14 − 𝑃04 𝑃22 + 𝑃04 𝑃12 𝛼9 −1
−20𝑃03 𝑃13 𝛼2 − 𝑃03 𝑃13 𝛼8 ) × (𝑃010 ) ,
Proof. Let us apply the change of base:
(12) 𝛿8 =
𝑦0 = 𝑃0 𝑥0 + 𝑃1 𝑥1 + 𝑃2 𝑥2 + 𝑃3 𝑥3 + 𝑃4 𝑥4 + 𝑃5 𝑥5 + 𝑃6 𝑥6 + 𝑃7 𝑥7 + 𝑃8 𝑥8 , 𝑦1 = 𝑄0 𝑥0 + 𝑄1 𝑥1 + 𝑄2 𝑥2 + 𝑄3 𝑥3 + 𝑄4 𝑥4 + 𝑄5 𝑥5 + 𝑄6 𝑥6 + 𝑄7 𝑥7 + 𝑄8 𝑥8 , 𝑦𝑖+1 = [𝑦0 , 𝑦𝑖 ] ,
𝛿9 =
(−20𝑃1 + 𝛼8 𝑃0 ) , 𝑃02
(8𝑃12 𝑃02 − 𝑃03 𝑃1 𝛼8 + 𝑃04 𝛼9 + 2𝑃02 𝑄3 + 2𝑃1 𝛼2 𝑃03 − 𝑄22 ) 𝑃06
𝑦8 = [𝑦1 , 𝑦6 ] . The subfamilies of laws are 𝜇𝑖 (𝛿2 , 𝛿3 , 𝛿4 , 𝛿5 , 𝛿8 , 𝛿9 ). The determinant of the change matrix is 𝑃018 (𝑃0 𝑄1 − 𝑃1 𝑄0 )9 ; thus 𝑃0 ≠ 0 and (𝑃0 𝑄1 − 𝑃1 𝑄0 ) ≠ 0. Since the coefficient of 𝑦3 and 𝑦4 in [𝑦1 , 𝑦2 ] must be null, then 𝑄0 = 0 and 𝑄1 = 𝑃02 . Let us
.
(14)
(8)
1 ⩽ 𝑖 ⩽ 6,
(13)
Let us select 𝑃1 , 𝑄3 , 𝑃3 , and 𝑃4 appropriately and the subfamilies of laws result in 𝜇𝑖 (0, 0, 0, 0, 𝛿8 , 𝛿9 ) [𝑦0 , 𝑦𝑖 ] = 𝑦𝑖+1 ,
1 ⩽ 𝑖 ⩽ 6,
[𝑦1 , 𝑦2 ] = 𝑦4 , [𝑦1 , 𝑦3 ] = 𝑦5 , [𝑦1 , 𝑦4 ] = 3𝑦6 + 𝛿8 𝑦7 + 𝛿9 𝑦8 ,
Journal of Applied Mathematics
5
No
𝛼1 ≠ 0
𝛼2 ≠ 0
Yes
Yes Yes
𝛼2 ≠ 𝛼8
𝛼3 ≠ 0
𝛼3 ≠ 0
Yes
Yes 𝜇14
𝛼2 ≠ 0
𝜇15
Yes 𝛼3 ≠ 0
𝛼4 ≠ 0
𝛼4 ≠ 0
Yes
Yes
𝜇16
𝛼3 ≠ 0
Yes
𝜇17
𝜇18
𝜇19
Yes
𝜇12
𝜇13
𝛼8 ≠ 0
𝛼8 ≠ 0
Yes
Yes
𝛼8 ≠ 4𝛼2
𝛼8 ≠ 4𝛼2
Yes
Yes
𝛼8 ≠ 10𝛼2
Yes
𝛼4 ≠ −𝛼2 𝛼9
Yes 𝛼4 ≠ 3𝛼2 𝛼9 Yes
Yes 𝜇1 (𝜆)
𝜇2
𝜇3
𝜇4 (𝜆)
𝜇5 (𝜆)
𝜇6 (𝜆)
𝜇7 (𝜆)
𝜇8
𝜇9
𝜇10
𝜇11
Figure 2: Case A: 𝛼11 ≠ 0; nonisomorphic subfamilies determination.
[𝑦1 , 𝑦5 ] = 5𝑦7 ,
= 1, and the algebra is described by Thus 𝛿9 𝜇2 (1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 1). Finally, the forth subcase is the algebra 𝜇3 (1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1).
[𝑦1 , 𝑦6 ] = 𝑦8 , [𝑦2 , 𝑦3 ] = −2𝑦6 − 𝛿8 𝑦7 − 𝛿9 𝑦8 ,
2.3.2. Case A.2. One has 𝛼11 ≠ 0 and 𝛼1 = 0.
[𝑦2 , 𝑦4 ] = −2𝑦7 ,
Proposition 5. The nilpotent QFLA of dimension 9 with 𝛼11 ≠ 0, 𝛼1 = 0, and 𝛼2 = 𝛼8 are nonisomorphic to the algebras with 𝛼11 ≠ 0, 𝛼1 = 0, and 𝛼2 ≠ 𝛼8 .
[𝑦2 , 𝑦5 ] = −𝑦8 , [𝑦3 , 𝑦4 ] = 𝑦8 . (15) The nullity of −10𝛼2 + 𝛼8 and the nullity of 31𝛼22 + 8𝛼9 − 4𝛼3 − 4𝛼2 𝛼8 are invariants. It can be proved by substituting (9) and (13) in the expressions −10𝛿2 + 𝛿8 and 31𝛿22 + 8𝛿9 − 4𝛿3 − 4𝛿2 𝛿8 ; then 𝑃0 is in the denominator and since it must be nonnull, the nullity of those expressions is invariant for the change (8). Then four subcases are determined by the nullity of 𝛿8 and 𝛿9 . If 𝛿8 ≠ 0, the subcases corresponding to 𝛿9 nullity or nonnullity can be reduced to a subfamily of algebras 𝜇𝜆1 (1, 0, 0, 0, 0, 0, 3, 1, 𝜆, 0, 1) with one parameter 𝜆 ∈ C, with 𝜆 = (1/8)(−31𝛼22 +8𝛼9 +4𝛼3 +4𝛼2 𝛼8 )/(−31𝛼2 +31𝛼8 )2 . If 𝛿8 = 0, in the subcase corresponding to 𝛿9 ≠ 0, 𝑃0 can be selected as 1/2 (−18𝛼22 + 16𝛼9 − 8𝛼3 ) /4 since 𝛼9 ≠ (−31𝛼22 +4𝛼3 +4𝛼2 𝛼8 )/8 = 10𝛼2 imply −9𝛼22 + 8𝛼9 − 4𝛼3 ≠ 0. and 𝛼8
Proof. The derived series is D1 g = ⟨𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 ⟩, D2 g = ⟨(𝛼2 − 𝛼8 )𝑥7 , 𝑥8 ⟩, and so forth; thus Dim[D2 g] = 2 if 𝛼2 ≠ 𝛼8 and Dim[D2 g] = 1 if 𝛼2 = 𝛼8 . The nullity of 𝛼2 − 𝛼8 constitutes a new classification criterion. Proposition 6. The nilpotent QFLA of dimension 9 with 𝛼11 ≠ 0, 𝛼1 = 0, and 𝛼2 ≠ 𝛼8 can be classified in ten nonisomorphic subfamilies 𝜇𝑗 with 𝑗 from 4 to 13, described in Figure 1, according to the conditions described in Figure 2. Proof. Let us apply the change of base (8). The subfamilies of laws are 𝜇𝑗 (𝛿2 , 𝛿3 , 𝛿4 , 𝛿5 , 𝛿8 , 𝛿9 ) and the restrictions 𝑃0 ≠ 0 and 𝑄1 ≠ 0. Thus, with the coefficient identifications, the new parameters are 𝛿2 =
𝑄1 𝛼2 , 𝑃03
(16)
6
Journal of Applied Mathematics 𝛿3 =
𝑄1 𝛼3 , 𝑃03
(17)
𝛿4 = − (2𝑄12 𝑃1 𝛼22 + 2𝑄3 𝑃0 𝑄1 𝛼2 − 𝑄22 𝑃0 𝛼2 + 𝛼2 𝑄12 𝑃1 𝛼8 − 𝑄12 𝑃0 𝛼4
(18) −1
−2𝑄1 𝑃0 𝑄3 𝛼8 + 𝑃0 𝑄22 𝛼8 ) × (𝑄1 𝑃06 ) ,
+ 2𝑃0 𝑃2 𝑄1 𝛼2 𝑄2 − 𝑃1 𝑃0 𝛼4 𝑄12 − 2𝑃1 𝑃0 𝑄1 𝑄3 𝛼8
+
𝑄12 𝑃02 𝛼5
−𝑄22 𝑃02 𝛼9
+
2𝑄1 𝑃02 𝑄3 𝛼9
−
2𝑄4 𝑃02 𝑄2 𝛿8 =
𝛿9 =
+
+
(19)
2𝑄5 𝑃02 𝑄1
𝑄32 𝑃02 )
𝛼8 𝑄1 , 𝑃03
×
−1 (𝑄12 𝑃06 ) ,
𝛿4 =
(20)
(−𝑄12 𝑃1 𝛼8 + 𝑄12 𝑃0 𝛼9 + 2𝑃0 𝑄3 𝑄1 + 2𝑄12 𝑃1 𝛼2 − 𝑃0 𝑄22 ) 𝑃03 𝑄12
Proposition 7. The nilpotent QFLA of dimension 9 with 𝛼11 ≠ 0, 𝛼1 = 0, and 𝛼2 = 𝛼8 can be classified in six nonisomorphic subfamilies 𝜇𝑘 with 𝑘 from 14 to 19, described in Figure 1, according to the conditions described in Figure 2. Proof. Let us apply the change of base (8). The subfamilies of laws are 𝜇𝑘 (𝛿2 , 𝛿3 , 𝛿4 , 𝛿5 , 𝛿8 , 𝛿9 ) and the restrictions 𝑃0 ≠ 0 and 𝑄1 ≠ 0. Thus, with the coefficient identifications, the new parameters are
𝛿5 = (−𝛼2 𝑄12 𝑃1 𝑃0 𝛼9 − 2𝛼2 𝑄12 𝑃3 𝑃0 + 𝛼2 𝑄12 𝑃12 𝛼8
− 2𝑃1 𝑃0 𝛼2 𝑄22 + 𝑃1 𝑃0 𝑄22 𝛼8 + 2𝑃1 𝑃0 𝛼2 𝑄1 𝑄3
𝛼3 = 0, selecting 𝑄1 = 𝑃03 /𝛼8 ≠ 0 and 𝑃1 = 𝑃0 (−𝛼4 + 𝛼9 𝛼8 + 𝑃02 𝛼8 )/𝛼82 ≠ 0, the subfamily is 𝜇13 (0, 0, 0, 1, 0, 0, 1, 0).
.
(21)
𝛿2 =
𝑄1 𝛼2 , 𝑃03
(22)
𝛿3 =
𝑄1 𝛼3 , 𝑃04
(23)
𝑄1 (−3𝛼22 𝑃1 + 𝑃0 𝛼4 ) 𝑃06
,
(24)
𝛿5 = (−𝛼2 𝑄12 𝑃1 𝑃0 𝛼9 − 2𝛼2 𝑄12 𝑃3 𝑃0 + 𝛼22 𝑄12 𝑃12 + 2𝑃2 𝑃0 𝑄1 𝛼2 𝑄2 − 𝑃0 𝑄12 𝑃1 𝛼4 + 𝑄12 𝑃02 𝛼5
(25)
+ 2𝑄1 𝑃02 𝑄3 𝛼9 + 2𝑄5 𝑃02 𝑄1 − 𝑄22 𝑃02 𝛼9 Selecting 𝑄5 and 𝑄3 appropriately the subfamilies of laws result in 𝜇𝑗 (𝛿2 , 𝛿3 , 𝛿4 , 0, 𝛿8 , 0). From (16), (17), and (20) the invariance of the nullities of 𝛼2 , 𝛼3 , and 𝛼8 is clear. The nullity of 4𝛼2 − 𝛼8 is also invariant. It can be proved by substituting (16) and (20) in the expression 4𝛿2 −𝛿8 considering the change restrictions. If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 ≠ 0, 𝛼3 ≠ 0, 𝛼8 ≠ 0, and 𝛼8 ≠ 4𝛼2 , selecting 𝑄1 = 𝑃03 /𝛼8 ≠ 0, 𝑃0 = 𝛼3 /𝛼8 ≠ 0, and 𝑃1 = 𝛼3 (−𝛼8 𝛼9 + 𝛼4 + 𝛼2 𝛼9 )/(𝛼82 (−𝛼8 + 4𝛼2 )), the subfamily is 𝜇𝜆4 (0, 𝜆, 1, 0, 0, 0, 1, 0), with 𝛿2 = 𝛼2 /𝛼8 = 𝜆 ∈ C − {0, 1, 1/4}. If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 ≠ 0, 𝛼3 ≠ 0, 𝛼8 ≠ 0, and 𝛼8 = 4𝛼2 , then 𝛿4 is invariant, and the subfamily is 𝜇𝜆5 (0, 1/4, 1, 𝜆, 0, 0, 1, 0), 𝜆 ∈ C. If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 ≠ 0, 𝛼3 ≠ 0, and 𝛼8 = 0, selecting 𝑄1 = 𝑃04 /𝛼3 ≠ 0 and 𝑃0 = 𝛼3 /𝛼2 ≠ 0, the subfamily is 𝜇𝜆6 (0, 1, 1, 𝜆, 0, 0, 0, 0), 𝜆 ∈ C. If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 ≠ 0, 𝛼3 = 0, 𝛼8 ≠ 0, and 𝛼8 ≠ 4𝛼2 , selecting 𝑄1 = 𝑃03 /𝛼8 ≠ 0 and 𝑃1 = 𝑃0 (−𝛼9 𝛼8 + 𝛼2 𝛼9 + 𝛼4 − 𝑃02 𝛼8 )/(𝛼8 (−𝛼8 + 4𝛼2 )), the subfamily is 𝜇𝜆7 (0, 𝜆, 0, 1, 0, 0, 1, 0), 𝜆 ∈ C − {0, 1, 1/4}. If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 ≠ 0, 𝛼3 = 0, 𝛼8 ≠ 0, and 𝛼8 = 4𝛼2 , selecting 𝑄1 = 𝑃03 /𝛼2 , the nullity of −𝛼4 + 3𝛼2 𝛼9 is invariant. If 𝛼4 ≠ 3𝛼2 𝛼9 , selecting 𝑃02 = −𝛼2 (−𝛼4 + 3𝛼2 𝛼9 )/𝛼22 , the subfamily is 𝜇8 (0, 1, 0, 1, 0, 0, 4, 0), else the subfamily is 𝜇9 (0, 1, 0, 0, 0, 0, 4, 0). If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 ≠ 0, 𝛼3 = 0, and 𝛼8 = 0, selecting 𝑄1 = 𝑃03 /𝛼2 , the nullity of 𝛼2 𝛼9 + 𝛼4 is invariant. If 𝛼4 ≠ − 𝛼2 𝛼9 , selecting 𝑃02 = 𝛼2 (𝛼2 𝛼9 + 𝛼4 )/𝛼22 ≠ 0, the subfamily is 𝜇10 (0, 1, 0, 1, 0, 0, 0, 0); else with 𝑄1 = 𝑃03 /𝛼2 the subfamily is 𝜇11 (0, 1, 0, 0, 0, 0, 0, 0). If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 = 0, and 𝛼3 ≠ 0, selecting 𝑄1 = 𝑃04 /𝛼3 ≠ 0, 𝑃0 = 𝛼3 /𝛼8 ≠ 0, and 𝑃1 = −𝛼3 (𝛼4 − 𝛼9 𝛼8 )/𝛼83 , the subfamily is 𝜇12 (0, 0, 1, 0, 0, 0, 1, 0). If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 ≠ 𝛼8 , 𝛼2 = 0, and
−1
−2𝑄4 𝑃02 𝑄2 − 𝑃0 𝑃1 𝑄22 𝛼2 + 𝑄32 𝑃02 ) × (𝑄12 𝑃06 ) , 𝛿9 =
(𝑃1 𝑄12 𝛼2 + 𝑄12 𝑃0 𝛼9 + 2𝑃0 𝑄3 𝑄1 − 𝑃0 𝑄22 ) (𝑃03 𝑄12 )
.
(26)
Let us select 𝑄5 and 𝑄3 appropriately and the subfamilies of laws result in 𝜇𝑘 (𝛿2 , 𝛿3 , 𝛿4 , 0, 0). From (22), (23), and (24) the invariance of the nullities of 𝛼2 , 𝛼3 , and 𝛼4 is clear. If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 = 𝛼8 , 𝛼2 ≠ 0, and 𝛼3 ≠ 0, selecting 𝑄1 = 𝑃03 /𝛼2 ≠ 0, 𝑃0 = 𝛼3 /𝛼2 ≠ 0, and 𝑃1 = 𝛼3 𝛼4 /(3𝛼23 ), the subfamily is 𝜇14 (0, 1, 1, 0, 0, 0, 1, 0). If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 = 𝛼8 , 𝛼2 ≠ 0, and 𝛼3 = 0, selecting 𝑄1 = 𝑃03 /𝛼2 ≠ 0 and 𝑃1 = 𝑃0 𝛼4 /(3𝛼22 ), the subfamily is 𝜇15 (0, 1, 0, 0, 0, 0, 1, 0). If 𝛼11 ≠ 0 ∧ 𝛼1 = 0, 𝛼2 = 𝛼8 , 𝛼2 = 0, and 𝛼3 ≠ 0, selecting 𝑄1 = 𝑃04 /𝛼3 ≠ 0, the nullity of 𝛼4 is invariant. If 𝛼4 ≠ 0 and 𝑃0 = 𝛼4 /𝛼3 ≠ 0, the subfamily is 𝜇16 (0, 0, 1, 1, 0, 0, 0, 0). If 𝛼4 = 0 the subfamily is 𝜇17 (0, 0, 1, 0, 0, 0, 0, 0). If 𝛼11 ≠ 0, 𝛼1 = 0, 𝛼2 = 𝛼8 , 𝛼2 = 0, and 𝛼3 = 0, selecting 𝑄1 = 𝑃05 /𝛼4 ≠ 0, the subfamilies are 𝜇18 (0, 0, 0, 1, 0, 0, 0, 0) with 𝛼4 ≠ 0, and 𝜇19 (0, 0, 0, 0, 0, 0, 0, 0) with 𝛼4 = 0. 2.4. Case B. One has 𝛼11 = 0. Proposition 8. The nilpotent QFLA of dimension 9 and 𝛼11 = 0 and 𝛼9 ≠ 0 are nonisomorphic to the algebras with 𝛼11 = 0 and 𝛼9 = 0. Proof. For the family described by (3a)–(3o) and (4a)–(4q), C3 g = ⟨𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝛼9 𝑥8 ⟩; therefore its dimension is Dim[C3 g] = 5, if 𝛼9 ≠ 0 or Dim[C3 g] = 4, if 𝛼9 = 0, and the nullity of 𝛼9 constitutes a new classification criterion.
Journal of Applied Mathematics
7
2.4.1. Case B.1. One has 𝛼11 = 0 and 𝛼9 ≠ 0. Proposition 9. The nilpotent QFLA of dimension 9 with 𝛼11 = 0, 𝛼9 ≠ 0, and 𝛼6 ≠ 0 are nonisomorphic to the algebras with 𝛼11 = 0, 𝛼9 ≠ 0, and 𝛼6 = 0. Proof. If 𝛼9 ≠ 0, it can be deduced that 𝛼12 = 𝛼13 = 𝛼14 = 0. By computing the Jacobi equations, the family of laws is reduced to
Case B.1.1: 𝛼11 = 0, 𝛼9 ≠ 0 and 𝛼6 ≠ 0. Figure 3 provides the classification in 18 subfamilies in this case. Case B.1.2: 𝛼11 = 0, 𝛼9 ≠ 0, and 𝛼6 = 0. Figure 4 provides the classification in 26 subfamilies in this case. 2.4.2. Case B.2. One has 𝛼11 = 0 and 𝛼9 = 0. The restrictions in the family (4a)–(4q) are reduced to
(27)
𝛼5 𝛼12 = 0,
(41)
(28)
𝛼6 𝛼12 = 0,
(42)
[𝑥1 , 𝑥3 ] = 𝛼1 𝑥5 + 𝛼2 𝑥6 + 𝛼3 𝑥7 ,
(29)
𝛼6 𝛼13 = 0,
(43)
[𝑥1 , 𝑥4 ] = 𝛼6 𝑥5 + 𝛼7 𝑥6 + 𝛼8 𝑥7 + 𝛼9 𝑥8 ,
(30)
𝛼10 𝛼12 = 0,
(44)
[𝑥1 , 𝑥5 ] = 2𝛼6 𝑥6 + (2𝛼7 − 𝛼1 ) 𝑥7 ,
(31)
𝛼12 (𝛼1 − 𝛼7 ) = 0,
(45)
[𝑥1 , 𝑥6 ] = 𝛼10 𝑥7 ,
(32)
𝛼5 𝛼13 − 2𝛼62 = 0,
(46)
[𝑥1 , 𝑥8 ] = 𝛼15 𝑥6 + 𝛼16 𝑥7 ,
(33)
[𝑥0 , 𝑥𝑖 ] = 𝑥𝑖+1 ,
1 ⩽ 𝑖 ⩽ 6,
[𝑥1 , 𝑥2 ] = 𝛼1 𝑥4 + 𝛼2 𝑥5 + 𝛼3 𝑥6 + 𝛼4 𝑥7 + 𝛼5 𝑥8 ,
[𝑥2 , 𝑥3 ] = −𝛼6 𝑥5 + (𝛼1 − 𝛼7 ) 𝑥6 + (𝛼2 − 𝛼8 ) 𝑥7 − 𝛼9 𝑥8 , (34) [𝑥2 , 𝑥4 ] = −𝛼6 𝑥6 + (𝛼1 − 𝛼7 ) 𝑥7 ,
(35)
[𝑥2 , 𝑥5 ] = (2𝛼6 − 𝛼10 ) 𝑥7 ,
(36)
[𝑥2 , 𝑥8 ] = 𝛼15 𝑥7 ,
(37)
[𝑥3 , 𝑥4 ] = (−3𝛼6 + 𝛼10 ) 𝑥7
(38)
with two restrictions −2𝛼62 − 𝛼9 𝛼15 = 0, −4𝛼1 𝛼6 + 3𝛼1 𝛼10 − 2𝛼6 𝛼7 − 𝛼7 𝛼10 − 𝛼9 𝛼16 = 0.
(39)
Since 𝛼9 ≠ 0, the application of the elementary change of base CB 𝑦0 = 𝑦1 𝑦𝑖 =
𝑥𝑖 , 𝛼9
1⩽𝑖⩽𝑛−1
with |CB| =
(40)
1 ≠ 0 𝛼98
permits us to suppose that 𝛼9 = 1. Then from (39), 𝛼15 = −2𝛼62 and 𝛼16 = −4𝛼1 𝛼6 + 3𝛼1 𝛼10 − 2𝛼6 𝛼7 − 𝛼7 𝛼10 . This implies that (33) and (37) are changed to [𝑥1 , 𝑥8 ] = −2𝛼62 𝑥6 + (𝛼1 (−4𝛼6 + 3𝛼10 ) − 𝛼7 (2𝛼6 + 𝛼10 ))𝑥7 and [𝑥2 , 𝑥8 ] = −2𝛼62 𝑥7 , respectively, and the subfamily of laws 𝜇𝑙 (𝛼1 , 𝛼2 , 𝛼3 , 𝛼4 , 𝛼5 , 𝛼6 , 𝛼7 , 𝛼8 , 1, 𝛼10 , 0, 0, 0, 0, 0, 0), with 𝑙 from 20 to 63, has no restrictions (4a)–(4q). Its derived series is D1 g = ⟨𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 ⟩, D2 g = ⟨−𝛼6 𝑥5 + (𝛼1 − 𝛼7 )𝑥6 + (𝛼2 − 𝛼8 )𝑥7 , 𝑥8 , −𝛼6 𝑥6 + (𝛼1 − 𝛼7 )𝑥7 , (2𝛼6 − 𝛼10 )𝑥7 , −2𝛼62 𝑥7 , (−3𝛼6 + 𝛼10 )𝑥7 ⟩, and so forth. Thus Dim[D2 g] = 4 if 𝛼6 ≠ 0 and Dim[D2 g] ⩽ 3 if 𝛼6 = 0. Therefore the nullity of 𝛼6 constitutes a new classification criterion.
2 (𝛼2 − 𝛼8 ) 𝛼12 + 3 (𝛼1 − 𝛼7 ) 𝛼13 + 2 (𝛼6 − 𝛼10 ) 𝛼14 = 0, (47) 𝛼5 𝛼14 − 2 (2𝛼1 + 𝛼7 ) 𝛼6 − 𝛼9 𝛼16 + (3𝛼1 − 𝛼7 ) 𝛼10 = 0.
(48)
Proposition 10. The nilpotent QFLA of dimension 9 with 𝛼11 = 0, 𝛼9 = 0, and 𝛼5 ≠ 0 are nonisomorphic to the algebras with 𝛼11 = 0, 𝛼9 = 0, and 𝛼5 = 0. Proof. Equations (43) and (46) imply that 𝛼6 = 0. By computing the Jacobi equations, the subfamily of laws is 𝜇𝑚 (𝛼1 , 𝛼2 , 𝛼3 , 𝛼4 , 𝛼5 , 0, 𝛼7 , 𝛼8 , 0, 𝛼10 , 0, 𝛼12 , 𝛼13 , 𝛼14 , 𝛼15 , 𝛼16 ), with 𝑚 from 64 to 263, and the restrictions are reduced to 6. Its descending central series is C1 g = ⟨𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝛼5 𝑥8 ⟩, and so forth. Thus the nullity of 𝛼5 constitutes a new classification criterion. An exhaustive and extensive process of analysis with the same methodology shown in the previous subsections leads to the final subclassification, which is summarized in the following Figures. Case B.2.1: 𝛼11 = 0, 𝛼9 = 0, and 𝛼5 ≠ 0. Figure 5 provides the classification in 55 subfamilies in this case. Case B.2.2: 𝛼11 = 0, 𝛼9 = 0, and 𝛼5 = 0. Figures 6 and 7 provide the classification in 145 subfamilies in this case.
3. 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
8
Journal of Applied Mathematics 𝜇 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
𝛼1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0
𝛼2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼5 𝜆1 0 2/3 2/3 1 1 1 0 0 0 0 0 2/3 2/3 1 1 0 0
𝛼6 𝛼7 𝛼8 𝛼9 𝛼10 𝛼11 𝜆2 0 1 1 1 0 1 0 1 3 0 𝜆 1 0 0 1 3 0 1 7/9 0 1 3 0 1 0 1 ≠3 0 𝜆 1 1 0 1 3 0 1 0 0 1 3 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 1 3 0 1 0 0 1 3 0 1 0 1 0 0 𝜆 1 0 0 1 0 0 1 −1/3 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0
𝛼12 𝛼13 𝛼14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼15 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2
𝛼16 −1 − 3𝜆2 5 − 5𝜆 5 10/9 −3𝜆 −5 0 −3 0 −5 0 −4 − 2𝜆 −4 −10/3 −2 0 −2 0
Classifcation criterion Classifcation criterion from previous cases Additional result from previous cases Direct result
Aa Aa Aa Aa
Figure 3: Case B.1.1: 𝛼11 = 0, 𝛼9 ≠ 0, and 𝛼6 ≠ 0; classification of the QFLA of dimension 9. 𝜇
𝛼1
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
Aa Aa Aa Aa Aa
𝛼2 𝛼3 𝜆1 𝜆2 1 𝜆 −2 1 −2 5 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0
𝛼4
𝛼5
𝛼6
𝛼7
𝛼8
𝛼9
𝛼10 𝛼11 𝛼12 𝛼13 𝛼14 𝛼15 𝛼16
0 0 0 0 0 0 0 0 0 0 0 0 0 𝜆 1 0 0 0 0 0 0 1 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 3 3 3 𝜆 1 0 1 0 1 0 𝜆 𝜆 1 1 1 l 0 1 0 1 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 0 0 0 −𝜆 −1 0 −1 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Classification criterion Classifcation criterion from previous cases Additional result Additional result from previous cases Direct result
Figure 4: Case B.1.2: 𝛼11 = 0; 𝛼9 ≠ 0, and 𝛼6 = 0; classification of the QFLA of dimension 9.
Journal of Applied Mathematics 𝜇
𝛼1
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 Aa Aa Aa Aa Aa
𝛼2
9 𝛼3
1 1 𝜆1 1 0 𝜆1 1 𝜆1 − 2 𝜆2 1 𝜆−2 0 1 15/8 1 1 𝜆 − 2 31 − 8𝜆 1 1 𝜆 1 1 0 1 1 7 1 𝜆1 𝜆2 𝜆 1 0 1 0 −2 1 31 −2 0 𝜆2 𝜆1 0 1 𝜆 0 0 0 0 1 0 0 1 𝜆 0 0 0 −1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 𝜆2 1 𝜆1 1 1 𝜆1 1 1 0 1 0 0 1 0 𝜆1 1 1 0 1 0 0 0 0 𝜆1 𝜆 0 0 0 1 𝜆 0 0 𝜆 0 0 𝜆 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0
𝛼4
𝛼5 𝛼6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
𝛼7
0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝜆1 ≠ 1/2 0 𝜆1 ≠ 1/2 0 𝜆1 ≠ 0 0 0 0 𝜆≠0 0 𝜆≠0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1/2 0 1/2 0 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼8 𝛼9 𝛼10 𝛼11 𝛼12 𝛼13 𝛼14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝜆2 1 𝜆2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼15
𝛼16
0 𝜆2 ≠ 0, 3 𝜆3 0 3 𝜆2 0 0 𝜆1 ≠ 0, 3 0 𝜆 ≠ 0, 3, 31/8 −5𝜆 0 31/8 −155/8 0 𝜆 0, 3 −5𝜆 0 3 0 −15 0 3 −15 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 𝜆3 ≠ 2 0 0 𝜆2 ≠ 2 0 2 0 0 2 𝜆 0 2 0 0 2 0 0 2 1 0 2 0 0 𝜆1 𝜆2 0 1 𝜆 0 0 𝜆 𝜆3 0 1 0 0 𝜆2 0 0 𝜆 0 0 𝜆 0 1 𝜆2 0 0 𝜆 0 0 𝜆 𝜆2 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
Classification criterion Classifcation criterion from previous cases Additional result Additional result from previous cases Direct result
Figure 5: Case B.2.1: 𝛼11 = 0, 𝛼9 = 0, 𝛼5 ≠ 0; classification of the QFLA of dimension 9.
10
Journal of Applied Mathematics 𝜇 𝛼1 𝛼2 𝛼3 119 1 0 0 120 1 0 0 121 1 0 0 122 1 0 0 123 1 −2 0 124 1 −2 0 125 1 −2 0 126 1 0 0 127 1 −2 0 128 1 0 −2 129 1 0 0 130 1 −2 0 131 1 0 −2 132 0 0 1 133 0 0 1 134 0 0 0 135 0 0 0 136 0 1 1 137 0 1 1 138 0 1 1 139 0 1 0 140 0 1 0 141 0 1 0 142 0 1 0 143 0 0 1 144 0 0 1 145 0 0 1 146 0 0 1 147 0 0 0 148 0 0 0 149 0 0 0 150 0 0 0 151 𝜆1 ≠ 0 0 0 152 𝜆1 ≠ 0 0 0 153 𝜆 ≠ 0 0 0 154 𝜆 ≠ 0 0 0 155 𝜆1 ≠ 0 0 0 156 𝜆 ≠ 0 0 0 157 𝜆 ≠ 0 0 0 158 𝜆 ≠ 0 0 0 159 𝜆 ≠ 0 0 0 160 𝜆 ≠ 0 0 0 161 𝜆 ≠ 0 0 0 162 0 1 0 163 0 1 0 164 0 1 0 165 0 1 0 166 0 0 0 167 0 0 0 168 0 0 0 169 0 0 0 170 0 0 0 171 0 0 0 172 0 0 0 173 1 0 0 174 1 0 0 175 1 0 𝜆1 176 1 0 𝜆 177 1 0 𝜆 178 0 0 𝜆 179 0 0 2/3 180 0 0 1 181 0 0 0 182 0 0 0 183 0 𝜆 ≠ 1/2 0 184 0 1/2 0 185 0 𝜆 ≠ 1/2 0 186 0 1/2 0 187 0 1/2 0 188 0 𝜆1 𝜆2 189 0 1 𝜆 190 0 0 𝜆
Aa Aa Aa Aa Aa
𝛼4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼6 𝛼7 𝛼8 0 3 0 0 3 0 0 3 0 0 3 𝜆1 0 3 𝜆 0 3 0 0 3 −31/2 0 3 𝜆 0 3 0 0 3 −31/2 0 3 𝜆 0 3 0 0 3 −31/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝜆1 − 1 1 0 𝜆1 − 1 0 0 𝜆−1 0 0 𝜆−1 0 0 𝜆1 − 1 𝜆2 0 𝜆−1 1 0 𝜆−1 0 0 𝜆−1 1 0 𝜆−1 0 0 𝜆−1 1 0 𝜆−1 0 0 0 −1 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 −1 0 0 1 −1 0 1 −1 −1 0 1 −1 0 1 0 1 −1 0 0 −1 −1 0 0 −1 0 0 0 0 −1 −1 0 0 0 0 𝜆−1 0 0 −1/2 0 0 𝜆−1 0 0 −1/2 0 0 −1/2 0 0 𝜆1 − 1 0 0 𝜆−1 0 0 𝜆−1
𝛼9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼13 𝛼14 1 −3 1 −3 1 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −3 1 −3 1 −3 1 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 2 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0
𝛼15 1 9/2 9/2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 𝜆2 1 0 0 1 1 1 0 0 0 0 𝜆1 1 0 0 1 0 0 1 1 0 0 𝜆1 𝜆1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1
𝛼16 𝜆 0 15/2 𝜆2 0 −5 −5 1 1 1 0 0 0 0 -1 1 0 𝜆 1 0 1 0 1 0 1 0 1 0 1 0 1 0 𝜆3 𝜆2 1 0 1 0 0 1 1 0 0 𝜆2 𝜆 1 0 𝜆 1 0 1 0 1 0 𝜆2 𝜆2 𝜆2 1 0 0 𝜆 0 1 0 0 𝜆 0 1 0 1 0 0
Classification criterion Classifcation criterion from previous cases Additional result Additional result from previous cases Direct result
Figure 6: Case B.2.2. First part: 𝛼11 = 0, 𝛼9 = 0, and 𝛼5 = 0. Classification of the QFLA of dimension 9.
Journal of Applied Mathematics
11 𝜇 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263
𝛼1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Aa Aa Aa Aa Aa
Classification criterion Classifcation criterion from previous cases Additional result Additional result from previous cases Direct result
𝛼2 𝜆 𝜆 𝜆 𝜆 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0
𝛼3 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝜆 1 0 1 0 0 1 0 0 𝜆 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0
𝛼4 0 0 0 0 𝜆1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 𝜆1 1 0 0 𝜆 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 𝜆 1 0 𝜆 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0
𝛼5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼7 𝛼8 𝛼9 0 𝜆−1 0 0 𝜆−1 0 0 𝜆−1 0 0 𝜆−1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼12 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼13 𝛼14 0 0 0 0 0 0 0 0 0 𝜆2 0 0 0 0 0 0 0 0 0 𝜆1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
𝛼15 0 0 0 0 0 𝜆1 1 0 0 𝜆2 0 0 0 𝜆 1 0 𝜆1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
𝛼16 1 1 0 0 𝜆3 𝜆2 𝜆 1 0 0 𝜆 1 0 0 0 0 𝜆2 𝜆 1 0 𝜆2 𝜆 1 0 1 𝜆 𝜆 0 0 1 0 1 0 𝜆 1 0 𝜆 𝜆 𝜆 1 0 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0
Figure 7: Case B.2.2. Second part: 𝛼11 = 0, 𝛼9 = 0, and 𝛼5 = 0. Classification of the QFLA of dimension 9.
12 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 (𝑎1 ⋅ ⋅ ⋅ 𝑎16).
Journal of Applied Mathematics
[11]
[12]
[13]
[14]
[15]
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
[16]
[17]
Acknowledgment [18]
The authors appreciate the aid of Jos´e Ram´on G´omez Martn, Professor of the University of Seville, advisor of the Ph.D. thesis of one of the authors, which constitutes the basis of these works.
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