Classification of the Quasifiliform Nilpotent Lie Algebras of Dimension 9

Research Article Classification of the Quasifiliform Nilpotent Lie Algebras of Dimension 9 Mercedes Pérez,1Francisco P. Pérez,2and Emilio Jiménez3 1Department of Mechanical Engineering, University of La Rioja, 26004 Logro˜ no, Spain 2Department of Applied Mathematics I, University of Sevilla, 41012 Seville, Spain 3Department of Electrical Engineering, University of La Rioja, 26004 Logro˜ no, Spain Correspondence should be addressed to Emilio Jim´ enez; emilio[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´ udezandGoze[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]developanalgorithmfor 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 Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 173072, 12 pages http://dx.doi.org/10.1155/2014/173072 2Journal of Applied Mathematics 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. 2. The Subfamilies of Laws 2.1. Preliminaries. Let gbe a nilpotent Lie algebra; the characteristic sequence of 𝑎𝑑(𝑋)is denoted by 𝑐(𝑋) = (𝑐1,...,𝑐 𝑘,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 𝑛is the only one with Goze’s invariant (1,...,1),inmetabelianalgebrasthecharacteristic series is (2,...2,1,...1), in Heisenberg algebras it is (2,1,...,1),infiliformalgebrasitis(𝑛 − 1,1),andin quasifiliform algebras it is (𝑛 − 2,1,1).ALiealgebragis nilpotent if and only if the characteristic polynomial of the matrix 𝑎𝑑(𝑥) is 𝜆9,foreveryvector𝑥of g.Anywaythis 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 [𝑥0,𝑥𝑖]=𝑥 𝑖+1, 1⩽𝑖⩽6; [𝑥 0,𝑥𝑖]=0, 7⩽𝑖⩽8. (1) On the whole all the bracket products can be described by [𝑥𝑖,𝑥𝑗]=𝑛−1 ∑ 𝑘=0𝐶𝑘 𝑖𝑗 ⋅𝑥 𝑘, 0⩽𝑖,𝑗⩽𝑛−1, (2) where 𝐶𝑘 𝑖𝑗 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]=𝑥 𝑖+1,1⩽𝑖⩽6, (3a) [𝑥1,𝑥2]=𝛼 1𝑥4+𝛼 2𝑥5+𝛼 3𝑥6+𝛼 4𝑥7+𝛼 5𝑥8,(3b) [𝑥1,𝑥3]=𝛼 1𝑥5+𝛼 2𝑥6+𝛼 3𝑥7,(3c) [𝑥1,𝑥4]=𝛼 6𝑥5+𝛼 7𝑥6+𝛼 8𝑥7+𝛼 9𝑥8,(3d) [𝑥1,𝑥5]=2𝛼 6𝑥6+(2𝛼7−𝛼 1)𝑥7,(3e) [𝑥1,𝑥6]=𝛼 10𝑥7+𝛼 11𝑥8,(3f) [𝑥1,𝑥8]=𝛼 12𝑥3+𝛼 13𝑥4+𝛼 14𝑥5+𝛼 15𝑥6+𝛼 16𝑥7,(3g) [𝑥2,𝑥3]=−𝛼 6𝑥5+(𝛼 1−𝛼 7)𝑥6+(𝛼 2−𝛼 8)𝑥7−𝛼 9𝑥8, (3h) [𝑥2,𝑥4]=−𝛼 6𝑥6+(𝛼 1−𝛼 7)𝑥7,(3i) [𝑥2,𝑥5]=(2𝛼6−𝛼 10)𝑥7−𝛼 11𝑥8,(3j) [𝑥2,𝑥8]=𝛼 12𝑥4+𝛼 13𝑥5+𝛼 14𝑥6+𝛼 15𝑥7,(3k) [𝑥3,𝑥4]=(−3𝛼6+𝛼 10)𝑥7+𝛼 11𝑥8,(3l) [𝑥3,𝑥8]=𝛼 12𝑥5+𝛼 13𝑥6+𝛼 14𝑥7,(3m) [𝑥4,𝑥8]=𝛼 12𝑥6+𝛼 13𝑥7,(3n) [𝑥5,𝑥8]=𝛼 12𝑥7(3o) subject to 𝛼5𝛼12 =0, (4a) 𝛼6𝛼12 =0, (4b) 𝛼6𝛼13 =0, (4c) 𝛼9𝛼12 =0, (4d) 𝛼9𝛼13 =0, (4e) 𝛼9𝛼14 =0, (4f) 𝛼10𝛼12 =0, (4g) 𝛼11𝛼12 =0, (4h) 𝛼11𝛼13 =0, (4i) 𝛼11𝛼14 =0, (4j) 𝛼11𝛼15 =0, (4k) 𝛼11𝛼16 =0, (4l) 𝛼11 (3𝛼1−𝛼 7)=0, (4m) 𝛼12 (𝛼1−𝛼 7)=0, (4n) 𝛼5𝛼13 −2𝛼2 6−𝛼 9𝛼15 =0, (4o) 2(𝛼2−𝛼 8)𝛼12 +3(𝛼1−𝛼 7)𝛼13 +2(𝛼6−𝛼 10)𝛼14 =0, (4p) 𝛼5𝛼14 −2(2𝛼1+𝛼 7)𝛼6−𝛼 9𝛼16 +(3𝛼1−𝛼 7)𝛼10 =0 (4q) with the application of the Jacobi identity to the 3-tuple (𝑥0,𝑥𝑖,𝑥𝑗),where𝑥𝑖,𝑥𝑗arebasevectorsdifferentfrom Journal of Applied Mathematics 3 Table 1: Notation for the QFLA parameters. 𝛼1=𝐶 4 12 𝛼2=𝐶 5 12 𝛼3=𝐶 6 12 𝛼4=𝐶 7 12 𝛼5=𝐶 8 12 𝛼6=𝐶 5 14 𝛼7=𝐶 6 14 𝛼8=𝐶 7 14 𝛼9=𝐶 8 14 𝛼10 =𝐶 7 16 𝛼11 =𝐶 8 16 𝛼12 =𝐶 3 18 𝛼12 =𝐶 4 18 𝛼14 =𝐶 5 18 𝛼15 =𝐶 6 18 𝛼16 =𝐶 7 18 𝑥0vector. 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 =0and 𝛼1=0), B.1.1 (𝛼11 =0,𝛼9=0, and 𝛼6=0), B.1.2 (𝛼11 =0,𝛼9=0,and𝛼6=0), B.2.1 (𝛼11 =0, 𝛼9=0,and𝛼5=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. ThenilpotentQFLAofdimension9and𝛼11 =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 C1g=⟨𝑥 2,𝑥3,𝑥4, 𝑥5,𝑥6,𝑥7,𝛼5𝑥8,𝛼9𝑥8,𝛼11𝑥8⟩,C2g=⟨𝑥 3,𝑥4,𝑥5,𝑥6,𝑥7,𝛼5𝑥8, 𝛼9𝑥8,𝛼11𝑥8⟩,C3g=⟨𝑥 4,𝑥5,𝑥6,𝑥7,𝛼9𝑥8,𝛼11𝑥8⟩,C4g=⟨𝑥 5, 𝑥6,𝑥7,𝛼9𝑥8,𝛼11𝑥8⟩,C5g=⟨𝑥 6,𝑥7,𝛼11𝑥8⟩,andsoforth.Thus Dim[C5g]=3if 𝛼11 =0and Dim[C5g]=2if 𝛼11 =0. Therefore the nullity of 𝛼11 constitutes the first classification criterion. 2.3. Case A. 𝛼11 =0. Proposition 2. The nilpotent QFLA of dimension 9 with 𝛼11 =0∧𝛼1=0are nonisomorphic to the algebras with 𝛼11 =0∧ 𝛼1=0. Proof. If 𝛼11 =0, from restrictions (4a)–(4q)itcanbededuced that 𝛼6=𝛼 12 =𝛼 13 =𝛼 14 =𝛼 15 =𝛼 16 =0and 𝛼7=3𝛼 1.By computing the Jacobi equations, the family of laws is reduced to [𝑥0,𝑥𝑖]=𝑥 𝑖+1,1⩽𝑖⩽6, (5a) [𝑥1,𝑥2]=𝛼 1𝑥4+𝛼 2𝑥5+𝛼 3𝑥6+𝛼 4𝑥7+𝛼 5𝑥8,(5b) [𝑥1,𝑥3]=𝛼 1𝑥5+𝛼 2𝑥6+𝛼 3𝑥7,(5c) [𝑥1,𝑥4]=3𝛼 1𝑥6+𝛼 8𝑥7+𝛼 9𝑥8,(5d) [𝑥1,𝑥5]=5𝛼 1𝑥7,(5e) [𝑥1,𝑥6]=𝛼 10𝑥7+𝛼 11𝑥8,(5f) [𝑥2,𝑥3]=−2𝛼 1𝑥6+(𝛼 2−𝛼 8)𝑥7−𝛼 9𝑥8,(5g) [𝑥2,𝑥4]=−2𝛼 1𝑥7,(5h) [𝑥2,𝑥5]=−𝛼 10𝑥7−𝛼 11𝑥8,(5i) [𝑥3,𝑥4]=𝛼 10𝑥7+𝛼 11𝑥8(5j) without restrictions derived from the Jacobi identity (4a)– (4q).Itcanbeobservedthat𝑥7and 𝑥8are now central; thus the application of the elementary change of base 𝑦𝑖=𝑥 𝑖,𝑖=8, 𝑦8=𝛼 10 ⋅𝑥 7+𝛼 11 ⋅𝑥 8 (6) permitsustosupposethat𝛼10 =0and 𝛼11 =1.Then (5f), (5i), and (5j) are simplified and the derived series is D1g=⟨𝑥 2,𝑥3,𝑥4,𝑥5,𝑥6,𝑥7,𝑥8⟩,D2g=⟨−2𝛼 1𝑥6,(𝛼2− 𝛼8)𝑥7,−𝛼9𝑥8,−2𝛼1𝑥7,𝑥8⟩,andsoforth.ThusDim[D2g]=3 if 𝛼1=0and Dim[D2g]⩽2if 𝛼1=0. Therefore the nullity of 𝛼1constitutes a new classification criterion. In this subsection (case A), the notation to describe the parameters of the subfamily 𝑖is 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 =0and 𝛼1=0. Proposition 3. Case A.1 permits us to suppose that 𝛼1=1. Proof. With the elementary change of base CB, 𝑦0=𝑥 0, 𝑦𝑖=𝑥𝑖 𝛼1, 1⩽𝑖⩽7, 𝑦8=𝑥8 𝛼2 1, (7) |CB|=1/𝛼 9 1=0and 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 𝛿the new parameters obtained from the changes of base and 𝜇the 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). 4Journal of Applied Mathematics 1100003 1 2100003 01 3100003 00 40 1 00 0 10 50 1 00 10 60 1 1 00 00 70 0 10 0 10 80 1 0 1 00 40 90 1 0 0 00 40 10 0 1 0 1 00 00 11 0 1 0 0 00 00 12 0 0 1 0 00 10 13 0 0 0 10 0 10 14 0 1 1 00 0 10 15 0 1 0 00 0 10 16 0 0 1 1 00 00 17 0 0 1 0 00 00 18 0 0 0 1 00 00 19 0 0 0 0 00 00 Aa Direct result 𝜆 𝜆 𝜆 𝜆 𝜆 1/4 𝜇𝛼1𝛼2𝛼3𝛼4𝛼5𝛼7𝛼8𝛼9 Aa Classification criterion Aa Additional result Aa Aa Classifcation criterion from previous cases Additional result from previous cases 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). Proposition 4. ThenilpotentQFLAofdimension9with 𝛼11 =0 and 𝛼1=0 canbeclassifiedinthreenonisomorphic subfamilies 𝜇𝑖with 𝑖from 1 to 3, described in Figure 1, according to the conditions described in Figure 2. Proof. Let us apply the change of base: 𝑦0=𝑃 0𝑥0+𝑃 1𝑥1+𝑃 2𝑥2+𝑃 3𝑥3+𝑃 4𝑥4 +𝑃 5𝑥5+𝑃 6𝑥6+𝑃 7𝑥7+𝑃 8𝑥8, 𝑦1=𝑄 0𝑥0+𝑄 1𝑥1+𝑄 2𝑥2+𝑄 3𝑥3+𝑄 4𝑥4 +𝑄 5𝑥5+𝑄 6𝑥6+𝑄 7𝑥7+𝑄 8𝑥8, 𝑦𝑖+1 =[𝑦 0,𝑦𝑖], 1⩽𝑖⩽6, 𝑦8=[𝑦 1,𝑦6]. (8) The subfamilies of laws are 𝜇𝑖(𝛿2,𝛿3,𝛿4,𝛿5,𝛿8,𝛿9). The determinantofthechangematrixis𝑃18 0(𝑃0𝑄1−𝑃 1𝑄0)9;thus𝑃0=0 and (𝑃0𝑄1−𝑃 1𝑄0) =0. Since the coefficient of 𝑦3and 𝑦4 in [𝑦1,𝑦2]must be null, then 𝑄0=0and 𝑄1=𝑃 2 0.Letus apply (8) again and the only restriction is 𝑃0=0.Thus,with the coefficient identifications, the new parameters are 𝛿2=(−2𝑃1+𝛼 2𝑃0) 𝑃2 0,(9) 𝛿3=(−2𝑄2 2+4𝑃2 0𝑄3−7𝑃 1𝛼2𝑃3 0+7𝑃2 1𝑃2 0+𝛼 3𝑃4 0) 𝑃6 0,(10) 𝛿4=(𝛼 4𝑃6 0−2𝛼2 2𝑃5 0𝑃1−2𝛼 2𝑃4 0𝑄3+𝛼 2𝑃2 0𝑄2 2 −𝛼 2𝑃5 0𝑃1𝛼8+2𝑄3 2−12𝑃1𝛼3𝑃5 0 −42𝑃1𝑃3 0𝑄3−44𝑃3 1𝑃3 0+56𝑃4 0𝑃2 1𝛼2 +18𝑃1𝑄2 2𝑃0−6𝑃2 0𝑄3𝑄2+6𝑃4 0𝑄4 +6𝑃 2𝑃3 0𝑄2+𝑃4 0𝑃2 1𝛼8−6𝑃5 0𝑃3 +2𝛼8𝑃4 0𝑄3−𝑄2 2𝑃2 0𝛼8)×(𝑃9 0)−1, (11) 𝛿5=(2𝑃 0𝑃1𝑄3𝑄2+2𝑃 2𝛼2𝑃3 0𝑄2+6𝑃2 1𝛼3𝑃4 0 −2𝑃 2𝑃0𝑄2 2−2𝑃2 0𝑄4𝑄2+6𝑃4 0𝑃3𝑃1 −8𝑃2 1𝑄2 2−2𝑃 0𝑃1𝑄2 2𝛼2−4𝑃 1𝑃2𝑃2 0𝑄2 +2𝑃 2𝑃3 0𝑄3+2𝑃 3𝑃3 0𝑄2−4𝑃 1𝑃3 0𝑄4 +20𝑃2 0𝑃2 1𝑄3−𝑄2 2𝑃2 0𝛼9+2𝑄 3𝑃4 0𝛼9 +𝑃 1𝑄2 2𝑃0𝛼8−𝑃 1𝑃5 0𝛼4+𝑃2 0𝑄2 3+2𝑃4 0𝑄5 +𝑃6 0𝛼5+2𝑃 1𝑃3 0𝑄3𝛼2−2𝑃 1𝛼8𝑃3 0𝑄3 +𝛼 2𝑃4 0𝑃2 1𝛼8−𝛼 2𝑃5 0𝑃1𝛼9−2𝑃5 0𝑃3𝛼2 −2𝑃5 0𝑃4+ 15𝑃2 0𝑃4 1−𝑃4 0𝑃2 2+𝑃4 0𝑃2 1𝛼9 −20𝑃3 0𝑃3 1𝛼2−𝑃3 0𝑃3 1𝛼8)×(𝑃10 0)−1, (12) 𝛿8=(−20𝑃1+𝛼 8𝑃0) 𝑃2 0,(13) 𝛿9=(8𝑃2 1𝑃2 0−𝑃3 0𝑃1𝛼8+𝑃4 0𝛼9+2𝑃2 0𝑄3+2𝑃 1𝛼2𝑃3 0−𝑄2 2) 𝑃6 0. (14) Let us select 𝑃1,𝑄3,𝑃3,and𝑃4appropriately and the subfamiliesoflawsresultin𝜇𝑖(0,0,0,0,𝛿8,𝛿9) [𝑦0,𝑦𝑖]=𝑦 𝑖+1, 1⩽𝑖⩽6, [𝑦1,𝑦2]=𝑦 4, [𝑦1,𝑦3]=𝑦 5, [𝑦1,𝑦4]=3𝑦 6+𝛿 8𝑦7+𝛿 9𝑦8, Journal of Applied Mathematics 5 No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 𝜇1(𝜆) 𝜇2𝜇3𝜇4(𝜆) 𝜇5(𝜆) 𝜇6(𝜆) 𝜇7(𝜆) 𝜇8𝜇9𝜇10 𝜇11 𝛼10≠ ≠ 𝛼20≠ 𝛼20≠ 𝛼30≠ 𝛼30 ≠ 𝛼30≠ 𝛼30≠ 𝛼80≠ ≠ ≠ 𝛼80≠ 𝛼40≠ 𝛼40≠ 𝜇14 𝜇15 𝜇16 𝜇17 𝜇18 𝜇19 𝜇13 𝜇12 𝛼2𝛼8 𝛼810𝛼2 𝛼84𝛼2≠ ≠ ≠ 𝛼84𝛼2𝛼4−𝛼2𝛼9 𝛼43𝛼2𝛼9 Figure 2: Case A: 𝛼11 =0; nonisomorphic subfamilies determination. [𝑦1,𝑦5]=5𝑦 7, [𝑦1,𝑦6]=𝑦 8, [𝑦2,𝑦3]=−2𝑦 6−𝛿 8𝑦7−𝛿 9𝑦8, [𝑦2,𝑦4]=−2𝑦 7, [𝑦2,𝑦5]=−𝑦 8, [𝑦3,𝑦4]=𝑦 8. (15) The nullity of −10𝛼2+𝛼 8and the nullity of 31𝛼2 2+8𝛼 9−4𝛼 3−4𝛼 2𝛼8are invariants. It can be proved by substituting (9)and(13) in the expressions −10𝛿2+𝛿 8and 31𝛿2 2+8𝛿 9−4𝛿 3−4𝛿 2𝛿8;then𝑃0is 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 𝛿8and 𝛿9.If𝛿8=0, the subcases corresponding to 𝛿9nullity or nonnullity can be reduced to a subfamily of algebras 𝜇𝜆 1(1,0,0,0,0,0,3,1,𝜆,0,1)with one parameter 𝜆∈C,with 𝜆 = (1/8)(−31𝛼2 2+8𝛼9+4𝛼3+4𝛼2𝛼8)/(−31𝛼2+31𝛼8)2.If𝛿8= 0, in the subcase corresponding to 𝛿9=0,𝑃0canbeselectedas (−18𝛼2 2+16𝛼9−8𝛼 3)1/2/4since 𝛼9=(−31𝛼2 2+4𝛼3+4𝛼2𝛼8)/8 and 𝛼8= 10𝛼2imply −9𝛼2 2+8𝛼 9−4𝛼 3=0. Thus 𝛿9=1, and the algebra is described by 𝜇2(1,0,0,0,0,0,3,0,1,0,1).Finally,theforthsubcase is the algebra 𝜇3(1,0,0,0,0,0,3,0,0,0,1). 2.3.2. Case A.2. One has 𝛼11 =0and 𝛼1=0. Proposition 5. ThenilpotentQFLAofdimension9with 𝛼11 =0,𝛼1=0,and𝛼2=𝛼 8are nonisomorphic to the algebras with 𝛼11 =0,𝛼1=0,and𝛼2=𝛼8. Proof. The derived series is D1g=⟨𝑥 2,𝑥3,𝑥4,𝑥5,𝑥6,𝑥7,𝑥8⟩, D2g=⟨(𝛼 2−𝛼 8)𝑥7,𝑥8⟩,andsoforth;thusDim[D2g]=2if 𝛼2=𝛼8and Dim[D2g]=1if 𝛼2=𝛼 8.Thenullityof𝛼2−𝛼 8 constitutes a new classification criterion. Proposition 6. The nilpotent QFLA of dimension 9 with 𝛼11 =0,𝛼1=0,and𝛼2=𝛼8canbeclassifiedinten nonisomorphic subfamilies 𝜇𝑗with 𝑗from4to13,describedin Figure 1, according to the conditions described in Figure 2. Proof. Letusapplythechangeofbase(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=𝑄1𝛼2 𝑃3 0,(16) 6Journal of Applied Mathematics 𝛿3=𝑄1𝛼3 𝑃3 0,(17) 𝛿4=−(2𝑄 2 1𝑃1𝛼2 2+2𝑄 3𝑃0𝑄1𝛼2−𝑄2 2𝑃0𝛼2 +𝛼 2𝑄2 1𝑃1𝛼8−𝑄2 1𝑃0𝛼4 −2𝑄1𝑃0𝑄3𝛼8+𝑃 0𝑄2 2𝛼8)×(𝑄 1𝑃6 0)−1, (18) 𝛿5=(−𝛼 2𝑄2 1𝑃1𝑃0𝛼9−2𝛼 2𝑄2 1𝑃3𝑃0+𝛼 2𝑄2 1𝑃2 1𝛼8 +2𝑃 0𝑃2𝑄1𝛼2𝑄2−𝑃 1𝑃0𝛼4𝑄2 1−2𝑃 1𝑃0𝑄1𝑄3𝛼8 −2𝑃 1𝑃0𝛼2𝑄2 2+𝑃 1𝑃0𝑄2 2𝛼8+2𝑃 1𝑃0𝛼2𝑄1𝑄3 +𝑄2 1𝑃2 0𝛼5+2𝑄 1𝑃2 0𝑄3𝛼9+2𝑄 5𝑃2 0𝑄1 −𝑄2 2𝑃2 0𝛼9−2𝑄 4𝑃2 0𝑄2+𝑄2 3𝑃2 0)×(𝑄2 1𝑃6 0)−1, (19) 𝛿8=𝛼8𝑄1 𝑃3 0,(20) 𝛿9=(−𝑄2 1𝑃1𝛼8+𝑄2 1𝑃0𝛼9+2𝑃 0𝑄3𝑄1+2𝑄2 1𝑃1𝛼2−𝑃 0𝑄2 2) 𝑃3 0𝑄2 1. (21) Selecting 𝑄5and 𝑄3appropriately 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,and𝛼8is clear. The nullity of 4𝛼2−𝛼8is also invariant. It can be proved by substituting (16)and(20) in the expression 4𝛿2−𝛿8considering the change restrictions. If 𝛼11 =0,𝛼1=0,𝛼2=𝛼8,𝛼2=0,𝛼3=0,𝛼8=0, and 𝛼8=4𝛼2,selecting𝑄1=𝑃 3 0/𝛼8=0,𝑃0=𝛼 3/𝛼8=0, and 𝑃1=𝛼 3(−𝛼8𝛼9+𝛼 4+𝛼 2𝛼9)/(𝛼2 8(−𝛼8+4𝛼 2)),the subfamily is 𝜇𝜆 4(0,𝜆,1,0,0,0,1,0),with𝛿2=𝛼2/𝛼8= 𝜆∈C−{0,1,1/4}.If𝛼11 =0,𝛼1=0,𝛼2=𝛼8,𝛼2=0, 𝛼3=0,𝛼8=0,and𝛼8=4𝛼2,then𝛿4is invariant, and the subfamily is 𝜇𝜆 5(0,1/4,1,𝜆,0,0,1,0), 𝜆∈C.If𝛼11 =0, 𝛼1=0,𝛼2=𝛼8,𝛼2=0,𝛼3=0,and𝛼8=0,selecting 𝑄1=𝑃 4 0/𝛼3=0 and 𝑃0=𝛼 3/𝛼2=0,thesubfamilyis 𝜇𝜆 6(0,1,1,𝜆,0,0,0,0),𝜆∈C.If𝛼11 =0,𝛼1=0,𝛼2=𝛼8, 𝛼2=0,𝛼3=0,𝛼8=0,and𝛼8=4𝛼2,selecting𝑄1=𝑃 3 0/𝛼8=0 and 𝑃1=𝑃 0(−𝛼9𝛼8+𝛼 2𝛼9+𝛼 4−𝑃 2 0𝛼8)/(𝛼8(−𝛼8+4𝛼 2)), the subfamily is 𝜇𝜆 7(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,and𝛼8= 4𝛼2,selecting𝑄1=𝑃 3 0/𝛼2,thenullityof−𝛼4+3𝛼 2𝛼9is invariant. If 𝛼4=3𝛼2𝛼9,selecting𝑃2 0=−𝛼 2(−𝛼4+3𝛼 2𝛼9)/𝛼2 2, the subfamily is 𝜇8(0,1,0,1,0,0,4,0),elsethesubfamilyis 𝜇9(0,1,0,0,0,0,4,0).If𝛼11 =0,𝛼1=0,𝛼2=𝛼8,𝛼2=0,𝛼3= 0,and𝛼8=0,selecting𝑄1=𝑃 3 0/𝛼2,thenullityof𝛼2𝛼9+𝛼4is invariant. If 𝛼4=−𝛼2𝛼9,selecting𝑃2 0=𝛼 2(𝛼2𝛼9+𝛼4)/𝛼2 2=0, the subfamily is 𝜇10(0,1,0,1,0,0,0,0);elsewith𝑄1=𝑃 3 0/𝛼2 the subfamily is 𝜇11(0,1,0,0,0,0,0,0).If𝛼11 =0,𝛼1=0, 𝛼2=𝛼8,𝛼2=0,and𝛼3=0,selecting𝑄1=𝑃 4 0/𝛼3=0,𝑃0= 𝛼3/𝛼8=0,and𝑃1=−𝛼 3(𝛼4−𝛼 9𝛼8)/𝛼3 8,thesubfamilyis 𝜇12(0,0,1,0,0,0,1,0).If𝛼11 =0,𝛼1=0,𝛼2=𝛼8,𝛼2=0,and 𝛼3=0,selecting𝑄1=𝑃 3 0/𝛼8=0and 𝑃1=𝑃 0(−𝛼4+𝛼 9𝛼8+ 𝑃2 0𝛼8)/𝛼2 8=0,thesubfamilyis𝜇13(0,0,0,1,0,0,1,0). Proposition 7. The nilpotent QFLA of dimension 9 with 𝛼11 =0,𝛼1=0,and𝛼2=𝛼 8can be classified in six nonisomorphic subfamilies 𝜇𝑘with 𝑘from 14 to 19, described in Figure 1, according to the conditions described in Figure 2. Proof. Letusapplythechangeofbase(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 𝛿2=𝑄1𝛼2 𝑃3 0,(22) 𝛿3=𝑄1𝛼3 𝑃4 0,(23) 𝛿4=𝑄1(−3𝛼2 2𝑃1+𝑃 0𝛼4) 𝑃6 0,(24) 𝛿5=(−𝛼 2𝑄2 1𝑃1𝑃0𝛼9−2𝛼 2𝑄2 1𝑃3𝑃0+𝛼2 2𝑄2 1𝑃2 1 +2𝑃 2𝑃0𝑄1𝛼2𝑄2−𝑃 0𝑄2 1𝑃1𝛼4+𝑄2 1𝑃2 0𝛼5 +2𝑄 1𝑃2 0𝑄3𝛼9+2𝑄 5𝑃2 0𝑄1−𝑄2 2𝑃2 0𝛼9 −2𝑄4𝑃2 0𝑄2−𝑃 0𝑃1𝑄2 2𝛼2+𝑄2 3𝑃2 0)×(𝑄2 1𝑃6 0)−1, (25) 𝛿9=(𝑃1𝑄2 1𝛼2+𝑄2 1𝑃0𝛼9+2𝑃 0𝑄3𝑄1−𝑃 0𝑄2 2) (𝑃3 0𝑄2 1).(26) Let us select 𝑄5and 𝑄3appropriately 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,and𝛼4is clear. If 𝛼11 =0, 𝛼1=0,𝛼2=𝛼8,𝛼2=0,and𝛼3=0,selecting𝑄1=𝑃 3 0/𝛼2=0, 𝑃0=𝛼 3/𝛼2=0,and𝑃1=𝛼 3𝛼4/(3𝛼3 2),thesubfamilyis 𝜇14(0,1,1,0,0,0,1,0).If𝛼11 =0,𝛼1=0,𝛼2=𝛼8,𝛼2=0, and 𝛼3=0,selecting𝑄1=𝑃 3 0/𝛼2=0and 𝑃1=𝑃 0𝛼4/(3𝛼2 2), the subfamily is 𝜇15(0,1,0,0,0,0,1,0).If𝛼11 =0∧𝛼1=0, 𝛼2=𝛼8,𝛼2=0,and𝛼3=0,selecting𝑄1=𝑃 4 0/𝛼3=0,the nullity of 𝛼4is invariant. If 𝛼4=0and 𝑃0=𝛼 4/𝛼3=0,the subfamily is 𝜇16(0,0,1,1,0,0,0,0).If𝛼4=0the subfamily is 𝜇17(0,0,1,0,0,0,0,0).If𝛼11 =0,𝛼1=0,𝛼2=𝛼8,𝛼2=0, and 𝛼3=0,selecting𝑄1=𝑃 5 0/𝛼4=0,thesubfamiliesare 𝜇18(0,0,0,1,0,0,0,0)with 𝛼4=0,and𝜇19(0,0,0,0,0,0,0,0) with 𝛼4=0. 2.4. Case B. One has 𝛼11 =0. Proposition 8. ThenilpotentQFLAofdimension9and𝛼11 = 0and 𝛼9=0are nonisomorphic to the algebras with 𝛼11 =0 and 𝛼9=0. Proof. For the family described by (3a)–(3o)and(4a)–(4q), C3g=⟨𝑥 4,𝑥5,𝑥6,𝑥7,𝛼9𝑥8⟩; therefore its dimension is Dim[C3g]=5,if𝛼9=0or Dim[C3g]=4,if𝛼9=0,and the nullity of 𝛼9constitutes a new classification criterion. Journal of Applied Mathematics 7 2.4.1. Case B.1. One has 𝛼11 =0and 𝛼9=0. Proposition 9. ThenilpotentQFLAofdimension9with𝛼11 = 0,𝛼9=0,and𝛼6=0are nonisomorphic to the algebras with 𝛼11 =0,𝛼9=0,and𝛼6=0. Proof. If 𝛼9=0,itcanbededucedthat𝛼12 =𝛼 13 =𝛼 14 =0.By computing the Jacobi equations, the family of laws is reduced to [𝑥0,𝑥𝑖]=𝑥 𝑖+1,1⩽𝑖⩽6, (27) [𝑥1,𝑥2]=𝛼 1𝑥4+𝛼 2𝑥5+𝛼 3𝑥6+𝛼 4𝑥7+𝛼 5𝑥8,(28) [𝑥1,𝑥3]=𝛼 1𝑥5+𝛼 2𝑥6+𝛼 3𝑥7,(29) [𝑥1,𝑥4]=𝛼 6𝑥5+𝛼 7𝑥6+𝛼 8𝑥7+𝛼 9𝑥8,(30) [𝑥1,𝑥5]=2𝛼 6𝑥6+(2𝛼7−𝛼 1)𝑥7,(31) [𝑥1,𝑥6]=𝛼 10𝑥7,(32) [𝑥1,𝑥8]=𝛼 15𝑥6+𝛼 16𝑥7,(33) [𝑥2,𝑥3]=−𝛼 6𝑥5+(𝛼 1−𝛼 7)𝑥6+(𝛼 2−𝛼 8)𝑥7−𝛼 9𝑥8, (34) [𝑥2,𝑥4]=−𝛼 6𝑥6+(𝛼 1−𝛼 7)𝑥7,(35) [𝑥2,𝑥5]=(2𝛼 6−𝛼 10)𝑥7,(36) [𝑥2,𝑥8]=𝛼 15𝑥7,(37) [𝑥3,𝑥4]=(−3𝛼6+𝛼 10)𝑥7(38) with two restrictions −2𝛼2 6−𝛼 9𝛼15 =0, −4𝛼1𝛼6+3𝛼 1𝛼10 −2𝛼 6𝛼7−𝛼 7𝛼10 −𝛼 9𝛼16 =0. (39) Since 𝛼9=0, the application of the elementary change of base CB 𝑦0=𝑦 1 𝑦𝑖=𝑥𝑖 𝛼9,1⩽𝑖⩽𝑛−1 with |CB|=1 𝛼8 9=0 (40) permits us to suppose that 𝛼9=1.Thenfrom(39), 𝛼15 = −2𝛼2 6and 𝛼16 =−4𝛼 1𝛼6+3𝛼 1𝛼10 −2𝛼 6𝛼7−𝛼 7𝛼10.This implies that (33)and(37)arechangedto[𝑥1,𝑥8]=−2𝛼 2 6𝑥6+ (𝛼1(−4𝛼6+3𝛼 10)−𝛼 7(2𝛼6+𝛼 10))𝑥7and [𝑥2,𝑥8]= −2𝛼2 6𝑥7, respectively, and the subfamily of laws 𝜇𝑙(𝛼1,𝛼2,𝛼3, 𝛼4,𝛼5,𝛼6,𝛼7,𝛼8,1,𝛼10,0,0,0,0,0,0),with𝑙from 20 to 63, has no restrictions (4a)–(4q). Its derived series is D1g= ⟨𝑥2,𝑥3,𝑥4,𝑥5,𝑥6,𝑥7,𝑥8⟩,D2g=⟨−𝛼 6𝑥5+(𝛼1−𝛼7)𝑥6+(𝛼2− 𝛼8)𝑥7,𝑥8,−𝛼6𝑥6+(𝛼1−𝛼7)𝑥7,(2𝛼6−𝛼10)𝑥7,−2𝛼2 6𝑥7,(−3𝛼6+ 𝛼10)𝑥7⟩,andsoforth.ThusDim[D2g]=4if 𝛼6=0 and Dim[D2g]⩽3if 𝛼6=0. Therefore the nullity of 𝛼6 constitutes a new classification criterion. Case B.1.1: 𝛼11 =0,𝛼9=0and 𝛼6=0.Figure 3 provides the classification in 18 subfamilies in this case. Case B.1.2: 𝛼11 =0,𝛼9=0,and𝛼6=0.Figure 4 provides the classification in 26 subfamilies in this case. 2.4.2. Case B.2. One has 𝛼11 =0and 𝛼9=0. The restrictions in the family (4a)–(4q) are reduced to 𝛼5𝛼12 =0, (41) 𝛼6𝛼12 =0, (42) 𝛼6𝛼13 =0, (43) 𝛼10𝛼12 =0, (44) 𝛼12 (𝛼1−𝛼 7)=0, (45) 𝛼5𝛼13 −2𝛼2 6=0, (46) 2(𝛼2−𝛼 8)𝛼12 +3(𝛼1−𝛼 7)𝛼13 +2(𝛼6−𝛼 10)𝛼14 =0, (47) 𝛼5𝛼14 −2(2𝛼1+𝛼 7)𝛼6−𝛼 9𝛼16 +(3𝛼1−𝛼 7)𝛼10 =0. (48) Proposition 10. The nilpotent QFLA of dimension 9 with 𝛼11 =0,𝛼9=0,and𝛼5=0are nonisomorphic to the algebras with 𝛼11 =0,𝛼9=0,and𝛼5=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 𝑚from 64 to 263, and the restrictions are reduced to 6. Its descending central series is C1g= ⟨𝑥2,𝑥3,𝑥4,𝑥5,𝑥6,𝑥7,𝛼5𝑥8⟩,andsoforth.Thusthenullityof 𝛼5constitutes 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,and𝛼5=0.Figure 5 provides the classification in 55 subfamilies in this case. Case B.2.2: 𝛼11 =0,𝛼9=0,and𝛼5=0.Figures 6and 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 8Journal of Applied Mathematics 20 1000 101 1 0 000 21 10000 1 01 3 0 000 22 1000 1 0 01 3 0 000 5 23 1000 101 3 0 000 24 00001 1 01 0 000 25 00001 1 1 01 3 0 000 26 00001 1 0 01 3 0 000 0 27 00000 1 1 01 1 0 000 28 00000 1 0 01 1 0 000 0 29 00000 1 1 01 3 0 000 30 00000 1 0 01 3 0 000 0 31 10000 1 01 0 0 000 32 1000 1 0 01 0 0 000 33 1000 101 0 0 000 34 00001 1 1 01 0 0 000 35 00001 1 0 01 0 0 000 0 36 00000 1 1 01 0 0 000 37 00000 1 0 01 0 0 000 0 Aa Classifcation criterion Aa Classifcation criterion from previous cases Aa Additional result from previous cases Aa Direct result 𝜆 𝜆 𝜆 −1/3 2/3 2/3 2/3 2/3 𝜆1𝜆2 −3𝜆 10/9 7/9 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −2 −4 −5 −5 −3 −2 −2 ≠3 −1 − 3𝜆2 5−5𝜆 −4 − 2𝜆 −10/3 𝜇𝛼1𝛼2𝛼3𝛼4𝛼5𝛼6𝛼7𝛼8𝛼9𝛼10 𝛼11 𝛼12 𝛼13 𝛼14 𝛼15 𝛼16 Figure 3: Case B.1.1: 𝛼11 =0,𝛼9=0,and𝛼6=0; classification of the QFLA of dimension 9. 38 1000 1 011000002 39 1 1 000 3 011000000 40 1 1 000 3 011000000 41 1 5 000 3 011000000 42 0 1 1 00001100000 43 0 1 0 000 1 01100000 44 0 1 0 000 0 011000000 45 0 0 1 000 1 01100000 46 0 0 1 000 0 011000000 47 0 0 0 000 1 01100000 48 0 0 0 000 0 011000000 49 101000010000000 50 100000010000000 51 10100 1 010000000 52 100 1 00 1 010000000 53 100 0 00 1 010000000 54 0 1 1 000 l 010000000 55 0 1 1 000 0 010000000 56 0 1 0 000 1 010000000 57 0 1 0 0 0 00 0 010000000 58 0 0 1 00 1 010000000 59 0 0 1 1 00 0 010000000 60 0 0 1 0 00 0 010000000 61 0 0 0 000 1 010000000 62 0 0 0 1 00 0 010000000 63 0 0 0 0 00 0 010000000 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆1𝜆2 −2 −2 −1 −1 −1 −𝜆 𝜇𝛼1𝛼2𝛼3𝛼4𝛼5𝛼6𝛼7𝛼8𝛼9𝛼10 𝛼11 𝛼12 𝛼13 𝛼14 𝛼15 𝛼16 Aa Direct result Aa Classification criterion Aa Additional result Aa Aa Classifcation criterion from previous cases Additional result from previous cases Figure 4: Case B.1.2: 𝛼11 =0;𝛼9=0,and𝛼6=0; classification of the QFLA of dimension 9. Journal of Applied Mathematics 9 64 11 01030010000 65 10 01030010000 3 66 101030010000 0 67 1001030010000 68 1101030010000 31/8 69 101030010000 70 11 01030010000 30 71 11 001030010000 3 72 11 701030010000 3 73 101030010000 01 74 10 01030010000 00 75 1001030010000 00 76 13101030010000 00 77 001000010000 11 78 0101000010000 10 79 00 001000010000 10 80 01 001000010000 10 81 01 01000010000 01 82 00 001000010000 01 83 00 01000010000 01 84 01 101000010000 00 85 01 001000010000 00 86 00 101000010000 00 87 00 001000 0 010000 00 88 100010000000 1 89 100010000000 0 90 11 0010000000 2 91 11 00100 000000 2 92 10 00101000000 20 0 93 10 00100000000 20 94 10 001000000000 21 95 10 001000000000 20 96 01 00100000000 97 00 00101 1 0000000 1 98 00 001010000000 0 99 101010000000 1 100 110100000000 0 101 11001010000000 0 102 10001010000000 0 103 100101/2 0000000 1 104 11 00101/2 0000000 0 105 10 0 0 0 0101/2 0000000 0 106 00100 1000000 1 107 00100 1000000 10 108 010100 1000000 00 109 000100 1000000 00 110 000100 0000000 11 111 0010100 0000000 10 112 0000100 0000000 10 113 01 000 0000000 011 114 01 10100 0000000 00 115 01 00100 0000000 00 116 00 0 0 0100 0000000 01 117 00 10100 0000000 00 118 00 00100 0000000 00 𝜆1 1 𝜆2 𝜆2 1/2 ≠ 𝜆1 1/2 ≠ ≠ 𝜆1 0 ≠ 𝜆 𝜆 𝜆 𝜆 𝜆 2𝜆1 𝜆2 𝜆2 𝜆3 𝜆2 𝜆1 𝜆1 𝜆1 𝜆 𝜆2 𝜆1 𝜆2 𝜆2 𝜆2 𝜆3 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆 𝜆2 𝜆1 𝜆1 𝜆1 𝜆1 𝜆 𝜆 𝜆 𝜆 𝜆0 ≠𝜆0 𝜆2 2 ≠ 𝜆3 2 ≠ 𝜆1 − 2 𝜆−2 𝜆−2 15/8 31 − 8𝜆 𝜆1 0, 3 𝜆2 0, 3 𝜆<>0,3 ≠ ≠ ≠−5𝜆 −5𝜆 −155/8 𝜆 0, 3, 31/8 Aa Direct result −1 −2 −2 −15 −15 𝜇𝛼1𝛼2𝛼3𝛼4𝛼5𝛼6𝛼7𝛼8𝛼9𝛼10 𝛼11 𝛼12 𝛼13 𝛼14 𝛼15 𝛼16 Aa Classification criterion Aa Additional result Aa Aa Classifcation criterion from previous cases Additional result from previous cases Figure 5: Case B.2.1: 𝛼11 =0,𝛼9=0,𝛼5=0; classification of the QFLA of dimension 9.