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Complex filiform Lie algebras of dimension 11
Applied Mathematics and Computation 141 (2003) 611–630 www.elsevier.com/locate/amc
Complex filiform Lie algebras of dimension 11 ~ez Luis Boza a, Eugenio M. Fedriani b, Juan N un
c,*
a
Departamento de Matem atica Aplicada I. Escuela T ecnica Superior de Arquitectura, Universidad deSevilla, Avda. Reina Mercedes 2, 41012 Sevilla, Spain b Departamento de Economıa y Empresa, Universidad Pablo de Olavide, Carretera de Utrera, Km. 1, 41013 Sevilla, Spain c Facultad de Matematica, Departamento de Geometrıa y Topologıa, Universidad de Sevilla, Apartado Correos 1160, E-41080 Sevilla, Spain
Abstract In this paper we give the explicit classification of complex filiform Lie algebras of dimension 11. To do this, we use a method previously obtained by us in an earlier paper, which is based on the concept of isomorphism between Lie algebras. At present, this explicit classification is not known, although G omez, Jimenez and Khakimdjanov gave a list of these algebras in a non-explicit way, but in terms of cocycles. We find that there exist 188 families of complex filiform Lie algebras of dimension 11. 2002 Elsevier Science Inc. All rights reserved.
1. Introduction At present, the research on Lie theory is very extended. It is due to several reasons: the main one is that this theory is being actually applied not only in different branches of mathematics, such as topology, differential equations, differential or algebraic geometry, or arithmetic, for instance, but in many other sciences, such as engineering or physics, mainly. On the sake of an example, the exceptional group of Killing plays an important role in the recent superstring theory, constituting an important approximation to the relation
*
Corresponding author. E-mail addresses: [email protected] (L. Boza), [email protected] (E.M. Fedriani), [email protected] (J. Nu´n˜ez). 0096-3003/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00280-1
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between quantic mechanic and relativity theory. Moreover, one of the Lie classic group, the simplectic group of n-dimensional space is being actually very used in quantic mechanic and in optics. Apart from that, the Lie theory is also important by itself, from a strictly mathematical point of view. Indeed, from its origins by S. Lie, distinguished mathematicians, like Killing, Cartan, Ado, Levi and others, have studied this subject. However, sometimes, the research worker on Lie theory needs lots of examples of finite Lie algebras of high dimensions to check if a particular property which is verified for those of low dimensions is also satisfied on them. They are also needed, for instance, to verify if a previous conjecture, observed in the examples already known, is really true. But, at present, it is difficult to have those examples of Lie algebras of dimension, say, non-excessively low. Indeed, if we take into no consideration simple and semisimple Lie algebras, whose classification was obtained by Killing, Cartan and others, in the last decade of XIX century, only the classifications of solvable, nilpotent and filiform complex Lie algebras of dimension n, with n 6 5, n 6 7 and n 6 12, respectively, are actually known (see [2,8,11], for instance). Moreover, this problem worsens if we note that not only the classification problem of Lie algebras is still unsolved, but, in fact, it seems to be accepted by experts (like Shalev and Zelmanov, see [15]) that one will not totally classify Lie algebras of finite dimension. Indeed, it is reasonable for research workers in this theory to impose further conditions to these algebras to obtain, at least, partial results. In this way, the main goal of this paper is to give the explicit classification of complex filiform Lie algebras of dimension 11. To do this, we use a method previously obtained by us in an earlier paper, which is based on the concept of isomorphism between Lie algebras. At present, this explicit classification is not known, even if it is convenient to note that G omez, Jimenez and Khakimdjanov (see [12]) already gave a list of these algebras in a non-explicit way, but in terms of cocycles. However, this way of giving the list does not result operative to be handled. We would like now to explain why we are dealing with complex filiform Lie algebras. These algebras were introduced by M. Vergne in the late 60s of the past century (see [16]). However, before that, Blackburn (see [5]) studied the analogous class of finite p-groups and used the term maximal class to call them, which is also now used for Lie algebras. In fact, both terms filiform and maximal class are synonyms. Vergne showed that, within the variety of nilpotent Lie multiplications on a fixed vector space, non-filiforms can be relegated to small-dimensional components; thus, from some intuitive point of view, it is possible to consider that quite a lot nilpotent Lie algebras are filiform, in spite of this last subset not being dense in the space of nilpotent Lie algebras. Apart from that, complex
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filiform Lie algebras are the most structured subset of nilpotent Lie algebras, with respect to an adapted basis (this concept can be checked in the next section). In this sense, we can study and classify these algebras easier than the set of nilpotent Lie algebras. So, from now on, we will only deal with the evolution of the problem of classifying complex filiform Lie algebras, which are the aim of our study. In earlier papers, several authors have deeply studied these algebras and have obtained quite a lot results about them. So, in 1988, Goze and Ancochea, by using an invariant of these algebras, named the characteristic sequence, which corresponds to the maximal dimensions of Jordan blocks of a certain nilpotent matrix, obtained the classification of complex nilpotent Lie algebras of dimension 7 [2]. However, the use of that method to higher dimensions did not result very suitable and great difficulties appeared. So, both authors treated to classify first filiform Lie algebras, due to the reasons above mentioned. So, by the same way, they obtained the classification of complex filiform Lie algebras of dimension 8 [1]. It is convenient to say that the importance of these authors’ work is not only obtaining these classifications, but, above all, devising techniques which can be applied in any dimension. So, Echarte and G omez [10] classified, by using this method, filiform Lie algebras of dimension 9. Certainly, in that classification some two-parameter families of filiform Lie algebras appear for the first time. However, the classification given in [1] were uncomplete and some errors appeared. It was later corrected by the same authors [3] and independently by Seeley [14]. ~ez classified complex filiform Lie algebras of Later, Boza, Echarte and N un dimension 10 by the introduction of another invariant for these algebras, which they call the pair (i,h) [7]. One difference from the case of the dimension 9 is that some three-parameter families of filiform Lie algebras appeared in this last classification. In 1998, G omez, Jimenez and Khakimdjanov, by the method of the elementary changes of basis gave a correct classification of complex filiform Lie algebras of dimension n with n 6 11 [12]. To do this, they use a result by ~ez, which presented a polynomial time recursive G omez, Jimenez and N un algorithm which generates families of filiform Lie algebras of dimension n and by using it, they obtained a parametrization of the affine algebraic set of filiform Lie algebras of dimension 11 [13]. However, we had previously prepublished in [6] the classification of filiform Lie algebras of dimension 12. In that classification, four, five or even sixparameter families of filiform Lie algebras can be already found. The structure of this paper is the following: In Section 2 some definitions and notations related to complex filiform Lie algebras are given. In Section 3 we explain, in a schematic way, the application of our method for classifying complex filiform Lie algebras to the particular case of dimension 11, in which starting from the structure theorem of these algebras we obtain a representant
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of each class of isomorphism. Finally, in Section 4, we give the complete classification of these algebras. The main result of this paper is the Theorem 4.2.
2. Definitions and notations In this paper, all the Lie algebras which appear will be considered over the complex field C. It is convenient to note that by classifying a set, we mean to find a certain property of it which let us define an equivalence relation among its elements. The equivalence relation which we will use to classify complex filiform Lie algebras will be being isomorphic. So, we will explicitly compute a representant of each class of complex filiform Lie algebras in dimension 11. To do this, most of the calculations needed were made by using the Mathematica package. 2.1. Homomorphisms between Lie algebras Let g and g0 be two Lie algebras. A map U : g7!g0 is said to be an homomorphism between Lie algebras if U is a linear application such that U : ½X ; Y 7!½UðX Þ; UðY Þ; 8X ; Y 2 g. In the case of being U a bijection, it is called isomorphism. As this concept of isomorphism between filiform Lie algebras is the main idea of our method, we next give an example: let g be a complex Lie algebra of dimension 2, defined by the bracket ½e1 ; e2 ¼ e1 , with respect to an adapted basis fe1 ; e2 g and let g0 be another Lie algebra of the same dimension, defined by the bracket ½e01 ; e02 ¼ e02 , with respect to an adapted basis fe01 ; e02 g. The mapping U : g0 7!g defined by Uðe01 Þ ¼ e2 , Uðe02 Þ ¼ e1 is an isomorphism, since Uð½e01 ; e02 Þ ¼ Uðe02 Þ ¼ e1 and ½Uðe01 Þ; Uðe02 Þ ¼ ½ e2 ; e1 ¼ ½e1 ; e2 ¼ e1 . 2.2. Filiform Lie algebras Let g ¼ ðCn ; lÞ be a Lie algebra of dimension n, with l the associated law. We consider the lower central series of g defined by C 1 g ¼ g, C i g ¼ lðg; C i 1 gÞ. The Lie algebra g is filiform if dimC C i g ¼ n i for 2 6 i 6 n. These algebras were defined by M. Vergne (see [16]). If x 2 g we denote by adðxÞ the adjoint mapping associated to x (i.e. the map y7!lðx; yÞ). Let g be a filiform Lie algebra of dimension n. Then there exists a basis B ¼ fe1 ; . . . ; en g of g such that e1 2 g n C 2 g, the matrix of adðe1 Þ with respect to B has a Jordan block of order n 1 and C i g is the vector space generated by fe2 ; . . . ; en ði 1Þ g with 2 6 i 6 n 1. Such a basis is called an adapted basis. Sometimes, on the sake of simplicity, we will use ½x; y instead of lðx; yÞ for the Lie bracket in a Lie algebra.
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3. A method to classify the filiform Lie algebras of dimension 11 In this Section we apply a method (previously obtained by us [8] for filiform Lie algebras of any dimension, in general) to the particular case of dimension 11, to give the explicit classification of these algebras. First, we remind that to classify filiform Lie algebras of any dimension with this method, it is necessary to use the structure theorem of those algebras. These theorems are already known for dimension n, with n 6 14 ([4,13], for instance). They consist on giving explicitly the rest of the non-null brackets. These brackets are expressed as function of some structure constants ai;j . These constants, which define the law of the algebra, are related themselves by polynomial relations coming from the Jacobi identities. Let g be a filiform Lie algebra of dimension n with an adapted basis fe1 ; . . . ; en g. So, ½e1 ; eh ¼ eh 1 , 3 6 h 6 n. We denote the subindexes of the structure constants according to the first bracket in which they appear. For instance, in dimension 11, the first bracket we consider is ½e4 ; e11 ¼ a4;11 e2 . In order to decide which is the first bracket, we have set a certain order relation in the set of the subindexes pairs ði; jÞ with i < j. This order is defined by: 8
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Let consider the isomorphism U : g0 7!g, this is, the bases change given by 11 11 X X v1 ¼ si ui ; v11 ¼ r i ui i¼1
i¼1
and by the property of filiformity of the algebra. Step 2: By straightforward computations in the previous step, we can deduce the following general expression: 7
v2 ¼ s1 ðs1 r11 s11 r1 Þðs1 s11 a4;10 Þ u2 ; where a4;10 is the coefficient of u9 in the bracket ½u10 ; u11 . In fact, it can be easily proved that a4;10 ¼ 0 due to 11 is odd. Then, as v2 belongs to a basis, necessarily v2 6¼ 0 and thus, we have s1 6¼ 0 and s1 r11 s11 r1 6¼ 0. Step 3: Now, to make next computations easier, by using again the property of filiformity, we have: 0 ¼ ½v3 ; v11 ¼ r1 ðs1 r11 s11 r1 Þs71 u2 : Then, by taking the previous step, we have r1 ¼ 0. P11 into consideration P11 So, v1 ¼ i¼1 si ui , v11 ¼ i¼2 ri ui . We also deduce, in this step, the following conditions between the elements of the law of the algebra: s1 6¼ 0 and r11 6¼ 0. Step 4: As the bases change U is an isomorphism between Lie algebras, then ½Uðvk Þ; Uðvl Þ Uð½vk ; vl Þ ¼ 0, 8k; l ¼ 1; . . . ; 11. We obtain some vector equations by using this last expressions. Step 5: It is easy to prove in each dimension that we do not lose any piece of information although we operate with second components only. So, from now on, we will take into consideration these components only. Step 6: In this step, we already have new equations from the previous vector equations and the equations consisting on Pk ¼ 0 (these last ones are the restrictions coming from the Jacobi identities in the corresponding structure theorem). So, we have a set of four equations involving the coefficients which appear in the laws of each algebra and the coefficients ri and sj from the isomorphisms. We wish that these equations give specific values (that is, constants) for the coefficients of the law or of the isomorphisms or, in other case, relations between them. In this way, for instance, if we find an expression like the following: a1 ¼ s1 a01 , we would distinguish two non-isomorphic cases: a1 ¼ 0 ¼ a01 and a1 6¼ 0 6¼ a01 (where a1 represents any coefficient ai;j of the law). In this method, we use some kind of particular sets to describe algebras having the same starting point in our method, in a non-redundant way and by using a unique family. For dimension 11, we have considered the following kind of set: 2p Cm ¼ reiu : r > 0; u 2 0; : m
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Now, we will explain something more about the set Cm , which has been already used in earlier classifications to solve some situations as following: As we described in the general explanation, it can be convenient to distinguish two cases depending on a2 ¼ 0 or not. We are interested in the second case. Then, let suppose that we find an expression like a2 sm1 a02 ¼ 0. Then, if 1=m we consider an isomorphism with s1 ¼ 1=ða02 Þ , any algebra of the family which we are considering is isomorphic to one with a2 ¼ 1. So, from then on, we must suppose a2 ¼ 1 ¼ a02 and we must also suppose that s1 is a mth root of the unit. When continuing the study of this same case, if it appears an equation of the kind a3 s1 a03 ¼ 0, we would have to distinguish two new cases: • a3 ¼ 0 ¼ a03 and s1 is a mth root of the unit. • a3 ¼ s1 a03 6¼ 0. In this case, the sector Cm is sufficient to represent in a unique way all possible values of a3 of the algebras of that family, as a convenient value of s1 relates both a3 and a03 by one isomorphism. So, from then on, we can suppose a3 ¼ a03 2 Cm and s1 ¼ 1. 4. The classification of filiform Lie algebras of dimension 11 As we are considering the particular case of the dimension 11, we start from ~ez in [9] and later the following result, previously obtained by Castro and N un ~ez (although by using a different noconfirmed by G omez, Jimenez and N un tation) in [13]: Theorem 4.1 (Structure theorem of filiform Lie algebras of dimension 11). The set of filiform Lie algebras laws over C11 can be parametrized (up to isomorphism) by the points of an affine algebraic set V C11 of Krull dimension 12. Furthermore, if g ¼ ðC11 ; lÞ is a filiform Lie algebra, then there exists an adapted basis b fe1 ; . . . ; e11 g of g such that: • • • • • • • • • • •
½e1 ; eh ¼ eh 1 ð3 6 h 6 11Þ ½e4 ; e11 ¼ a4;11 e2 ½e5 ; e10 ¼ a5;10 e2 ½e6 ; e9 ¼ a6;9 e2 ½e7 ; e8 ¼ a7;8 e2 ½e5 ; e11 ¼ a5;11 e2 þ ða4;11 þ a5;10 Þe3 ½e6 ; e10 ¼ a6;10 e2 þ ða5;10 þ a6;9 Þe3 ½e7 ; e9 ¼ a7;9 e2 þ ða6;9 þ a7;8 Þe3 ½e6 ; e11 ¼ a6;11 e2 þ ða5;11 þ a6;10 Þe3 þ ða4;11 þ 2a5;10 þ a6;9 Þe4 ½e7 ; e10 ¼ a7;10 e2 þ ða6;10 þ a7;9 Þe3 þ ða5;10 þ 2a6;9 þ a7;8 Þe4 ½e8 ; e9 ¼ a8;9 e2 þ a7;9 e3 þ ða6;9 þ a7;8 Þe4
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• ½e7 ; e11 ¼ a7;11 e2 þ ða6;11 þ a7;10 Þe3 þ ða5;11 þ 2a6;10 þ a7;9 Þe4 þ ða4;11 þ 3a5;10 þ 3a6;9 þ a7;8 Þe5 • ½e8 ; e10 ¼ a8;10 e2 þ ða7;10 þ a8;9 Þe3 þ ða6;10 þ 2a7;9 Þe4 þ ða5;10 þ 3a6;9 þ 2a7;8 Þe5 • ½e8 ; e11 ¼ a8;11 e2 þ ða7;11 þ a8;10 Þe3 þ ða6;11 þ 2a7;10 þ a8;9 Þe4 þ ða5;11 þ 3a6;10 þ 3a7;9 Þe5 þ ða4;11 þ 4a5;10 þ 6a6;9 þ 3a7;8 Þe6 • ½e9 ; e10 ¼ a9;10 e2 þ a8;10 e3 þ ða7;10 þ a8;9 Þe4 þ ða6;10 þ 2a7;9 Þe5 þ ða5;10 þ 3a6;9 þ 2a7;8 Þe6 • ½e9 ; e11 ¼ a9;11 e2 þ ða8;11 þ a9;10 Þe3 þ ða7;11 þ 2a8;10 Þe4 þ ða6;11 þ 3a7;10 þ 2a8;9 Þe5 þ ða5;11 þ 4a6;10 þ 5a7;9 Þe6 þ ða4;11 þ 5a5;10 þ 9a6;9 þ 5a7;8 Þe7 • ½e10 ; e11 ¼ a10;11 e2 þ a9;11 e3 þ ða8;11 þ a9;10 Þe4 þ ða7;11 þ 2a8;10 Þe5 þ ða6;11 þ 3a7;10 þ 2a8;9 Þe6 þ ða5;11 þ 4a6;10 þ 5a7;9 Þe7 þ ða4;11 þ 5a5;10 þ 9a6;9 þ 5a7;8 Þe8 with the coefficients a4;11 ; a5;10 ; . . . ; a10;11 2 C verifying the following four equations: Pi ¼ 0, with 1 6 i 6 4 where • P1 ¼ 3a25;10 þ 2a4;11 a6;9 3a5;10 a6;9 6a5;10 a7;8 9a6;9 a7;8 5a27;8 • P2 ¼ 4a5;10 a6;9 6a26;9 þ 2a4;11 a7;8 þ 5a5;10 a7;8 þ 6a6;9 a7;8 þ 5a27;8 • P3 ¼ 3a5;11 a6;9 7a5;10 a6;10 6a6;9 a6;10 þ 3a5;11 a7;8 þ a6;10 a7;8 þ 2a4;11 a7;9 3a5;10 a7;9 þ 5a7;8 a7;9 • P4 ¼ 4a26;10 þ 4a6;9 a6;11 þ 2a6;11 a7;8 þ 3a5;11 a7;9 a6;10 a7;9 þ 5a27;9 8a5;10 a7;10
6a6;9 a7;10 5a7;8 a7;10 þ 2a4;11 a8;9 þ 3a5;10 a8;9 þ 11a6;9 a8;9 þ 5a7;8 a8;9 . From now on, the law of this algebra will be denoted by lða4;11 ; a5;10 ; a6;9 ; a7;8 ; a5;11 ; a6;10 ; a7;9 ; a6;11 ; a7;10 ; a8;9 ; a7;11 ; a8;10 ; a8;11 ; a9;10 ; a9;11 ; a10;11 Þ: However, to gain space, we will use the notation to indicate n consecutive parameters equal 0. Moreover, we call pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðaÞ ¼ 2 3 3a þ 2a2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi B ¼ 2 26 þ 15 3; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi C ¼ 2 26 15 3; pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi
4710 1500 10 þ 18370a þ 5840 10a þ 1395a2 þ 441 10a2 pffiffiffiffiffi DðaÞ ¼ 1380 þ 456 10 and EðaÞ ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi
4710 þ 1500 10 þ 18370a 5840 10a þ 1395a2 441 10a2 pffiffiffiffiffi : 1380 456 10
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Under these notations, our method gives the following families of algebras (where parameters belong to C, unless another thing is indicated):
10a 8a2 þ8a3 þAðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ 1 ; ; a; • l11 6 3
24þ14a 7AðaÞ 12 9b 14abþ16a2 b 8AðaÞb 8aAðaÞb ; 1; 0; ; 9 6þ12a pffiffiffi pffiffiffi b; 0; c; 0; d; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 7 ; 1 ; 2 2 2 5 ; 5 ; 5 ; 1; 38 ; 1; 0; 84 55a ; a; 0; b; 0; c; l2 , with c 2 C n f0g; 11 21 21 7 9 18 l3 5 ; 5 ; 5 ; 1; 38 ; 1; 0; 84 55a ; a; 0; b; ; c ; 7 11 21 21 9 18
10a 8a2 þ8a3 þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ ; ; a; l4 11 6 3
9 14aþ16a2 8AðaÞ 8aAðaÞ 1; ; ; 1; 0; b; 0; c; , with 6þ12a pffiffiffi pffiffiffi a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 ; 1 ; b 2 C2 [ f0g; 7 2 2 2
5 5 5 5 55 l ; ; 7 ; 1; ; ; 1; 0; a; 0; b; 0; 0; c , with a 2 C2 [ f0g; 11 21 21 18
2 3
10a 8a þ8a þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ ; ;a; l6 6 3 11 pffiffiffi pffiffiffi 1; ;1; 0;b; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 7 ; 1 ; b 2 C3 [ f0g; 2 2 2
5 ; 5 ; 5 ; 1; ; 1; 0; a; 0; 0; b , with a 2 C [ f0g; l7 3 7 11 21 21
10a 8a2 þ8a3 þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ ; ; a; l8 11 6 3 pffiffiffi pffiffiffi 1; ; 1; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 7 ; 1 ; 2 2 2
5 ; 5 ; 5 ; 1; ; 1; 0; 0; a , with a 2 C [ f0g; l9 2 7 11 21 21
10a 8a2 þ8a3 þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ l10 ; ; 11 6 3 pffiffiffi pffiffiffi a; 1; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 ; 1 ; 7 2 2 2
5 5 5 l11 11 21 ; 21 ; 7 ; 1; ; 1 ; 1;
• • •
• •
• •
• •
•
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5 5 5 12 ; ; 7 ; 1; • l11 ; 21 21
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; • l13 537 310 3 þ 105C 359C 11 2 2 2 4 3 2 3 pffiffiffi pffiffiffi
303þ150 3 36Cþ10 3C ; 0; 1; 9 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 144 75 3þ18C 5 3Cþ342a 195 3 aþ36Ca 22 3 Ca pffiffiffi ; a; 0; b; 0; c; ; 30 18 3
pffiffiffi pffiffiffi pffiffiffi 105B 359B 49 11B pffiffiffi pffiffiffi 9 • l14 11 537 þ 310 3 2 4 3 ; 2 þ 14 3 3B 2 3 ; 2 þ 3 3; 1; pffiffiffi pffiffiffi
303 150 3þ36Bþ10 3B ; 0; 1; 9 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 144þ75 3 18B 5 3Bþ342aþ195 3a 36Ba 22 3Ba ; a; 0; b; 0; c; pffiffiffi ; 30þ18 3
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; ; • l15 537 310 3 þ 105C 359C 2 11 4 3 2 2 3 2 pffiffiffi pffiffiffi 684 390 3þ72C 44 3C ; 1; 0; a; 0; b; pffiffiffi , with a 2 C2 [ f0g; 60 36 3
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 þ 14 3 3B 11B pffiffiffi ; 9 þ 3 3; 1; ; • l16 537 þ 310 3 105B 359B 2 2 2 11 4 3 2 3 pffiffiffi pffiffiffi 342þ195 3 36B 22 3 B pffiffiffi ; 1; 0; a; 0; b; , with a 2 C2 [ f0g; 6ð5þ3 3
pffiffiffi pffiffiffi pffiffiffi 17 pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; ; • l 537 310 3 þ 105C 359C 11 2 2 2 4 3 2 3 1; 0; a; , with a 2 C3 [ f0g;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 þ 14 3 3B 11B pffiffiffi ; 9 þ 3 3; 1; ; • l18 537 þ 310 3 105B 359B 11 2 4 3 2 2 3 2 1; 0; a; , with a 2 C3 [ f0g;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; ; • l19 537 310 3 þ 105C 359C 2 2 2 11 4 3 2 3 1; ;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 þ 14 3 3B 11B pffiffiffi ; 9 þ 3 3; 1; • l20 537 þ 310 3 105B 359B 2 2 2 11 4 3 2 3 ; 1; ;
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pffiffiffi pffiffiffi pffiffiffi 105C 359C 49 11C 9 21
pffiffiffi ; 14 3 þ 3C pffiffiffi ; 3 3; 1; • l11 537 310 3 þ ; 2 4 3 2 2 3 2
pffiffiffi pffiffiffi pffiffiffi 105B 359B 49 11B pffiffiffi ; pffiffiffi ; 9 þ 3 3; 1;
þ14 3
3B
• l22 537 þ 310 3
; 11 2 4 3 2 2 3 2
10a 8a2 þ8a3 5AðaÞ aAðaÞþ4a2 AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ ; ; • l23 11 6 3
24þ14aþ7AðaÞ 12 9b 14abþ16a2 bþ8AðaÞbþ8aAðaÞb ; 1; 0; ; b; 0; c; 0; 9 6þ12a pffiffiffiffiffi pffiffiffiffiffi d; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; 2 4 4
10a 8a2 þ8a3 AðaÞ aAðaÞþ4a2 AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ ; ; a; 1; l24 11 6 3
9 14aþ16a2 þ8AðaÞþ8aAðaÞ ;1; 0; b; 0; c; ; , with a 2 C n 1 ; 6þ12a 2 pffiffiffiffiffi pffiffiffiffiffi 3 33 ; 3þ 33 ; 1 ; b 2 C [ f0g; 2 4 4
2 3
10a 8a þ8a 5AðaÞ aAðaÞþ4a2 AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ l25 ; ; 11 6 3 pffiffiffiffiffi pffiffiffiffiffi a; 1; ; 1; 0; b; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; b 2 C3 [ f0g; 4 4 2
2 3 2 2
10a 8a þ8a
5AðaÞ aAðaÞþ4a AðaÞ
3 4aþ2a þAðaÞþaAðaÞ ; ; l26 11 6 3 pffiffiffiffiffi pffiffiffiffiffi a; 1; ; 1; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; 4 4 2
2 3 2
10a 8a þ8a 5AðaÞ aAðaÞþ4a AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ l27 ; ; 11 6 3 pffiffiffiffiffi pffiffiffiffiffi a; 1; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; 2 4 4
pffiffiffi pffiffiffi pffiffiffi pffiffiffi l28 4þ7i 2 ; 1 i 2 ; 1 ; 1; 31 7i 2; 1; 0; a; 4 8i 2 ; 0; b; 0; 0; c; 11 12 6 6 2 9 9 0; 0 ;
pffiffiffi pffiffiffi 4þ7i 2
1 i 2 29 1 l ; ; ; 1; ; 1; 0; 0; a; 0; 0; b; 0; 0 , with a 2 C2 [ f0g; 11 12 6 2
pffiffiffi pffiffiffi l30 4þ7i 2 ; 1 i 2 ; 1 ; 1; ; 1; 0; 0; a; 0; 0 , with a 2 C3 [ f0g; 11 12 6 2
pffiffiffi pffiffiffi l31 4þ7i 2 ; 1 i 2 ; 1 ; 1; ; 1; 0; 0 ; 12 6 11 2 a; 1;
•
•
•
•
•
• • •
622
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
pffiffiffi pffiffiffi 4þ7i 2
1 i 2 1 32 ; ; ; 1; • l ; 11 12 6 2
pffiffiffi pffiffiffi pffiffiffi pffiffiffi • l33 4 7i 2 ; 1þi 2 ; 1 ; 1; i 31i þ 7 2 ; 1; 0; a; 4þ8i 2 ; 0; b; 0; 0; 11 12 6 6 2 9 c; 0; 0 ;
pffiffiffi pffiffiffi • l34 4 7i 2 ; 1þi 2 ; 1 ; 1; ; 1; 0; 0; a; 0; 0; b; 0; 0 , with a 2 C2 [ f0g; 11 2 12 6
pffiffiffi pffiffiffi 4 7i 2
1þi 2 1 35 • l ; ; ; 1; ; 1; 0; 0; a; 0; 0 , with a 2 C3 [ f0g; 11 2 12 6
pffiffiffi pffiffiffi • l36 4 7i 2 ; 1þi 2 ; 1 ; 1; ; 1; 0; 0 ; 12 6 2 11
pffiffiffi pffiffiffi 4 7i 2
1þi 2 1 37 ; ; ; 1; • l ; 12 6 2 11
2 5þ3a b 4b 7c ; c; 0; d; ; e; 0 , with a 6¼ • l38 1; 1; 1; 1; a; b; 1; 2 11
•
•
• •
•
• •
3 5b ; b 2 C n f 2g; 2
9þ3a 7b ; b; 0; c; 0; d; l39 1; 1; 1; 1; a; 2; 1; , with b 2 C n f1g; 2 11 n o a 2Cn 7 ; 2
, with c 2 C n f0g; a 2 l40 1; 1; 1; 1; a; 2; 1; 16þ3a ; 1; 0; b; 0; c; 11 2 n o Cn 7 ; 2
n o
16þ3a ; 1; 0; b; ; c; 0 , with a 2 C n 7 ; l41
1; 1;
1; 1; a;
2; 1; 11 2 2
2 1 17a 8a 14b
3 5a l42 , with c 2 C ; b; 0; 0; c; 11 1; 1; 1; 1; 2 ; a; 1; 4 pffiffiffiffiffi pffiffiffiffiffi nf0g; a 2 C n 13 2 10 ; 13þ2 10 ; 2 ; 3 3
1 17a 8a2 14b l43 1; 1; 1; 1; 3 5a ; a; 1; ; b; ; c; 0 , with a 2 Cn 11 4 2 pffiffiffiffiffi pffiffiffiffiffi
13 2 10 ; 13þ2 10 ; 2 ; 3 3
7 ; 2; 1; 3 14a ; a; 0; 0; b; c; l44
1; 1;
1; 1; , with a 2 C n f1g; 11 2 4
, with a 6¼ b; l45 1; 1; 1; 1; 7 ; 2; 1; 11 ; 1; 0; 0; a; b; 2 4 11
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
• •
•
•
•
•
•
• • • • • •
623
7 11 46 l
1; 1; 1; 1; ; 2; 1; ; 1; 0; 0; a; a; 0; b; 0 ; 11 2 4
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l47 1; 1; 1; 1; 28 5 10 ; 13þ2 10 ; 1; 500þ157 10 63a ; a; 0; b; c; 11 3 3 18 , with c 2 C n f0g;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l48 1; 1; 1; 1; 28þ5 10 ; 13 2 10 ; 1; 500 157 10 63a ; a; 0; b; c; 11 3 3 18 , with c 2 C n f0g;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l49 1; 1; 1; 1; 28 5 10 ; 13þ2 10 ; 1; 500þ157 10 63a ; a; 0; b; 3 3 18 11 , with b 2 C n fDðaÞ; EðaÞg;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l50 1; 1; 1; 1; 28þ5 10 ; 13 2 10 ; 1; 500 157 10 63a ; a; 0; b; 3 3 18 11 , with b 2 C n fDðaÞ; EðaÞg;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 28 5 10
13þ2 10
500þ157 10 63a ; a; 0; EðaÞ; 51 ; ; 1; l
1; 1; 1; 1; 3 3 18 11 ; b; 0 ;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l52 1; 1; 1; 1; 28þ5 10 ; 13 2 10 ; 1; 500 157 10 63a ; a; 0; DðaÞ; 11 3 3 18 ; b; 0 ;
n o
4 7b 53 ; b; 0; c; ; d; 0 , with a 2 C n 5 ; l
1; 1; 1; 1; a; 1; 0; 11 2 2
l54 1; 1; 1; 1; 5 ; 1; 0; 4 7a ; a; 0; 0; b; , with b 2 C n f0g; 11 2 2
l55 1; 1; 1; 1; 5 ; 1; 0; 4 7a ; a; ; b; 0 ; 2 2 11
7a 56 ; a; 0; 0; b; c; l
1; 1; 1; 1; 1; 0; 0; , with a 2 C n f0g; 11 2
, with a 6¼ b; l57 1; 1; 1; 1; 1; ; a; b; 11
l58 1; 1; 1; 1; 1; ; a; a; 0; b; 0 ; 11
624
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
7 59 • l
1; 1; 1; 1; ; ; 1; 0; a; b; c; , with b 2 C2 ; 11 2
• l60 1; 1; 1; 1; ; 7 ; 1; 0; a; 0; b; , with a 2 C2 [ f0g; 11 2 • l61 ð 1; 1; 1; 1; ; a; 1; b; Þ, with b 2 ðC3 [ f0gÞ n f1g; 11 62 • l ð 1; 1; 1; 1; ; a; 1; 1; 0; b; 0Þ; 11 • l63 ð 1; 1; 1; 1; ; 1; 0; a; Þ, with a 2 C3 ; 11 64 • l ð 1; 1; 1; 1; ; 1; ; a; 0Þ, with a 2 C3 [ f0g; 11 • l65 ð 1; 1; 1; 1; ; 1; Þ; 11 66 • l ð 1; 1; 1; 1; ; 1; 0Þ; 11 • l67 ð 1; 1; 1; 1; Þ; 11 68 9 ; 3 ; 1; 0; 3 ; 1; 0; 1 35a ; 0; a; b; 0; c; • l , with c 2 C n f0g; 11 8 2 2 16
• l69 9 ; 3 ; 1; 0; 3 ; 1; 0; 1 35a ; 0; a; b; ; c ; 11 8 2 2 16
• l70 9 ; 3 ; 1; ; 35 ; 0; 1; a; 0; b; 0; 0; c , with a 2 C2 [ f0g; 11 8 2 16
71 9 3 • l ; ; 1; ; 1; 0; a; 0; 0; b , with a 2 C3 [ f0g; 11 8 2
• l72 9 ; 3 ; 1; ; 1; 0; 0; a , with a 2 C4 [ f0g; 11 8 2
73 9 3 • l ; ; 1; ; 1 ; 11 8 2
• l74 9 ; 3 ; 1; ; 11 8 2 • l75 ð1; 11 • l76 ð1; 11 77 • l ð1; 11 • l78 ð1; 11 79 • l ð1; 11 • l80 ð1; 11 81 • l ð1; 11
; 1; 0; a; b; 2; 0; c; 0; d; 0; 0Þ; ; a; 1; 0; b; c; 0; d; 0; 0Þ,
with c 2 C2 ;
; a; 1; 0; b; 0; 0; c; 0; 0Þ,
with b 2 C2 [ f0g;
; 1; 0; 0; a; b; c; d; 0; 0Þ,
with b 2 C2 ;
; 1; 0; 0; a; 0; b; c; d; 0Þ,
with c 2 C n f0g; a 2 C2 ;
; 1; 0; 0; a; 0; b; 0; c; dÞ,
with a 2 C2 ;
; 1;
; a; b; c; 0Þ,
with b 2 C n f0g; c 2 C2 [ f0g;
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
• l82 ð1; 11 • l83 ð1; 11 • l84 ð1; 11 • l85 ð1; 11 • l86 ð1; 11 • l87 ð1; 11 88 • l ð1; 11 89 • l ð1; 11 • l90 ð1; 11 91 • l ð1; 11 • l92 ð1; 11 • l93 ð1; 11 94 • l ð1; 11 95 • l 11
• l96 11
97 • l 11
• l98 11
99 • l 11 • l100 ð 11 • l101 11
102 • l 11
• l103 11
104 • l 11
; 1;
; a; 0; b; cÞ,
Þ,
with b 2 C2 [ f0g; with c 2 C3 ;
; a; 1; b; c; 0; 0Þ, ; a; 1; b;
with b 2 C3 [ f0g;
; 1; 0; a; b; c; 0Þ,
with b 2 C3 ;
; 1; 0; a; 0; b; cÞ,
with a 2 C3 ;
; 1;
; a; bÞ,
625
with a 2 C3 [ f0g; with b 2 C4 ;
; a; 1; b; 0Þ, ; a; 1; 0; 0Þ; ; 1; 0; a; bÞ,
with a 2 C4 ;
; 1; 0; 0; aÞ,
with a 2 C2 [ f0g;
; 1; aÞ,
with a 2 C5 [ f0g;
; 1Þ; Þ; 2 ; 5þaþ4a ; a; 1; b; c; 1; 0; 0; d; 3 ; 2 ; 1; 1; a; b; 1; 0; 0; c; ; 3 ; 2 ; 1; 1; a; b; 1; ; c ; 3 ; 5 ; 3 ; 1; a; b; 1; 0; 0; c; 0; d; 0 ; 6 2 ; 21; 4; 1; a; b; 1; c; ;
; 3; 2; 1; a; b; 1; c; 0; d; ; 5þaþ4a ; a; 1; b; 1; 3 2
; 2 ; 1; 1; a; 1; 3 ; 2 ; 1; 1; a; 1; 3 ; 5 ; 3 ; a; 1; 6 2
; b; ;b ;
Þ; ; c; ,
; b; 0; c; 0 ;
,
,
n o with a 2 C n 1; 3 ; 2; 4 ; 2
n o with a 2 C n 1; 3 ; 2; 4 ; 2
with b 2 C n f0g;
626
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
n o • l105 ð ; 21; 4; 1; a; 1; 0; b; Þ, with a 2 C n 41 ; 11 4
; 21; 4; 1; 41 ; 1; 0; a; 0; b; ; • l106 11 4 • l107 ð 11
• l108 11
109 • l 11
• l110 11
• l111 11
; 3; 2; 1; a; 1; 0; b; 0; c;
• l112 11 ð • l113 ð 11
• l114 11
• l115 11
116 • l 11
• l117 11
• l118 11
119 • l 11
• l120 11
; 21; 4; 1; 1; 0; 0; a;
• l121 11 ð • l122 11 ð • l123 ð 11 • l124 11 ð • l125 ð 11
; 21; 4; 1;
; 1; 0; a;
; 21; 4; 1;
; 1;
; 21; 4; 1;
Þ;
; 3; 2; 1;
; 1; 0; a;
; 3; 2; 1;
; 1;
; 5þaþ4a ; a; 1; 1; 3 2
; 2 ; 1; 1; 1; 3 ; 2 ; 1; 1; 1; 3 ; 5 ; 3 ; 1; 1; 6 2
; a; ;a ;
Þ; n o ; b; , with a 2 C n 1; 3 ; 2; 4 ; 2 , with a 2 C n f0g;
; a; 0; b; 0 ; Þ;
; 3; 2; 1; 1; 0; 0; a; 0; b;
Þ;
n o ; 1; , with a 2 C n 3 ; 2; 4 ; 2 ; 5 ; 3 ; 1; ; 1; 0; a; 0 , with a 2 C3 [ f0g; 6 2 n o 2
5þaþ4a ; , with a 2 C n 1; 3 ; 2; 4 ; ; a; 1; 2 3 ; 2 ; 1; 1; ; 1 ; 3 ; 2 ; 1; 1; ; 3 5 3 ; ; ; 1; ; 1; 0 ; 6 2 ; 5 ; 3 ; 1; ; 6 2 ; 5þaþ4a ; a; 1; 3 2
Þ,
with a 2 C2 [ f0g;
Þ,
with a 2 C2 [ f0g;
Þ;
Þ;
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
627
• l126 ð ; 3; 2; 1; Þ; 11 n o • l127 ð ; 1; 0; 0; a; b; 1; 0; c; d; Þ, with b 2 C n 1 ; 1 ; 11 2
; 1; 0; 0; a; 1 ; 1; 0; b; c; 0; 0; d ; • l128 11 2 • l129 ð 11 • l130 ð 11 131 • l ð 11 • l132 ð 11 133 • l ð 11 • l134 ð 11 • l135 ð 11 136 • l ð 11 • l137 ð 11 • l138 ð 11 • l139 ð 11 140 • l ð 11 • l141 11 ð • l142 11 ð
143 • l 11
; 1; 0; 0; a; 1; 1; 0; b; c; 0; d; 0Þ; ; 1; 0; 0; a; 1; 0; 0; b; 0; c; 0; 0Þ; ; 1; 0; 0; 1;
; a; b; c; 0; 0Þ,
; 1; 0; 0; 1;
; a; b; c; 0Þ,
; 1; 0; 0; 1;
; a; 0; b; cÞ;
with a 2 C n f0g; with b 2 C n f0g;
with b 2 C2 ;
; 1;
; 1; a; b; 0; 0Þ,
; 1;
; 1; a;
; 1;
; a; 1; b; 0Þ,
with b 2 C3 [ f0g;
; 1;
; 1; 0; a; bÞ,
with a 2 C3 ;
; 1;
; 1; 0; 0; aÞ,
with a 2 C3 [ f0g;
; 1;
; 1; aÞ,
; 1;
; 1Þ;
; 1;
Þ;
Þ,
; a; b; 1; c; 1; d;
with a 2 C2 [ f0g;
with a 2 C4 [ f0g;
Þ,
; a; 1 ; 1; b; 1; c; 2
n o with b 2 C n 1 ; 1 ; 2 , with 4 þ 16a 16a2 þ 5b þ 18ab þ 2c
þ 4ac 6¼ 0;
1 144 ; a; ; 1; b; 1; c; 0; 0; d , with 4 þ 16a 16a2 þ 5b þ 18ab þ 2c • l11 2 þ 4ac ¼ 0; • l145 ð ; a; 1; 1; b; 1; c; Þ, with a 2 C n f1g; 11 146 • l ð ; 1; 1; 1; a; 1; b; 0; c; 0Þ; 11
628
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
n o • l147 ð ; a; b; 1; 1; 0; c; Þ, with b 2 C n 1 ; 1 ; 11 2
• l148 ; a; 1 ; 1; 1; 0; b; ; with 5 þ 9a þ b þ 2ab 6¼ 0; 11 2 2
• l149 ; a; 1 ; 1; 1; 0; b; 0; 0; c , with 5 þ 9a þ b þ 2ab ¼ 0; 11 2 2 • l150 ð 11 • l151 ð 11 • l152 ð 11
• l153 11
154 • l 11
; a; 1; 1; 1; 0; b;
• l155 ð 11 156 • l ð 11 • l157 ð 11
158 • l 11
• l159 11
; a; 1; 1; 0; 0; 1;
• l160 ð 11 • l161 ð 11 • l162 ð 11 • l163 ð 11 • l164 ð 11 165 • l ð 11 • l166 ð 11 • l167 ð 11 168 • l ð 11 • l169 ð 11
; a; 1; 1;
Þ,
; 1; 1; 1;
; 1; 0Þ;
Þ,
with a 2 C n f1g;
; 1; 1; 1; 1; 0; a; 0; b; 0Þ;
n o with b 2 C n 1 ; 1 ; 2 n o ; a; 1 ; 1; 0; 0; 1; , with a 2 C n 1 ; 2 2 ; 1 ; 1 ; 1; 0; 0; 1; 0; 0; a ; 2 2
; a; b; 1; 0; 0; 1;
Þ,
Þ,
with a 2 C n f1g;
; 1; 1; 1; 0; 0; 1; 0; a; 0Þ,
with a 2 C2 [ f0g; n o ; a; b; 1; Þ, with b 2 C n 1 ; 1 ; 2 ; a; 1 ; 1; ; 1 ; 2 ; a; 1 ; 1; ; 2 with a 2 C;
; a; 1; 0; b; 1; 0; c; 0; 0Þ; ; a; 1; 0; 1; 0; 0; b; 0; 0Þ; ; a; 1;
; 1; 0; 0Þ;
; a; 1;
Þ;
; 1; 0; 0; a; 1; b; c; 0; 0Þ; ; 1; 0; 0; 1; 0; a; b; 0; 0Þ; ; 1;
; a; 1; 0; 0Þ;
; 1;
; 1; 0; 0; aÞ;
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
• l170 ð ; 1; 11
; 1Þ;
• l171 ð ; 1; 11
Þ;
629
• l172 ð ; a; 1; b; 1; 0; 0Þ; 11 • l173 ð ; a; 1; 1; 11 • l174 ð ; a; 1; 11
Þ; Þ;
• l175 ð ; 1; 0; a; 1; b; 0Þ; 11 • l176 ð ; 1; 0; 1; 0; a; bÞ; 11 • l177 ð ; 1; 11
; 1; aÞ,
• l178 ð ; 1; 11 • l179 ð ; 1; 11
; 1Þ;
with a 2 C2 [ f0g;
Þ;
• l180 ð ; a; 1; 1; 0Þ; 11 • l181 ð ; a; 1; 0; 0Þ; 11 • l182 ð ; 1; 0; 1; aÞ; 11 • l183 ð ; 1; 0; 0; 1Þ; 11 • l184 ð ; 1; Þ; 11 • l185 ð ; 1; 1Þ; 11 • l186 ð ; 1; 0Þ; 11 • l187 ð ; 1Þ; 11 188 • l ð Þ: 11 So, in this way it is proved the following: Theorem 4.2. Let g be a filiform Lie algebra of dimension 11. Then, g is isomorphic to one and only one of the algebras belonging to one of the families lh11 ; 1 6 h 6 188, previously mentioned.
630
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
References [1] J.M. Ancochea, M. Goze, Classification des algebres de Lie filiformes de dimension 8, Archiv. Math. 50 (1988) 511–525. [2] J.M. Ancochea, M. Goze, Classification des algebres de Lie nilpotentes complexes de dimension 7, Arch. Math. 52 (1989) 175–185. [3] J.M. Ancochea, M. Goze, On the varietes of nilpotent Lie algebras of dimension 7 and 8, J. Pure Appl. Algebra 77 (1992) 431–440. ~ ez, The equations of the sets of filiform Lie [4] J.C. Benjumea, F.J. Castro, M.C. Marquez, J. N un lgebra Computacional y algebras of dimension 13 and 14, Actas del 2 Encuentro de A Aplicaciones (1996) 67–75. [5] N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958) 45–92. ~ez, Classification of complex filiform Lie algebras of dimension [6] L. Boza, E.M. Fedriani, J. N un lgebra, Computaci 12, Prepublicaci on 39, Dpto. A on, Geometrıa y Topologıa. Universidad de Sevilla, 1997. ~ez Valdes, Classification of complex filiform Lie [7] L. Boza Prieto, F.J. Echarte Reula, J. N un algebras of dimension 10, Algebras, Groups and Geometries 11 (3) (1994) 253–276. ~ez Valdes, A new method for classifying complex [8] L. Boza Prieto, E.M. Fedriani Martel, J. N un filiform Lie algebras, Appl. Math. Comput. 121 (2–3) (2001) 169–175. ~ ez, The equations of the sets of filiform Lie algebras of dimension 11 and [9] F.J. Castro, J. N un 12, in: Personal Communication, Universidad de Sevilla, Espa~ na, 1996. [10] F.J. Echarte, J.R. G omez, Classification of complex filiform nilpotent Lie algebras of dimension 9, Rendiconti Seminario Facolt Scienze Universit Cagliari 61 (1) (1991) 21–29. [11] E.M. Fedriani, Clasificaci on de las algebras de Lie filiformes complejas de dimensi on 12, in: Tesis de Licenciatura, Universidad de Sevilla, Espa~ na, 1997. [12] J.R. G omez, A. Jimenez, Y. Khakimdjanov, Low-dimensional filiform Lie algebras, J. Pure Appl. Algebra 130 (1998) 133–158. ~ ez, An algorithm to obtain laws families of filiform Lie [13] J.R. G omez, A. Jimenez, J. N un algebras, Linear Algebra Appl. 279 (1998) 1–12. [14] C. Seeley, Some nilpotent Lie algebras of even dimensions, Bull. Aust. Math. Soc. 45 (1992) 71–77. [15] A. Shalev, E.I. Zelmanov, Narrow Lie algebras: a coclass theory and a characterization of the Witt algebra, J. Algebra 189 (1997) 294–331. [16] M. Vergne, Cohomologie des algebres de Lie nilpotentes. Application a l’etude de la variete des algebres de Lie nilpotentes, Bull. Soc. Math. France 98 (1970) 81–116.
Complex filiform Lie algebras of dimension 11 ~ez Luis Boza a, Eugenio M. Fedriani b, Juan N un
c,*
a
Departamento de Matem atica Aplicada I. Escuela T ecnica Superior de Arquitectura, Universidad deSevilla, Avda. Reina Mercedes 2, 41012 Sevilla, Spain b Departamento de Economıa y Empresa, Universidad Pablo de Olavide, Carretera de Utrera, Km. 1, 41013 Sevilla, Spain c Facultad de Matematica, Departamento de Geometrıa y Topologıa, Universidad de Sevilla, Apartado Correos 1160, E-41080 Sevilla, Spain
Abstract In this paper we give the explicit classification of complex filiform Lie algebras of dimension 11. To do this, we use a method previously obtained by us in an earlier paper, which is based on the concept of isomorphism between Lie algebras. At present, this explicit classification is not known, although G omez, Jimenez and Khakimdjanov gave a list of these algebras in a non-explicit way, but in terms of cocycles. We find that there exist 188 families of complex filiform Lie algebras of dimension 11. 2002 Elsevier Science Inc. All rights reserved.
1. Introduction At present, the research on Lie theory is very extended. It is due to several reasons: the main one is that this theory is being actually applied not only in different branches of mathematics, such as topology, differential equations, differential or algebraic geometry, or arithmetic, for instance, but in many other sciences, such as engineering or physics, mainly. On the sake of an example, the exceptional group of Killing plays an important role in the recent superstring theory, constituting an important approximation to the relation
*
Corresponding author. E-mail addresses: [email protected] (L. Boza), [email protected] (E.M. Fedriani), [email protected] (J. Nu´n˜ez). 0096-3003/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved. doi:10.1016/S0096-3003(02)00280-1
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between quantic mechanic and relativity theory. Moreover, one of the Lie classic group, the simplectic group of n-dimensional space is being actually very used in quantic mechanic and in optics. Apart from that, the Lie theory is also important by itself, from a strictly mathematical point of view. Indeed, from its origins by S. Lie, distinguished mathematicians, like Killing, Cartan, Ado, Levi and others, have studied this subject. However, sometimes, the research worker on Lie theory needs lots of examples of finite Lie algebras of high dimensions to check if a particular property which is verified for those of low dimensions is also satisfied on them. They are also needed, for instance, to verify if a previous conjecture, observed in the examples already known, is really true. But, at present, it is difficult to have those examples of Lie algebras of dimension, say, non-excessively low. Indeed, if we take into no consideration simple and semisimple Lie algebras, whose classification was obtained by Killing, Cartan and others, in the last decade of XIX century, only the classifications of solvable, nilpotent and filiform complex Lie algebras of dimension n, with n 6 5, n 6 7 and n 6 12, respectively, are actually known (see [2,8,11], for instance). Moreover, this problem worsens if we note that not only the classification problem of Lie algebras is still unsolved, but, in fact, it seems to be accepted by experts (like Shalev and Zelmanov, see [15]) that one will not totally classify Lie algebras of finite dimension. Indeed, it is reasonable for research workers in this theory to impose further conditions to these algebras to obtain, at least, partial results. In this way, the main goal of this paper is to give the explicit classification of complex filiform Lie algebras of dimension 11. To do this, we use a method previously obtained by us in an earlier paper, which is based on the concept of isomorphism between Lie algebras. At present, this explicit classification is not known, even if it is convenient to note that G omez, Jimenez and Khakimdjanov (see [12]) already gave a list of these algebras in a non-explicit way, but in terms of cocycles. However, this way of giving the list does not result operative to be handled. We would like now to explain why we are dealing with complex filiform Lie algebras. These algebras were introduced by M. Vergne in the late 60s of the past century (see [16]). However, before that, Blackburn (see [5]) studied the analogous class of finite p-groups and used the term maximal class to call them, which is also now used for Lie algebras. In fact, both terms filiform and maximal class are synonyms. Vergne showed that, within the variety of nilpotent Lie multiplications on a fixed vector space, non-filiforms can be relegated to small-dimensional components; thus, from some intuitive point of view, it is possible to consider that quite a lot nilpotent Lie algebras are filiform, in spite of this last subset not being dense in the space of nilpotent Lie algebras. Apart from that, complex
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filiform Lie algebras are the most structured subset of nilpotent Lie algebras, with respect to an adapted basis (this concept can be checked in the next section). In this sense, we can study and classify these algebras easier than the set of nilpotent Lie algebras. So, from now on, we will only deal with the evolution of the problem of classifying complex filiform Lie algebras, which are the aim of our study. In earlier papers, several authors have deeply studied these algebras and have obtained quite a lot results about them. So, in 1988, Goze and Ancochea, by using an invariant of these algebras, named the characteristic sequence, which corresponds to the maximal dimensions of Jordan blocks of a certain nilpotent matrix, obtained the classification of complex nilpotent Lie algebras of dimension 7 [2]. However, the use of that method to higher dimensions did not result very suitable and great difficulties appeared. So, both authors treated to classify first filiform Lie algebras, due to the reasons above mentioned. So, by the same way, they obtained the classification of complex filiform Lie algebras of dimension 8 [1]. It is convenient to say that the importance of these authors’ work is not only obtaining these classifications, but, above all, devising techniques which can be applied in any dimension. So, Echarte and G omez [10] classified, by using this method, filiform Lie algebras of dimension 9. Certainly, in that classification some two-parameter families of filiform Lie algebras appear for the first time. However, the classification given in [1] were uncomplete and some errors appeared. It was later corrected by the same authors [3] and independently by Seeley [14]. ~ez classified complex filiform Lie algebras of Later, Boza, Echarte and N un dimension 10 by the introduction of another invariant for these algebras, which they call the pair (i,h) [7]. One difference from the case of the dimension 9 is that some three-parameter families of filiform Lie algebras appeared in this last classification. In 1998, G omez, Jimenez and Khakimdjanov, by the method of the elementary changes of basis gave a correct classification of complex filiform Lie algebras of dimension n with n 6 11 [12]. To do this, they use a result by ~ez, which presented a polynomial time recursive G omez, Jimenez and N un algorithm which generates families of filiform Lie algebras of dimension n and by using it, they obtained a parametrization of the affine algebraic set of filiform Lie algebras of dimension 11 [13]. However, we had previously prepublished in [6] the classification of filiform Lie algebras of dimension 12. In that classification, four, five or even sixparameter families of filiform Lie algebras can be already found. The structure of this paper is the following: In Section 2 some definitions and notations related to complex filiform Lie algebras are given. In Section 3 we explain, in a schematic way, the application of our method for classifying complex filiform Lie algebras to the particular case of dimension 11, in which starting from the structure theorem of these algebras we obtain a representant
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of each class of isomorphism. Finally, in Section 4, we give the complete classification of these algebras. The main result of this paper is the Theorem 4.2.
2. Definitions and notations In this paper, all the Lie algebras which appear will be considered over the complex field C. It is convenient to note that by classifying a set, we mean to find a certain property of it which let us define an equivalence relation among its elements. The equivalence relation which we will use to classify complex filiform Lie algebras will be being isomorphic. So, we will explicitly compute a representant of each class of complex filiform Lie algebras in dimension 11. To do this, most of the calculations needed were made by using the Mathematica package. 2.1. Homomorphisms between Lie algebras Let g and g0 be two Lie algebras. A map U : g7!g0 is said to be an homomorphism between Lie algebras if U is a linear application such that U : ½X ; Y 7!½UðX Þ; UðY Þ; 8X ; Y 2 g. In the case of being U a bijection, it is called isomorphism. As this concept of isomorphism between filiform Lie algebras is the main idea of our method, we next give an example: let g be a complex Lie algebra of dimension 2, defined by the bracket ½e1 ; e2 ¼ e1 , with respect to an adapted basis fe1 ; e2 g and let g0 be another Lie algebra of the same dimension, defined by the bracket ½e01 ; e02 ¼ e02 , with respect to an adapted basis fe01 ; e02 g. The mapping U : g0 7!g defined by Uðe01 Þ ¼ e2 , Uðe02 Þ ¼ e1 is an isomorphism, since Uð½e01 ; e02 Þ ¼ Uðe02 Þ ¼ e1 and ½Uðe01 Þ; Uðe02 Þ ¼ ½ e2 ; e1 ¼ ½e1 ; e2 ¼ e1 . 2.2. Filiform Lie algebras Let g ¼ ðCn ; lÞ be a Lie algebra of dimension n, with l the associated law. We consider the lower central series of g defined by C 1 g ¼ g, C i g ¼ lðg; C i 1 gÞ. The Lie algebra g is filiform if dimC C i g ¼ n i for 2 6 i 6 n. These algebras were defined by M. Vergne (see [16]). If x 2 g we denote by adðxÞ the adjoint mapping associated to x (i.e. the map y7!lðx; yÞ). Let g be a filiform Lie algebra of dimension n. Then there exists a basis B ¼ fe1 ; . . . ; en g of g such that e1 2 g n C 2 g, the matrix of adðe1 Þ with respect to B has a Jordan block of order n 1 and C i g is the vector space generated by fe2 ; . . . ; en ði 1Þ g with 2 6 i 6 n 1. Such a basis is called an adapted basis. Sometimes, on the sake of simplicity, we will use ½x; y instead of lðx; yÞ for the Lie bracket in a Lie algebra.
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3. A method to classify the filiform Lie algebras of dimension 11 In this Section we apply a method (previously obtained by us [8] for filiform Lie algebras of any dimension, in general) to the particular case of dimension 11, to give the explicit classification of these algebras. First, we remind that to classify filiform Lie algebras of any dimension with this method, it is necessary to use the structure theorem of those algebras. These theorems are already known for dimension n, with n 6 14 ([4,13], for instance). They consist on giving explicitly the rest of the non-null brackets. These brackets are expressed as function of some structure constants ai;j . These constants, which define the law of the algebra, are related themselves by polynomial relations coming from the Jacobi identities. Let g be a filiform Lie algebra of dimension n with an adapted basis fe1 ; . . . ; en g. So, ½e1 ; eh ¼ eh 1 , 3 6 h 6 n. We denote the subindexes of the structure constants according to the first bracket in which they appear. For instance, in dimension 11, the first bracket we consider is ½e4 ; e11 ¼ a4;11 e2 . In order to decide which is the first bracket, we have set a certain order relation in the set of the subindexes pairs ði; jÞ with i < j. This order is defined by: 8
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Let consider the isomorphism U : g0 7!g, this is, the bases change given by 11 11 X X v1 ¼ si ui ; v11 ¼ r i ui i¼1
i¼1
and by the property of filiformity of the algebra. Step 2: By straightforward computations in the previous step, we can deduce the following general expression: 7
v2 ¼ s1 ðs1 r11 s11 r1 Þðs1 s11 a4;10 Þ u2 ; where a4;10 is the coefficient of u9 in the bracket ½u10 ; u11 . In fact, it can be easily proved that a4;10 ¼ 0 due to 11 is odd. Then, as v2 belongs to a basis, necessarily v2 6¼ 0 and thus, we have s1 6¼ 0 and s1 r11 s11 r1 6¼ 0. Step 3: Now, to make next computations easier, by using again the property of filiformity, we have: 0 ¼ ½v3 ; v11 ¼ r1 ðs1 r11 s11 r1 Þs71 u2 : Then, by taking the previous step, we have r1 ¼ 0. P11 into consideration P11 So, v1 ¼ i¼1 si ui , v11 ¼ i¼2 ri ui . We also deduce, in this step, the following conditions between the elements of the law of the algebra: s1 6¼ 0 and r11 6¼ 0. Step 4: As the bases change U is an isomorphism between Lie algebras, then ½Uðvk Þ; Uðvl Þ Uð½vk ; vl Þ ¼ 0, 8k; l ¼ 1; . . . ; 11. We obtain some vector equations by using this last expressions. Step 5: It is easy to prove in each dimension that we do not lose any piece of information although we operate with second components only. So, from now on, we will take into consideration these components only. Step 6: In this step, we already have new equations from the previous vector equations and the equations consisting on Pk ¼ 0 (these last ones are the restrictions coming from the Jacobi identities in the corresponding structure theorem). So, we have a set of four equations involving the coefficients which appear in the laws of each algebra and the coefficients ri and sj from the isomorphisms. We wish that these equations give specific values (that is, constants) for the coefficients of the law or of the isomorphisms or, in other case, relations between them. In this way, for instance, if we find an expression like the following: a1 ¼ s1 a01 , we would distinguish two non-isomorphic cases: a1 ¼ 0 ¼ a01 and a1 6¼ 0 6¼ a01 (where a1 represents any coefficient ai;j of the law). In this method, we use some kind of particular sets to describe algebras having the same starting point in our method, in a non-redundant way and by using a unique family. For dimension 11, we have considered the following kind of set: 2p Cm ¼ reiu : r > 0; u 2 0; : m
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Now, we will explain something more about the set Cm , which has been already used in earlier classifications to solve some situations as following: As we described in the general explanation, it can be convenient to distinguish two cases depending on a2 ¼ 0 or not. We are interested in the second case. Then, let suppose that we find an expression like a2 sm1 a02 ¼ 0. Then, if 1=m we consider an isomorphism with s1 ¼ 1=ða02 Þ , any algebra of the family which we are considering is isomorphic to one with a2 ¼ 1. So, from then on, we must suppose a2 ¼ 1 ¼ a02 and we must also suppose that s1 is a mth root of the unit. When continuing the study of this same case, if it appears an equation of the kind a3 s1 a03 ¼ 0, we would have to distinguish two new cases: • a3 ¼ 0 ¼ a03 and s1 is a mth root of the unit. • a3 ¼ s1 a03 6¼ 0. In this case, the sector Cm is sufficient to represent in a unique way all possible values of a3 of the algebras of that family, as a convenient value of s1 relates both a3 and a03 by one isomorphism. So, from then on, we can suppose a3 ¼ a03 2 Cm and s1 ¼ 1. 4. The classification of filiform Lie algebras of dimension 11 As we are considering the particular case of the dimension 11, we start from ~ez in [9] and later the following result, previously obtained by Castro and N un ~ez (although by using a different noconfirmed by G omez, Jimenez and N un tation) in [13]: Theorem 4.1 (Structure theorem of filiform Lie algebras of dimension 11). The set of filiform Lie algebras laws over C11 can be parametrized (up to isomorphism) by the points of an affine algebraic set V C11 of Krull dimension 12. Furthermore, if g ¼ ðC11 ; lÞ is a filiform Lie algebra, then there exists an adapted basis b fe1 ; . . . ; e11 g of g such that: • • • • • • • • • • •
½e1 ; eh ¼ eh 1 ð3 6 h 6 11Þ ½e4 ; e11 ¼ a4;11 e2 ½e5 ; e10 ¼ a5;10 e2 ½e6 ; e9 ¼ a6;9 e2 ½e7 ; e8 ¼ a7;8 e2 ½e5 ; e11 ¼ a5;11 e2 þ ða4;11 þ a5;10 Þe3 ½e6 ; e10 ¼ a6;10 e2 þ ða5;10 þ a6;9 Þe3 ½e7 ; e9 ¼ a7;9 e2 þ ða6;9 þ a7;8 Þe3 ½e6 ; e11 ¼ a6;11 e2 þ ða5;11 þ a6;10 Þe3 þ ða4;11 þ 2a5;10 þ a6;9 Þe4 ½e7 ; e10 ¼ a7;10 e2 þ ða6;10 þ a7;9 Þe3 þ ða5;10 þ 2a6;9 þ a7;8 Þe4 ½e8 ; e9 ¼ a8;9 e2 þ a7;9 e3 þ ða6;9 þ a7;8 Þe4
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• ½e7 ; e11 ¼ a7;11 e2 þ ða6;11 þ a7;10 Þe3 þ ða5;11 þ 2a6;10 þ a7;9 Þe4 þ ða4;11 þ 3a5;10 þ 3a6;9 þ a7;8 Þe5 • ½e8 ; e10 ¼ a8;10 e2 þ ða7;10 þ a8;9 Þe3 þ ða6;10 þ 2a7;9 Þe4 þ ða5;10 þ 3a6;9 þ 2a7;8 Þe5 • ½e8 ; e11 ¼ a8;11 e2 þ ða7;11 þ a8;10 Þe3 þ ða6;11 þ 2a7;10 þ a8;9 Þe4 þ ða5;11 þ 3a6;10 þ 3a7;9 Þe5 þ ða4;11 þ 4a5;10 þ 6a6;9 þ 3a7;8 Þe6 • ½e9 ; e10 ¼ a9;10 e2 þ a8;10 e3 þ ða7;10 þ a8;9 Þe4 þ ða6;10 þ 2a7;9 Þe5 þ ða5;10 þ 3a6;9 þ 2a7;8 Þe6 • ½e9 ; e11 ¼ a9;11 e2 þ ða8;11 þ a9;10 Þe3 þ ða7;11 þ 2a8;10 Þe4 þ ða6;11 þ 3a7;10 þ 2a8;9 Þe5 þ ða5;11 þ 4a6;10 þ 5a7;9 Þe6 þ ða4;11 þ 5a5;10 þ 9a6;9 þ 5a7;8 Þe7 • ½e10 ; e11 ¼ a10;11 e2 þ a9;11 e3 þ ða8;11 þ a9;10 Þe4 þ ða7;11 þ 2a8;10 Þe5 þ ða6;11 þ 3a7;10 þ 2a8;9 Þe6 þ ða5;11 þ 4a6;10 þ 5a7;9 Þe7 þ ða4;11 þ 5a5;10 þ 9a6;9 þ 5a7;8 Þe8 with the coefficients a4;11 ; a5;10 ; . . . ; a10;11 2 C verifying the following four equations: Pi ¼ 0, with 1 6 i 6 4 where • P1 ¼ 3a25;10 þ 2a4;11 a6;9 3a5;10 a6;9 6a5;10 a7;8 9a6;9 a7;8 5a27;8 • P2 ¼ 4a5;10 a6;9 6a26;9 þ 2a4;11 a7;8 þ 5a5;10 a7;8 þ 6a6;9 a7;8 þ 5a27;8 • P3 ¼ 3a5;11 a6;9 7a5;10 a6;10 6a6;9 a6;10 þ 3a5;11 a7;8 þ a6;10 a7;8 þ 2a4;11 a7;9 3a5;10 a7;9 þ 5a7;8 a7;9 • P4 ¼ 4a26;10 þ 4a6;9 a6;11 þ 2a6;11 a7;8 þ 3a5;11 a7;9 a6;10 a7;9 þ 5a27;9 8a5;10 a7;10
6a6;9 a7;10 5a7;8 a7;10 þ 2a4;11 a8;9 þ 3a5;10 a8;9 þ 11a6;9 a8;9 þ 5a7;8 a8;9 . From now on, the law of this algebra will be denoted by lða4;11 ; a5;10 ; a6;9 ; a7;8 ; a5;11 ; a6;10 ; a7;9 ; a6;11 ; a7;10 ; a8;9 ; a7;11 ; a8;10 ; a8;11 ; a9;10 ; a9;11 ; a10;11 Þ: However, to gain space, we will use the notation to indicate n consecutive parameters equal 0. Moreover, we call pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðaÞ ¼ 2 3 3a þ 2a2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi B ¼ 2 26 þ 15 3; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi C ¼ 2 26 15 3; pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi
4710 1500 10 þ 18370a þ 5840 10a þ 1395a2 þ 441 10a2 pffiffiffiffiffi DðaÞ ¼ 1380 þ 456 10 and EðaÞ ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi
4710 þ 1500 10 þ 18370a 5840 10a þ 1395a2 441 10a2 pffiffiffiffiffi : 1380 456 10
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Under these notations, our method gives the following families of algebras (where parameters belong to C, unless another thing is indicated):
10a 8a2 þ8a3 þAðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ 1 ; ; a; • l11 6 3
24þ14a 7AðaÞ 12 9b 14abþ16a2 b 8AðaÞb 8aAðaÞb ; 1; 0; ; 9 6þ12a pffiffiffi pffiffiffi b; 0; c; 0; d; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 7 ; 1 ; 2 2 2 5 ; 5 ; 5 ; 1; 38 ; 1; 0; 84 55a ; a; 0; b; 0; c; l2 , with c 2 C n f0g; 11 21 21 7 9 18 l3 5 ; 5 ; 5 ; 1; 38 ; 1; 0; 84 55a ; a; 0; b; ; c ; 7 11 21 21 9 18
10a 8a2 þ8a3 þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ ; ; a; l4 11 6 3
9 14aþ16a2 8AðaÞ 8aAðaÞ 1; ; ; 1; 0; b; 0; c; , with 6þ12a pffiffiffi pffiffiffi a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 ; 1 ; b 2 C2 [ f0g; 7 2 2 2
5 5 5 5 55 l ; ; 7 ; 1; ; ; 1; 0; a; 0; b; 0; 0; c , with a 2 C2 [ f0g; 11 21 21 18
2 3
10a 8a þ8a þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ ; ;a; l6 6 3 11 pffiffiffi pffiffiffi 1; ;1; 0;b; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 7 ; 1 ; b 2 C3 [ f0g; 2 2 2
5 ; 5 ; 5 ; 1; ; 1; 0; a; 0; 0; b , with a 2 C [ f0g; l7 3 7 11 21 21
10a 8a2 þ8a3 þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ ; ; a; l8 11 6 3 pffiffiffi pffiffiffi 1; ; 1; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 7 ; 1 ; 2 2 2
5 ; 5 ; 5 ; 1; ; 1; 0; 0; a , with a 2 C [ f0g; l9 2 7 11 21 21
10a 8a2 þ8a3 þ5AðaÞþaAðaÞ 4a2 AðaÞ 3 4aþ2a2 AðaÞ aAðaÞ l10 ; ; 11 6 3 pffiffiffi pffiffiffi a; 1; , with a 2 C n 1 ; 9 6 3 ; 9þ6 3 ; 5 ; 1 ; 7 2 2 2
5 5 5 l11 11 21 ; 21 ; 7 ; 1; ; 1 ; 1;
• • •
• •
• •
• •
•
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5 5 5 12 ; ; 7 ; 1; • l11 ; 21 21
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; • l13 537 310 3 þ 105C 359C 11 2 2 2 4 3 2 3 pffiffiffi pffiffiffi
303þ150 3 36Cþ10 3C ; 0; 1; 9 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 144 75 3þ18C 5 3Cþ342a 195 3 aþ36Ca 22 3 Ca pffiffiffi ; a; 0; b; 0; c; ; 30 18 3
pffiffiffi pffiffiffi pffiffiffi 105B 359B 49 11B pffiffiffi pffiffiffi 9 • l14 11 537 þ 310 3 2 4 3 ; 2 þ 14 3 3B 2 3 ; 2 þ 3 3; 1; pffiffiffi pffiffiffi
303 150 3þ36Bþ10 3B ; 0; 1; 9 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 144þ75 3 18B 5 3Bþ342aþ195 3a 36Ba 22 3Ba ; a; 0; b; 0; c; pffiffiffi ; 30þ18 3
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; ; • l15 537 310 3 þ 105C 359C 2 11 4 3 2 2 3 2 pffiffiffi pffiffiffi 684 390 3þ72C 44 3C ; 1; 0; a; 0; b; pffiffiffi , with a 2 C2 [ f0g; 60 36 3
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 þ 14 3 3B 11B pffiffiffi ; 9 þ 3 3; 1; ; • l16 537 þ 310 3 105B 359B 2 2 2 11 4 3 2 3 pffiffiffi pffiffiffi 342þ195 3 36B 22 3 B pffiffiffi ; 1; 0; a; 0; b; , with a 2 C2 [ f0g; 6ð5þ3 3
pffiffiffi pffiffiffi pffiffiffi 17 pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; ; • l 537 310 3 þ 105C 359C 11 2 2 2 4 3 2 3 1; 0; a; , with a 2 C3 [ f0g;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 þ 14 3 3B 11B pffiffiffi ; 9 þ 3 3; 1; ; • l18 537 þ 310 3 105B 359B 11 2 4 3 2 2 3 2 1; 0; a; , with a 2 C3 [ f0g;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 14 3 þ 3C 11C pffiffiffi ; 9 3 3; 1; ; • l19 537 310 3 þ 105C 359C 2 2 2 11 4 3 2 3 1; ;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi ; 49 þ 14 3 3B 11B pffiffiffi ; 9 þ 3 3; 1; • l20 537 þ 310 3 105B 359B 2 2 2 11 4 3 2 3 ; 1; ;
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pffiffiffi pffiffiffi pffiffiffi 105C 359C 49 11C 9 21
pffiffiffi ; 14 3 þ 3C pffiffiffi ; 3 3; 1; • l11 537 310 3 þ ; 2 4 3 2 2 3 2
pffiffiffi pffiffiffi pffiffiffi 105B 359B 49 11B pffiffiffi ; pffiffiffi ; 9 þ 3 3; 1;
þ14 3
3B
• l22 537 þ 310 3
; 11 2 4 3 2 2 3 2
10a 8a2 þ8a3 5AðaÞ aAðaÞþ4a2 AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ ; ; • l23 11 6 3
24þ14aþ7AðaÞ 12 9b 14abþ16a2 bþ8AðaÞbþ8aAðaÞb ; 1; 0; ; b; 0; c; 0; 9 6þ12a pffiffiffiffiffi pffiffiffiffiffi d; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; 2 4 4
10a 8a2 þ8a3 AðaÞ aAðaÞþ4a2 AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ ; ; a; 1; l24 11 6 3
9 14aþ16a2 þ8AðaÞþ8aAðaÞ ;1; 0; b; 0; c; ; , with a 2 C n 1 ; 6þ12a 2 pffiffiffiffiffi pffiffiffiffiffi 3 33 ; 3þ 33 ; 1 ; b 2 C [ f0g; 2 4 4
2 3
10a 8a þ8a 5AðaÞ aAðaÞþ4a2 AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ l25 ; ; 11 6 3 pffiffiffiffiffi pffiffiffiffiffi a; 1; ; 1; 0; b; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; b 2 C3 [ f0g; 4 4 2
2 3 2 2
10a 8a þ8a
5AðaÞ aAðaÞþ4a AðaÞ
3 4aþ2a þAðaÞþaAðaÞ ; ; l26 11 6 3 pffiffiffiffiffi pffiffiffiffiffi a; 1; ; 1; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; 4 4 2
2 3 2
10a 8a þ8a 5AðaÞ aAðaÞþ4a AðaÞ 3 4aþ2a2 þAðaÞþaAðaÞ l27 ; ; 11 6 3 pffiffiffiffiffi pffiffiffiffiffi a; 1; , with a 2 C n 1 ; 3 33 ; 3þ 33 ; 1 ; 2 4 4
pffiffiffi pffiffiffi pffiffiffi pffiffiffi l28 4þ7i 2 ; 1 i 2 ; 1 ; 1; 31 7i 2; 1; 0; a; 4 8i 2 ; 0; b; 0; 0; c; 11 12 6 6 2 9 9 0; 0 ;
pffiffiffi pffiffiffi 4þ7i 2
1 i 2 29 1 l ; ; ; 1; ; 1; 0; 0; a; 0; 0; b; 0; 0 , with a 2 C2 [ f0g; 11 12 6 2
pffiffiffi pffiffiffi l30 4þ7i 2 ; 1 i 2 ; 1 ; 1; ; 1; 0; 0; a; 0; 0 , with a 2 C3 [ f0g; 11 12 6 2
pffiffiffi pffiffiffi l31 4þ7i 2 ; 1 i 2 ; 1 ; 1; ; 1; 0; 0 ; 12 6 11 2 a; 1;
•
•
•
•
•
• • •
622
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
pffiffiffi pffiffiffi 4þ7i 2
1 i 2 1 32 ; ; ; 1; • l ; 11 12 6 2
pffiffiffi pffiffiffi pffiffiffi pffiffiffi • l33 4 7i 2 ; 1þi 2 ; 1 ; 1; i 31i þ 7 2 ; 1; 0; a; 4þ8i 2 ; 0; b; 0; 0; 11 12 6 6 2 9 c; 0; 0 ;
pffiffiffi pffiffiffi • l34 4 7i 2 ; 1þi 2 ; 1 ; 1; ; 1; 0; 0; a; 0; 0; b; 0; 0 , with a 2 C2 [ f0g; 11 2 12 6
pffiffiffi pffiffiffi 4 7i 2
1þi 2 1 35 • l ; ; ; 1; ; 1; 0; 0; a; 0; 0 , with a 2 C3 [ f0g; 11 2 12 6
pffiffiffi pffiffiffi • l36 4 7i 2 ; 1þi 2 ; 1 ; 1; ; 1; 0; 0 ; 12 6 2 11
pffiffiffi pffiffiffi 4 7i 2
1þi 2 1 37 ; ; ; 1; • l ; 12 6 2 11
2 5þ3a b 4b 7c ; c; 0; d; ; e; 0 , with a 6¼ • l38 1; 1; 1; 1; a; b; 1; 2 11
•
•
• •
•
• •
3 5b ; b 2 C n f 2g; 2
9þ3a 7b ; b; 0; c; 0; d; l39 1; 1; 1; 1; a; 2; 1; , with b 2 C n f1g; 2 11 n o a 2Cn 7 ; 2
, with c 2 C n f0g; a 2 l40 1; 1; 1; 1; a; 2; 1; 16þ3a ; 1; 0; b; 0; c; 11 2 n o Cn 7 ; 2
n o
16þ3a ; 1; 0; b; ; c; 0 , with a 2 C n 7 ; l41
1; 1;
1; 1; a;
2; 1; 11 2 2
2 1 17a 8a 14b
3 5a l42 , with c 2 C ; b; 0; 0; c; 11 1; 1; 1; 1; 2 ; a; 1; 4 pffiffiffiffiffi pffiffiffiffiffi nf0g; a 2 C n 13 2 10 ; 13þ2 10 ; 2 ; 3 3
1 17a 8a2 14b l43 1; 1; 1; 1; 3 5a ; a; 1; ; b; ; c; 0 , with a 2 Cn 11 4 2 pffiffiffiffiffi pffiffiffiffiffi
13 2 10 ; 13þ2 10 ; 2 ; 3 3
7 ; 2; 1; 3 14a ; a; 0; 0; b; c; l44
1; 1;
1; 1; , with a 2 C n f1g; 11 2 4
, with a 6¼ b; l45 1; 1; 1; 1; 7 ; 2; 1; 11 ; 1; 0; 0; a; b; 2 4 11
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
• •
•
•
•
•
•
• • • • • •
623
7 11 46 l
1; 1; 1; 1; ; 2; 1; ; 1; 0; 0; a; a; 0; b; 0 ; 11 2 4
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l47 1; 1; 1; 1; 28 5 10 ; 13þ2 10 ; 1; 500þ157 10 63a ; a; 0; b; c; 11 3 3 18 , with c 2 C n f0g;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l48 1; 1; 1; 1; 28þ5 10 ; 13 2 10 ; 1; 500 157 10 63a ; a; 0; b; c; 11 3 3 18 , with c 2 C n f0g;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l49 1; 1; 1; 1; 28 5 10 ; 13þ2 10 ; 1; 500þ157 10 63a ; a; 0; b; 3 3 18 11 , with b 2 C n fDðaÞ; EðaÞg;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l50 1; 1; 1; 1; 28þ5 10 ; 13 2 10 ; 1; 500 157 10 63a ; a; 0; b; 3 3 18 11 , with b 2 C n fDðaÞ; EðaÞg;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 28 5 10
13þ2 10
500þ157 10 63a ; a; 0; EðaÞ; 51 ; ; 1; l
1; 1; 1; 1; 3 3 18 11 ; b; 0 ;
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi l52 1; 1; 1; 1; 28þ5 10 ; 13 2 10 ; 1; 500 157 10 63a ; a; 0; DðaÞ; 11 3 3 18 ; b; 0 ;
n o
4 7b 53 ; b; 0; c; ; d; 0 , with a 2 C n 5 ; l
1; 1; 1; 1; a; 1; 0; 11 2 2
l54 1; 1; 1; 1; 5 ; 1; 0; 4 7a ; a; 0; 0; b; , with b 2 C n f0g; 11 2 2
l55 1; 1; 1; 1; 5 ; 1; 0; 4 7a ; a; ; b; 0 ; 2 2 11
7a 56 ; a; 0; 0; b; c; l
1; 1; 1; 1; 1; 0; 0; , with a 2 C n f0g; 11 2
, with a 6¼ b; l57 1; 1; 1; 1; 1; ; a; b; 11
l58 1; 1; 1; 1; 1; ; a; a; 0; b; 0 ; 11
624
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
7 59 • l
1; 1; 1; 1; ; ; 1; 0; a; b; c; , with b 2 C2 ; 11 2
• l60 1; 1; 1; 1; ; 7 ; 1; 0; a; 0; b; , with a 2 C2 [ f0g; 11 2 • l61 ð 1; 1; 1; 1; ; a; 1; b; Þ, with b 2 ðC3 [ f0gÞ n f1g; 11 62 • l ð 1; 1; 1; 1; ; a; 1; 1; 0; b; 0Þ; 11 • l63 ð 1; 1; 1; 1; ; 1; 0; a; Þ, with a 2 C3 ; 11 64 • l ð 1; 1; 1; 1; ; 1; ; a; 0Þ, with a 2 C3 [ f0g; 11 • l65 ð 1; 1; 1; 1; ; 1; Þ; 11 66 • l ð 1; 1; 1; 1; ; 1; 0Þ; 11 • l67 ð 1; 1; 1; 1; Þ; 11 68 9 ; 3 ; 1; 0; 3 ; 1; 0; 1 35a ; 0; a; b; 0; c; • l , with c 2 C n f0g; 11 8 2 2 16
• l69 9 ; 3 ; 1; 0; 3 ; 1; 0; 1 35a ; 0; a; b; ; c ; 11 8 2 2 16
• l70 9 ; 3 ; 1; ; 35 ; 0; 1; a; 0; b; 0; 0; c , with a 2 C2 [ f0g; 11 8 2 16
71 9 3 • l ; ; 1; ; 1; 0; a; 0; 0; b , with a 2 C3 [ f0g; 11 8 2
• l72 9 ; 3 ; 1; ; 1; 0; 0; a , with a 2 C4 [ f0g; 11 8 2
73 9 3 • l ; ; 1; ; 1 ; 11 8 2
• l74 9 ; 3 ; 1; ; 11 8 2 • l75 ð1; 11 • l76 ð1; 11 77 • l ð1; 11 • l78 ð1; 11 79 • l ð1; 11 • l80 ð1; 11 81 • l ð1; 11
; 1; 0; a; b; 2; 0; c; 0; d; 0; 0Þ; ; a; 1; 0; b; c; 0; d; 0; 0Þ,
with c 2 C2 ;
; a; 1; 0; b; 0; 0; c; 0; 0Þ,
with b 2 C2 [ f0g;
; 1; 0; 0; a; b; c; d; 0; 0Þ,
with b 2 C2 ;
; 1; 0; 0; a; 0; b; c; d; 0Þ,
with c 2 C n f0g; a 2 C2 ;
; 1; 0; 0; a; 0; b; 0; c; dÞ,
with a 2 C2 ;
; 1;
; a; b; c; 0Þ,
with b 2 C n f0g; c 2 C2 [ f0g;
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
• l82 ð1; 11 • l83 ð1; 11 • l84 ð1; 11 • l85 ð1; 11 • l86 ð1; 11 • l87 ð1; 11 88 • l ð1; 11 89 • l ð1; 11 • l90 ð1; 11 91 • l ð1; 11 • l92 ð1; 11 • l93 ð1; 11 94 • l ð1; 11 95 • l 11
• l96 11
97 • l 11
• l98 11
99 • l 11 • l100 ð 11 • l101 11
102 • l 11
• l103 11
104 • l 11
; 1;
; a; 0; b; cÞ,
Þ,
with b 2 C2 [ f0g; with c 2 C3 ;
; a; 1; b; c; 0; 0Þ, ; a; 1; b;
with b 2 C3 [ f0g;
; 1; 0; a; b; c; 0Þ,
with b 2 C3 ;
; 1; 0; a; 0; b; cÞ,
with a 2 C3 ;
; 1;
; a; bÞ,
625
with a 2 C3 [ f0g; with b 2 C4 ;
; a; 1; b; 0Þ, ; a; 1; 0; 0Þ; ; 1; 0; a; bÞ,
with a 2 C4 ;
; 1; 0; 0; aÞ,
with a 2 C2 [ f0g;
; 1; aÞ,
with a 2 C5 [ f0g;
; 1Þ; Þ; 2 ; 5þaþ4a ; a; 1; b; c; 1; 0; 0; d; 3 ; 2 ; 1; 1; a; b; 1; 0; 0; c; ; 3 ; 2 ; 1; 1; a; b; 1; ; c ; 3 ; 5 ; 3 ; 1; a; b; 1; 0; 0; c; 0; d; 0 ; 6 2 ; 21; 4; 1; a; b; 1; c; ;
; 3; 2; 1; a; b; 1; c; 0; d; ; 5þaþ4a ; a; 1; b; 1; 3 2
; 2 ; 1; 1; a; 1; 3 ; 2 ; 1; 1; a; 1; 3 ; 5 ; 3 ; a; 1; 6 2
; b; ;b ;
Þ; ; c; ,
; b; 0; c; 0 ;
,
,
n o with a 2 C n 1; 3 ; 2; 4 ; 2
n o with a 2 C n 1; 3 ; 2; 4 ; 2
with b 2 C n f0g;
626
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
n o • l105 ð ; 21; 4; 1; a; 1; 0; b; Þ, with a 2 C n 41 ; 11 4
; 21; 4; 1; 41 ; 1; 0; a; 0; b; ; • l106 11 4 • l107 ð 11
• l108 11
109 • l 11
• l110 11
• l111 11
; 3; 2; 1; a; 1; 0; b; 0; c;
• l112 11 ð • l113 ð 11
• l114 11
• l115 11
116 • l 11
• l117 11
• l118 11
119 • l 11
• l120 11
; 21; 4; 1; 1; 0; 0; a;
• l121 11 ð • l122 11 ð • l123 ð 11 • l124 11 ð • l125 ð 11
; 21; 4; 1;
; 1; 0; a;
; 21; 4; 1;
; 1;
; 21; 4; 1;
Þ;
; 3; 2; 1;
; 1; 0; a;
; 3; 2; 1;
; 1;
; 5þaþ4a ; a; 1; 1; 3 2
; 2 ; 1; 1; 1; 3 ; 2 ; 1; 1; 1; 3 ; 5 ; 3 ; 1; 1; 6 2
; a; ;a ;
Þ; n o ; b; , with a 2 C n 1; 3 ; 2; 4 ; 2 , with a 2 C n f0g;
; a; 0; b; 0 ; Þ;
; 3; 2; 1; 1; 0; 0; a; 0; b;
Þ;
n o ; 1; , with a 2 C n 3 ; 2; 4 ; 2 ; 5 ; 3 ; 1; ; 1; 0; a; 0 , with a 2 C3 [ f0g; 6 2 n o 2
5þaþ4a ; , with a 2 C n 1; 3 ; 2; 4 ; ; a; 1; 2 3 ; 2 ; 1; 1; ; 1 ; 3 ; 2 ; 1; 1; ; 3 5 3 ; ; ; 1; ; 1; 0 ; 6 2 ; 5 ; 3 ; 1; ; 6 2 ; 5þaþ4a ; a; 1; 3 2
Þ,
with a 2 C2 [ f0g;
Þ,
with a 2 C2 [ f0g;
Þ;
Þ;
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
627
• l126 ð ; 3; 2; 1; Þ; 11 n o • l127 ð ; 1; 0; 0; a; b; 1; 0; c; d; Þ, with b 2 C n 1 ; 1 ; 11 2
; 1; 0; 0; a; 1 ; 1; 0; b; c; 0; 0; d ; • l128 11 2 • l129 ð 11 • l130 ð 11 131 • l ð 11 • l132 ð 11 133 • l ð 11 • l134 ð 11 • l135 ð 11 136 • l ð 11 • l137 ð 11 • l138 ð 11 • l139 ð 11 140 • l ð 11 • l141 11 ð • l142 11 ð
143 • l 11
; 1; 0; 0; a; 1; 1; 0; b; c; 0; d; 0Þ; ; 1; 0; 0; a; 1; 0; 0; b; 0; c; 0; 0Þ; ; 1; 0; 0; 1;
; a; b; c; 0; 0Þ,
; 1; 0; 0; 1;
; a; b; c; 0Þ,
; 1; 0; 0; 1;
; a; 0; b; cÞ;
with a 2 C n f0g; with b 2 C n f0g;
with b 2 C2 ;
; 1;
; 1; a; b; 0; 0Þ,
; 1;
; 1; a;
; 1;
; a; 1; b; 0Þ,
with b 2 C3 [ f0g;
; 1;
; 1; 0; a; bÞ,
with a 2 C3 ;
; 1;
; 1; 0; 0; aÞ,
with a 2 C3 [ f0g;
; 1;
; 1; aÞ,
; 1;
; 1Þ;
; 1;
Þ;
Þ,
; a; b; 1; c; 1; d;
with a 2 C2 [ f0g;
with a 2 C4 [ f0g;
Þ,
; a; 1 ; 1; b; 1; c; 2
n o with b 2 C n 1 ; 1 ; 2 , with 4 þ 16a 16a2 þ 5b þ 18ab þ 2c
þ 4ac 6¼ 0;
1 144 ; a; ; 1; b; 1; c; 0; 0; d , with 4 þ 16a 16a2 þ 5b þ 18ab þ 2c • l11 2 þ 4ac ¼ 0; • l145 ð ; a; 1; 1; b; 1; c; Þ, with a 2 C n f1g; 11 146 • l ð ; 1; 1; 1; a; 1; b; 0; c; 0Þ; 11
628
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
n o • l147 ð ; a; b; 1; 1; 0; c; Þ, with b 2 C n 1 ; 1 ; 11 2
• l148 ; a; 1 ; 1; 1; 0; b; ; with 5 þ 9a þ b þ 2ab 6¼ 0; 11 2 2
• l149 ; a; 1 ; 1; 1; 0; b; 0; 0; c , with 5 þ 9a þ b þ 2ab ¼ 0; 11 2 2 • l150 ð 11 • l151 ð 11 • l152 ð 11
• l153 11
154 • l 11
; a; 1; 1; 1; 0; b;
• l155 ð 11 156 • l ð 11 • l157 ð 11
158 • l 11
• l159 11
; a; 1; 1; 0; 0; 1;
• l160 ð 11 • l161 ð 11 • l162 ð 11 • l163 ð 11 • l164 ð 11 165 • l ð 11 • l166 ð 11 • l167 ð 11 168 • l ð 11 • l169 ð 11
; a; 1; 1;
Þ,
; 1; 1; 1;
; 1; 0Þ;
Þ,
with a 2 C n f1g;
; 1; 1; 1; 1; 0; a; 0; b; 0Þ;
n o with b 2 C n 1 ; 1 ; 2 n o ; a; 1 ; 1; 0; 0; 1; , with a 2 C n 1 ; 2 2 ; 1 ; 1 ; 1; 0; 0; 1; 0; 0; a ; 2 2
; a; b; 1; 0; 0; 1;
Þ,
Þ,
with a 2 C n f1g;
; 1; 1; 1; 0; 0; 1; 0; a; 0Þ,
with a 2 C2 [ f0g; n o ; a; b; 1; Þ, with b 2 C n 1 ; 1 ; 2 ; a; 1 ; 1; ; 1 ; 2 ; a; 1 ; 1; ; 2 with a 2 C;
; a; 1; 0; b; 1; 0; c; 0; 0Þ; ; a; 1; 0; 1; 0; 0; b; 0; 0Þ; ; a; 1;
; 1; 0; 0Þ;
; a; 1;
Þ;
; 1; 0; 0; a; 1; b; c; 0; 0Þ; ; 1; 0; 0; 1; 0; a; b; 0; 0Þ; ; 1;
; a; 1; 0; 0Þ;
; 1;
; 1; 0; 0; aÞ;
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
• l170 ð ; 1; 11
; 1Þ;
• l171 ð ; 1; 11
Þ;
629
• l172 ð ; a; 1; b; 1; 0; 0Þ; 11 • l173 ð ; a; 1; 1; 11 • l174 ð ; a; 1; 11
Þ; Þ;
• l175 ð ; 1; 0; a; 1; b; 0Þ; 11 • l176 ð ; 1; 0; 1; 0; a; bÞ; 11 • l177 ð ; 1; 11
; 1; aÞ,
• l178 ð ; 1; 11 • l179 ð ; 1; 11
; 1Þ;
with a 2 C2 [ f0g;
Þ;
• l180 ð ; a; 1; 1; 0Þ; 11 • l181 ð ; a; 1; 0; 0Þ; 11 • l182 ð ; 1; 0; 1; aÞ; 11 • l183 ð ; 1; 0; 0; 1Þ; 11 • l184 ð ; 1; Þ; 11 • l185 ð ; 1; 1Þ; 11 • l186 ð ; 1; 0Þ; 11 • l187 ð ; 1Þ; 11 188 • l ð Þ: 11 So, in this way it is proved the following: Theorem 4.2. Let g be a filiform Lie algebra of dimension 11. Then, g is isomorphic to one and only one of the algebras belonging to one of the families lh11 ; 1 6 h 6 188, previously mentioned.
630
L. Boza et al. / Appl. Math. Comput. 141 (2003) 611–630
References [1] J.M. Ancochea, M. Goze, Classification des algebres de Lie filiformes de dimension 8, Archiv. Math. 50 (1988) 511–525. [2] J.M. Ancochea, M. Goze, Classification des algebres de Lie nilpotentes complexes de dimension 7, Arch. Math. 52 (1989) 175–185. [3] J.M. Ancochea, M. Goze, On the varietes of nilpotent Lie algebras of dimension 7 and 8, J. Pure Appl. Algebra 77 (1992) 431–440. ~ ez, The equations of the sets of filiform Lie [4] J.C. Benjumea, F.J. Castro, M.C. Marquez, J. N un lgebra Computacional y algebras of dimension 13 and 14, Actas del 2 Encuentro de A Aplicaciones (1996) 67–75. [5] N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958) 45–92. ~ez, Classification of complex filiform Lie algebras of dimension [6] L. Boza, E.M. Fedriani, J. N un lgebra, Computaci 12, Prepublicaci on 39, Dpto. A on, Geometrıa y Topologıa. Universidad de Sevilla, 1997. ~ez Valdes, Classification of complex filiform Lie [7] L. Boza Prieto, F.J. Echarte Reula, J. N un algebras of dimension 10, Algebras, Groups and Geometries 11 (3) (1994) 253–276. ~ez Valdes, A new method for classifying complex [8] L. Boza Prieto, E.M. Fedriani Martel, J. N un filiform Lie algebras, Appl. Math. Comput. 121 (2–3) (2001) 169–175. ~ ez, The equations of the sets of filiform Lie algebras of dimension 11 and [9] F.J. Castro, J. N un 12, in: Personal Communication, Universidad de Sevilla, Espa~ na, 1996. [10] F.J. Echarte, J.R. G omez, Classification of complex filiform nilpotent Lie algebras of dimension 9, Rendiconti Seminario Facolt Scienze Universit Cagliari 61 (1) (1991) 21–29. [11] E.M. Fedriani, Clasificaci on de las algebras de Lie filiformes complejas de dimensi on 12, in: Tesis de Licenciatura, Universidad de Sevilla, Espa~ na, 1997. [12] J.R. G omez, A. Jimenez, Y. Khakimdjanov, Low-dimensional filiform Lie algebras, J. Pure Appl. Algebra 130 (1998) 133–158. ~ ez, An algorithm to obtain laws families of filiform Lie [13] J.R. G omez, A. Jimenez, J. N un algebras, Linear Algebra Appl. 279 (1998) 1–12. [14] C. Seeley, Some nilpotent Lie algebras of even dimensions, Bull. Aust. Math. Soc. 45 (1992) 71–77. [15] A. Shalev, E.I. Zelmanov, Narrow Lie algebras: a coclass theory and a characterization of the Witt algebra, J. Algebra 189 (1997) 294–331. [16] M. Vergne, Cohomologie des algebres de Lie nilpotentes. Application a l’etude de la variete des algebres de Lie nilpotentes, Bull. Soc. Math. France 98 (1970) 81–116.