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Combinatorial Structures Associated with Lie Algebras of Finite Dimension Linear Algebra and its Applications 389 (2004) 43–61 www.elsevier.com/locate/laa Combinatorial structures associated with Lie algebras of finite dimensionୋ A. Carriazo, L.M. Fernández, J. Núñez∗ Department of Geometry and Topology, Faculty of Mathematics, University of Seville, Apdo. Correos 1160, 41080 Seville, Spain Received 10 January 2003; accepted 23 February 2004 Submitted by G. de Oliveira Abstract Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not. © 2004 Elsevier Inc. All rights reserved. AMS classification: 05C20; 17B60 Keywords: Combinatorial structure; Weighted graph; Oriented digraph; Bracket product; Lie algebra Introduction At present, the research on Lie Theory is very extended. It is due not only to its purely theoretical study, but because its current applications to other sciences, particularly Applied Mathematics, Engineering and Physics, are contributing to the fast development of many new theories. However, the ignorance on many aspects of this subject is still remarkable. Indeed, with respect to Lie algebras, although the classifications of simple and semisimple ones have been obtained since 1890 (in fact, until the beginning of the last century ୋ The authors are partially supported by the PAI project (Junta de Andalucía, Spain, 2002). ∗ Corresponding author. Tel.: +34-954-557962; fax: +34-954-557970. E-mail addresses: [email protected] (J. Nuñez),´ [email protected] (A. Carriazo), [email protected] (L.M. Fernandez).´ 0024-3795/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2004.02.030 44 A. Carriazo et al. / Linear Algebra and its Applications 389 (2004) 43–61 they continued being improved), the classifications of solvable and nilpotent Lie algebras are actually an open problem. Indeed, they are only known for dimension n, with n 5andn 7, respectively. Apart from that, the study of Graph Theory is also running in a high level, for similar reasons as previously mentioned. Moreover, this theory is actually being con- sidered to be used as a tool to deal with the study of other subjects, due to its many applications. At this point, the main goal of this paper is to set up a link between these two the- ories: Lie and Graph Theory. There have already been some progresses in this sense. For example, we can mention the Dynkin diagrams for simple finite dimensional Lie algebras (see [1,3]). Nevertheless, we now present a new and general association. In fact, we associate a combinatorial structure, of dimension 2 in general, with a Lie algebra of given finite dimension, from which we know a selected basis. Later, we study this association when the two-dimensional elements of the combinatorial struc- ture disappear, i.e., when we really have a graph, by obtaining a particular class of Lie algebras. Our main results, Theorems 3.2 and 3.6 lead to a complete characterization of such graphs, according to they have 3-cycles or not. Previously, in Section 1 we introduce the definitions and notations of the Lie algebras theory which we will use in this subject. For a more general overview of this theory the reader can consult the Chow’s books (see [2]). In Section 2, we define the above mentioned general association, and we give many examples. Section 3 is devoted to establish and prove the main results. In last section, we study some applications concerning the solvability and nilpotency of Lie algebras associated with graphs. We think that the tools introduced in this paper are useful to give a little step forward in the classification of Lie algebras, by firstly getting the classification of the associated combinatorial structures. It could involve an easier method to solve this problem than considering Lie algebras by themselves. Finally, let us remark that we just include a few general references, given the novelty of our work. 1. Definitions and notations A Lie algebra L is a vector space with a second inner bilinear composition law (denoted by [,] and called bracket product or law of the algebra), which satisfies the conditions: 1. [X, X]=0 ∀X ∈ L, 2. [[X, Y ],Z]+[[Y, Z],X]+[[Z,X],Y]=0 ∀X, Y, Z ∈ L. The second condition is named Jacobi identity. From now on, it will be denoted by J(X,Y,Z) = 0. A. Carriazo et al. / Linear Algebra and its Applications 389 (2004) 43–61 45 Note that the fact of the bracket product being bilinear and the first condition imply the skew-symmetry or self-annihilation property for the bracket product: [X, Y ]=−[Y, X] for all X, Y ∈ L. The concepts of dimension, basis, subalgebra and ideal of a Lie algebra are defined as those corresponding to that algebra, considered itself as a vector space. Let {e1,...,en} be a basis of L. Such a basis can be characterized by the struc- ture constants or Maurer–Cartan constants of the algebra, defined by [ei,ej ]= h ci,j eh for any 1 i, j n. They determine, by linear extension, the whole alge- bra. The center of a Lie algebra L is defined by cen L ={X ∈ L|[X, Y ]=0 ∀Y ∈ L}. Let L and L be two Lie algebras (both defined over the same field K). A map- ping : L → L is called an isomorphism from L into L if is a bijective linear mapping, satisfying :[X, Y ] →[(X), (Y )]∀X, Y ∈ L. For convenience of subsequent discussions, the following notations are intro- duced: L1 = L, L2 =[L, L], L3 =[L2, L2],...,Li =[Li−1, Li−1],... and − L1 = L, L2 =[L, L], L3 =[L2, L],...,Li =[Li 1, L],... (the first sequence is called derived series and the second one lower central series of L). Then, the Lie algebra L is said to be solvable (respectively nilpotent)ifthere n exists a natural number n such that Ln ≡{0} (respectively L ≡{0}). The least n satisfying that condition is called the solvability index (respectively nilpotent index) of the algebra. Note that from both definitions, it follows that every nilpotent Lie i algebra is solvable, because Li ⊆ L for all i. The converse is not true. It is also proved that every nilpotent Lie algebra has a non-null center. 2. A combinatorial structure associated with a Lie algebra of finite dimension Let L be a n-dimensional Lie algebra and let B ={e1,...,en} be a basis of L. Then, we can write n [ ]= k ei,ej ci,j ek, k=1 k where ci,j are the structure constants of L. We associate the pair (L, B) with a combinatorial structure by the following method: (a) For each ei ∈ B,1 i n, we draw one point (vertex) labeled by index i. (b) Given three vertices i<j<k, we draw the full triangle with those vertices, assigning to each of its edges the weight indicated in Fig. 1. 46 A. Carriazo et al. / Linear Algebra and its Applications 389 (2004) 43–61 i k c j c i,k i,j k j i cj,k Fig. 1. Full triangle. However, from now on we will agree that: k = i = j = (b1) If ci,j cj,k ci,k 0, the triangle will not be drawn. (b2) If any of the structure constants is null, the corresponding edge will be drawn by using a discontinuous line and then, it will be called a ghost edge. (b3) If two triangles of vertices i, j, k and i, j, l with 1 i<j<k, l n k = l satisfy that ci,j ci,j , then we draw only one edge between the vertices i and j, shared by both triangles (see Fig. 2). (c) Given two vertices i, j, with 1 i<j n, then: i = (c1) If ci,j / 0, we draw the directed edge of Fig. 3. j = (c2) Si ci,j / 0, we draw the directed edge of Fig. 4. So it will be possible to have a so-called double edge. Observe that in cases (c1) and (c2), the edge is directed to the vertex labeled with the repeated index. Actually, these edges show the “triangles” i, i, j and i, j, j, respectively. Consequently, every Lie algebra with a selected basis is associated with such a combinatorial structure. However, it is necessary to point out that this association depends on the selected basis. Notice that if there appears an isolated vertex, the corresponding basic element is in the center of the algebra L. Example 2.1. Let us consider the Lie algebra of dimension 4, with selected basis {e1,e2,e3,e4} and non-null bracket products: i k l i ci,j= ci,j k l k l j j Fig. 2. Triangles with a shared edge. A. Carriazo et al. / Linear Algebra and its Applications 389 (2004) 43–61 47 i ci,j i j Fig. 3. Directed edge. j ci,j i j Fig. 4. Directed edge. [e1,e2]=−e1, [e2,e3]=e3 + e4, [e2,e4]=e1 + e4. This Lie algebra, which is solvable but non-nilpotent (as it is easy to check), has the associated combinatorial structure of Fig. 5. Remark 2.2. It is easy to show that if a Lie algebra (with a selected basis) is associ- ated with a non-connected combinatorial structure, it can be decomposed into a direct sum of Lie subalgebras, corresponding each of them to a connected component of the structure.
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