Classical properties of algebras using a new graph association

CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ Abstract. We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated algebras, mainly the nilpotency and solvability in the case of Leibniz algebras. AMS Subject Classifications (2010): 17A32, 17B30, 17B10. Key words: Algebra, Graph theory, directed acyclic graph, nilpotency, solvability, Lie algebra, Leibniz algebra. The study of the relationship between algebraic and discrete structures is fruitful. Numerous contributions to this topic have made, for instance, in Graph Theory, one may consider matrices associated to a graph in order to obtain properties of those graphs as a consequence of the study of the spectrum of the adjacency matrix or the Laplacian matrix of such graph. One may also recall the case of Cayley’s graphs associated to groups. We refer to [8] for further details. It is also well known the classification of semi-simple Lie algebras by Dynkin diagrams, for the case of finite dimensional Lie algebras over algebraically closed fields (see [7]). In [1] the studies of quiver algebras and representation theory of finite dimensional associative algebras over a field are shown. Finally it is worthwhile to point out Carriazo, Fern´andez and N´u˜nez’s paper [5], where the authors have developed a method to associate a discrete structure to a finite dimensional Lie algebra that lead them to presente interesting applications concerning solvability and nilpotency of those algebras. The problem of that work is that the authors strongly applied the identity of Jacobi as a fundamental tool to obtain their results, thus their results are limited to Lie algebras, in fact, they are limited to a specificaly family of Lie algebras. It is natural to ask if it is possible to extend the relationship between any algebra and a graph, in order to obtain algebraic properties of the algebra by looking the properties of the graph. The aim of this work is to answer this question in the affirmative. Specially, in this paper we are going to show that given a finite dimensional algebra over a field (and a fixed basis), it is possible to associate some weighted oriented finite graphs, which allow us to recover completely the algebra, i.e.,to recover the product structure of the algebra. But the main contribution of this work is that if we drop the information about the labels (obtaining thus more simple digraphs), one obtains interesting algebraic properties of the algebra by analysing the properties of these simple digraphs. Furthermore, we succeed to apply our results to an important family of algebras: the Leibniz algebras, obtaining in this case significant results about nilpotency and solvability. We remark that Leibniz algebras rise up in the context of the studies of homology of Lie algebras, mainly on the studies of the Chevalley–Eilenberg boundary map, we refer to [10] for the reader. Lie algebras are related to Lie groups which are closely related to Yang-Mills’ theory, one of the best reason to study Lie algebras hence Leibniz algebras. The authors were supported by MCTI/CNPQ/Universal 10/2014, CAPES, and individually L.M.Camacho by MTM2016-79661-P (European FEDER support included) and A. M´arquez by BFU2016-74975-P. 1 arXiv:1706.00482v1 [math.CO] 1 Jun 2017 2 R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ It is worthwhile to point out that the graph and results obtained in [5] have been encompassed by our work. Moreover we have obtained properties of the digraphs by analysing their associated algebras (see section 4 for more details). Throughout this work, we fix as algebra any finite dimensional algebra over an infinite fixed field Fand as digraph any oriented finite graph. In Section 1 we recollect the main definitions and results useful to introduce our results. In Section 2 we define two classes of digraphs associated to a finite dimensional algebra Aover a field, with a fixed basis and a given list of relations. We also show that the digraphs associated to Lie algebras in [5] are a particular case of the digraphs defined in our current work, in Subsection 2.1. In Section 3 some basic properties of general algebras and their associated digraphs are given. In Section 4 interesting relationships between algebraic properties (as stability of derived series and the lower central series) and digraphs are shown. A complete discussion of nilpotency and solvability of Leibniz algebras is given in that section, in particular, in Theorem 5 , Proposition 7 and Theorem 7. In Section 5 some new digraphs are introduced in order to preserve more information about the algebraic and discrete structures and achieve more general results. Finally, some interesting counterexamples have been presented to provide the strength of our hypotheses on those results. 1. Preliminaries Throughout this paper Fwill denote an algebraically closed field with characteristic zero. We remark that any graph Gthroughout this paper is directed in the sense to be a digraph. Particularly, we will denote a (directed) graph Gby a pair of sets (V(G), A(G)) where V(G) is the non-empty set of vertices and A(G) is a set of directed edges or arrows, presented by ordered pairs of vertices xiand xj, denoted by (x1, x2). We notice that hypothesis fix as two the number of arrows between two fixed vertices, one arrow to each direction. In this context, we will call two arrows between the same pair of vertices by parallel arrows or parallel edges. We recall that the outdegree (respectively the indegree) of a vertex v∈G(denoted δout G(v), or, respectively, δin G(v)) is the number of arrows in Gstarting at v(resp. ending at v). Throughout this work, we consider the definition of cycles in graphs as usual, meaning, a closed path over that graph (we ignore the orientation of the arrows). We will denote a cycle with length nby n-cycle. When necessary, we will consider cycles over oriented graphs, that is, a closed oriented walk over the digraph. Furthermore to fix our context we may call those cycle as oriented cycle. It is well known that every digraph with no oriented cycles has a topological sorting, that is, it admits an ordering of the vertices such that the starting endpoint of every arrow occurs earlier in the ordering than the ending endpoint of the edge. It is also called an admissible order of the digraph (see [8]). Given a digraph G, we define a digraph Hsuch that V(H)⊆V(G) and A(H)⊆A(G) subdigraph of G. This subdigraph is said induced by X⊆V(G) if V(H) = Xand every arrow between elements of Xappearing in A(G) appears in A(H) as well. Furthermore, we will call two digraphs Gand Hisomorphic if there is a bijective function f:V(G)→ V(H) satisfying that there exists (v1, v2)∈A(G) if and only if, there exists (f(v1), f(v2)) ∈A(H).We will denoted G≃H. From the next definitions we follow [6]. A vertex colouring of a graph Gis a map c:V→Ssuch that c(v)6=c(w) whenever vand ware adjacent. The elements of the set Sare called the available colours and cis called colour map. The set of all vertices with one colour is independent and is called a ”colour class”. An n-colouring of a graph Guses ncolours; it thereby partions Vinto ncolour classes. The chromatic index of G, denoted by χ(G), is defined as the minimum nfor which Ghas an n-colouring. Analogously, we have the edge colouring of a graph. Similarly one may define a function from the set of arrows of a graph to a set of numbers, for instance, natural, real or complex numbers. This function and the graph associated to it are called weight function and weighted graph respectively. We recall that an algebra Aover a field Fis a ring A= (A, +,∗) such that (A, +) has a structure as vector space over F. We will denote shortly by Aor by (A, ∗) when it is necessary to identify the product of that algebra. Let Abe an n-dimensional algebra over the field F. Let {x1, x2,· · · , xn}be a fixed basis of the vector space (A, +).Then Ais determined, up to isomorphisms, by the multiplication rule for the basis CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 3 elements: xi∗xj= n X k=1 γk ijxk,for γk ij ∈F.We will take the same notations as used in Lie algebras by calling those elements, the γk ij, as the structure constants of such algebra. Therefore, for a fixed basis of (A, +), we can regard each algebra of dimension nover a field Fas a point in the n3-dimensional space of structure constants endowed with the Zariski topology. Recall that the Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology. As usual in Lie theory, the sequences A1=A, Ak+1 =Ak∗A1, k ≥1, A[1] =A, A[s+1] =A[s]∗A[s], s ≥1, are said to be the lower central series and the derived series of A, respectively. We remark that those concepts can be applied to any algebra. We recall that an algebra (A, [−,−]) over a field Fis called a Leibniz algebra if for any x, y, z ∈A, the Leibniz identity Leib(x, y, z) : [x, y], z−[x, z], y−x, [y, z]= 0 holds. In presence of anti-commutativity, Leibniz identity becomes Jacobi identity and therefore Lie algebras are examples of Leibniz algebras. From the Leibniz identity we conclude that for any x, y ∈A the elements [x, x] and [x, y]+[y, x] lie in Annr(A), the right annihilator of the algebra A, defined as usual. Let gand hbe two Leibniz algebras, a homomorphism of Leibniz algebras is a F-linear map φ:g−→ hsuch that φ([x, y]g)=[φ(x), φ(y)]h,for all x, y ∈g. A Leibniz algebra (A, [−,−]) is called nilpotent (respectively, solvable) if there exists m∈N(t∈N) such that Am= 0 (respectively, A[t]= 0). The minimal number m(respectively, t) such that this property is valid is called the nilindex (respectively, index of solvability) of the algebra A. 2. Digraphs associated to algebraic structures In this section we will define two different weighted oriented finite graphs associated to a F-algebra (A, ∗) of dimension nwith a fixed basis {x1, . . . , xn}as follows. Definition 1. Let the algebra (A, ∗)over a field Fand {x1, . . . , xn}its basis. •Action to the right digraph: We denoted by GR(A)the following weighted oriented graph as described below: (1) The vertices of the digraph are {x1, . . . , xn}. (2) For each i∈ {1, . . . , n},we consider the products xi∗xj= n X l=1 cl ijxldifferent to zero for j∈ {1, . . . , n}.Hence there exists an edge αil, oriented from xito xl, if cl ij 6= 0,for j= 1, . . . , n. (3) The weight of the edge αil is the vector (cl ij)j. •We define dually the action to the left digraph, by considering the action on the left and we will denoted that digraph by GL(A). Remark 1. We notice that those digraphs are well defined for any finite dimensional F-algebra and even for locally finite dimensional algebras. Example 1. Group algebras. Let Gbe a finite group of order 2 and the group algebra A=FG associated to Gfor Fbeing the complex number field. We fix a basis for A, {g1, g2},where g1and g2 are the elements of the group Gsuch that g1∗gj=g1and g2∗gi=gifor any i, j = 1,2.Hence we have the associated digraphs to Ashown in the following figures. Figure 1. GR(A) Figure 2. GL(A) 4 R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ Example 2. 2-dimensional Leibniz algebras. In this example we will present a complete non-redundant list of digraphs (previosly defined) associated with a 2-dimensional Leibniz algebra over the field of the complex numbers. In order to achieve that, we need the following two results. Proposition 1. Let Abe a 2-dimensional Leibniz algebra over C.Then Ais isomorphic either the abelian algebra, denoted by A1or to one of the non isomorphic algebras: A2:n[x1, x1] = x2,A3:n[x1, x2] = x1,A4:([x1, x2] = x2, [x2, x1] = −x2. As the isomorphism between algebras is not transmitted to isomorphism between digraphs, we are going to show all 2-dimensional Leibniz algebras, that can be obtained from the above algebras by change of bases. Corollary 1. Let Abe a 2-dimensional Leibniz algebra over C.Then either Ais the abelian algebra A1or it is one of the following algebras: A2:n[x1, x1] = x2,A3:n[x1, x2] = x1,A4:([x1, x2] = x2, [x2, x1] = −x2. A2,1:n[x2, x2] = x1,A2,2:                [x1, x1] = −bx1+b2 ax2, [x1, x2] = −ax1+bx2, [x2, x1] = −ax1+bx2, [x2, x2] = −a2 bx1+ax2. a, b 6= 0 A3,1:([x1, x1] = x2, [x2, x1] = x2. A3,2:([x2, x2] = x1, [x1, x2] = x1.A3,3:n[x2, x1] = x2.A3,4:     [x2, x1] = −dx1+cx2, [x2, x2] = −d2 cx1+dx2, c, d 6= 0. A3,5:                [x1, x1] = −eg fx1+ex2, [x1, x2] = −eg2 fx1+eg fx2, [x2, x1] = −gx1+fx2, [x2, x2] = −g2 fx1+gx2, e, f, g 6= 0, g(f2−eg)6= 0. A3,6:     [x1, x2] = ix1−hx2, [x1, x1] = hx1−h2 ix2, h, i 6= 0. A4,1:([x2, x1] = x1, [x1, x2] = −x1. A4,2:     [x1, x2] = −jx1−kx2, [x2, x1] = jx1+kx2, j, k 6= 0. We notice that those classifications will follow straightforward from solving the space of solutions of the system of equations given by the structure constants and the equations provided by the Leibniz identity. Remark 2. Note that the algebra A2,1and the algebras of the family A2,2are isomorphic to A2,; the algebras A3,1, A3,2, A3,3and the algebras of the families A3,4, A3,5and A3,6are isomorphic to A3;and A4,1and the algebras of the family A4,2are isomorphic to A4. The table below shows the action to the right digraphs associate to the 2-dimensional Leibniz algebras. Note that in this example we have omitted the weight of the digraphs. Thus, any 2- dimensional Leibniz algebra has associated as action to the right digraph one of the showing. 2.1. Remarks on Carriazo-Fern´andez-N´u˜nez’s graphs. One of the motivations of this work was founded in [5] which leads us to improve the link between Lie algebras and Graph Theory given by the authors in that paper. The digraphs defined above in Definition 1 show us that the link between Lie algebras and Graph Theory can be generalized for any algebraic structure. Furthermore if we consider the action to the left or to the right digraphs, results concerning solvability and nilpotency have been obtained for general classes of algebras, by considering the algebraic structure and the properties of the digraphs instead of using the structure constants. We refer to Section 3, 4 and 5 to the reader. CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 5 A1A2A3A4 A2,1A3,1A4,1 A2,2A3,2A4,2 A3,3 A3,4 A3,5 A3,6 Table 1. Classes of the action to the right grahps of 2-dimensional Leibniz algebras In fact, if we assume that the algebra Ahas the same hypothesis given in [5], that is, Ais a Lie algebra defined by the law [xi, xj] = aixi+bjxj,for ai, bj∈F,and we consider the action to the right digraphs GR(A),similar results like the obtained in [5] can be achieved by following theirs proofs. We notice that the most important tool used to prove those results is the Jacoby identity, thus we follow the same steps to obtain the same kind of results as given by the authors in [5]. Precisely, given a Lie algebra Aunder the hypotheses considered in [5], both the digraph GR(A) and GL(A) are a generalization of the graph presented in [5], provides that they represent the relationships between each operand and the result of the product, the same relationships investigated in [5]. Under these hypothesis, the operands and the result of product lead us to the same elements, therefore the digraphs contains the same information. We remark that for any Lie algebra A, GR(A)≃GL(A), see Proposition 5. Furthermore, recognizing that GR(A) is equivalent to digraphs in [5] except for loops occurring at the going-in vertex and zero coordinates in weight of digraphs action, we can see that equivalent results to those found in [5] can be obtained by considering these digraphs. In order to clear up that the digraphs defined in previous section are the generalization of the graphs defined in [5], we show as examples the following results, by denoting into parentheses the corresponding results of [5]. It is worthwhile to note that in the following results of this subsection we are going to consider Aas is a Lie algebra defined by the law [xi, xj] = aixi+bjxj,for ai, bj∈F. Lemma 1. (Lemma 3.1 in [5]) Let Abe a Lie algebra associated to a digraph GR(A).Then, the configurations of Figure 3 are allowed in GRfor any three different vertices i, j, k. 6 R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ Figure 3. Allowed configurations in action to the right digraph. We observe that each edge of the digraphs from [5] appears in GR(A). Furthermore, double edges in [5] creates in GR(A) two loops, one for each vertex linked by double edge. The weight in [5] graphs is equivalent to the weights in GR(A) except for the signal, that is opposite in edges directed from the vertex with the highest index to lowest. Theorem 2. ( Theorem 3.2 in [5]) Let GR(A)be the action to the right digraph associated with a Lie algebra A. If GR(A)has no 3-cycles, then GR(A)is given as follow. (1) Two vertices a,b, two oriented edges with opposite directions and one loop to each vertex. (2) A well-oriented graph with loops in incoming vertex. Furthermore, any digraph satisfying one of those conditions above is associated with a Lie algebra since that the one single coordinate nonzero of weights provided to each edge be the coordinate that corresponds to incoming vertex. Theorem 3. (Theorem 3.6 in [5]) Let GR(A)be a digraph containing 3-cycles associated with a Lie algebra. Then the following conditions holds: (1) The double edges of GR(A)lie on 3-cycles and there are no 3-cycles without double edges. (2) The adjacent vertices to the extreme vertices of the double edges are not mutually adjacent. Moreover, they appear in one of the configurations of Fig. 4. (3) The subdigraph obtained from GR(A)by removing its double edges and loops attached to it satisfies condition (2) of Theorem 2 . Figure 4. Allowed configurations with 3-cycles in digraph action to the right Theorem 4. (Theorem 4.1 in [5]) Any Lie algebra associated with a weighted digraph that does not contain the second configuration of Figure 4 is solvable, but not nilpotent. 3. Algebras and digraphs In section 2 we have associated two digraphs with an algebra but, as we saw in Example 2, isomorphic algebras can be associated to non isomorphic digraphs under these definitions. Hence, the following CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 7 natural question rises up: when a digraph determines an algebraic structure? In this section, we will present basic relations between an algebra and their associated digraphs to attempt to answer this question. Proposition 2. Let Gbe a non-weighted digraph. Then, there exists an algebra whose action to the right digraph (or action to the left digraph) is isomorphic to Gwith an associated weight. Proof Let us see the proof for the action to the right digraph, for the action to left digraph is analogous. Given a digraph G, define Aas the linear envelope of V(G). From the definition of linear envelope, Ais a vector space and V(G) is a basis of that vector space. Now, for each arrow (xi, xj)∈A(G), we define the following multiplication xi∗xj=xjin A. It is clear that if we extend that operation by bilinearity we will obtain an algebra such that its associated action to the right digraph is isomorphic to Gwith the corresponding weight. Some basic properties related to an algebra and its associated digraphs are given by the following result. Lemma 2. Let Abe an algebra, and let GR(A), and GL(A)be their associated digraphs. Then, the following properties hold: •An element xof the basis of Abelongs to Ann(A)if and only if δout GR(A)(x) = δout GL(A)(x) = 0. •An element xof the basis of Abelongs to Annr(A)if and only if δout GL(A)(x) = 0. Proof The proof follows straightforward from the definitions of the annihilator of A, the right annihilator of A, and the considered digraphs. Definition 2. Let Abe an algebra and B={x1, ..., xn}a basis to A. We will use the notation GR(A)2to indicate the subdigraph of GR(A)induced by the vertices that belong to W={xk∈ B : ∃y∈A2such that y=Pn i=1 αixiwith αk6= 0}. Respectively, GL(A)2will be denote the subdigraph of GL(A)induced by vertices that belong to W. This definition is equivalent to the subdigraph induced by all the vertices with nonzero indegree in GR(A)or, respectively, in GL(A). Example 3. Let Abe the 4-dimensional algebra given by the following table of multiplication. A:         [x1, x2]=2x3−x2, [x1, x1] = −x2, [x2, x4] = x1, [x4, x1] = −x1. The action to the left and to the right digraphs and its subdigraphs GL(A)2and GR(A)2are presented in the next figure. (a) GL(A) (b) GR(A) (c) GL(A)2(d) GR(A)2 8 R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ GR(A)2and GL(A)2are useful to obtain an upper and a lower bound of the chromatic index of GR(A) and GL(A). Proposition 3. Let Abe an algebra. The following statements about its associated digraphs are true: χ(GR(A)2)≤χ(GR(A)) ≤χ(GR(A)2)+1 χ(GL(A)2)≤χ(GL(A)) ≤χ(GL(A)2)+1 Proof It is enough to prove the first statement, the second one follows analogously. The bound χ(GR(A)2)≤χ(GR(A)) is clear because GR(A)2is a subdigraph of GR(A). On the one hand, from digraph colouring, if χ(GR(A)) = n, there is a complete subdigraph of GR(A) with nvertices. On the other hand, note that the vertices that have only output arrows do not belong to GR(A)2,but it is impossible that neither of two connected vertices have an input arrow. Therefore, we have a complete subdigraph with n−1 vertices in GR(A)2,i.e., χ(GR(A)) ≤χ(GR(A)2) + 1 . 4. Non-associative algebras The goal of this work is to study some classical properties of algebras by analysing the structure of the associated graphs. It is worthwhile to point out that although the relationship between algebraic and discrete structures has been worked in other papers as we have referenced in the Introduction of this work, it has not been studied thoroughly the fact of obtaining algebraic properties of an algebra by analysing, mainly, the structure of its graphs. In this section, we will consider the defined digraphs in Section 2 by removing their weights, to deal only with the combinatorial structure of that digraphs. With these non-weight digraph, properties such as nilpotency and solvability will be tackled. We recall that the nilpotency and solvability study of an algebras is a main classical problem in Algebra, particularly, in Leibniz algebras case. It is known that every finite dimensional Lie algebra is decomposed into a semi-direct sum of a semisimple subalgebra and its solvable radical, we refer Levi’s Theorem in [9] to the reader. Moreover, the study of solvable Lie algebras is reduced to the study of nilpotent algebras, see Mal’cev’s results in [11]. Recently, Barnes in [2] has proved an analogue of Levi’s Theorem for Leibniz algebras, namely, a Leibniz algebra is decomposed into a semi-direct sum of its solvable radical and a semisimple Lie algebra. Although Proposition 2 proves that any digraph is associated to a certain algebra, we have verified that there exist digraphs that can not be realised as a digraph associated to some algebra under some specific conditions. We have the following result. Proposition 4. There are digraphs which are not action to the right digraph (respectively, action to the left digraph) associated to Leibniz algebras. In particular, let Gbe the digraph with nvertices and arrows given by loops, one for each vertex. Then there is no Leibniz algebra whose associated action to the right digraph(respectively, action to the left digraph) is G. Proof Suppose Ais a Leibniz algebra and G(as above) its associated action to the right digraph. We have: [xj, xi] = cj jixj,1≤i6=j≤n, where, for each jit must exist ijsuch that cj j ij6= 0. By applying the Leibniz identity Leib(xk, xj, xij),for every jand j+ 1 ≤k≤n, we obtain that ck k j = 0 for all k. Therefore we have proved that the digraph Ghas not a loop in the vertex xk,which is a contradiction. Considering Gthe action to the left digraph of A, the proof is analogous. Proposition 5. Let Abe a Lie algebra. Then GL(A)≈GR(A). Proof We observe that [x, x] = 0 and [x, y] = −[y, x] for x, y ∈Asince Ais a Lie algebra. Therefore, the anty-symmetric property of the product in Atell us that GL(A)≈GR(A) since edges leaving xireach the same set of vertices that the edges leaving xjwhenever [xi, xj] is a nonzero element of A, for any pair (xi, xj) in the fixed basis of A. CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 9 Remark 3. In general a digraph verifying the above conditions does not correspond to a Lie algebra. For instance, the non–Lie Leibniz algebra presented below satisfies those hypothesis. Example 4. Let Abe a 3-dimensional Leibniz algebra given by the following relations with respect to the basis {x1, x2, x3}: [x1, x2] = αx3, [x2, x1] = βx3, αβ 6= 0, β 6=−α. We have that Ais a non-Lie Leibniz algebra and it satisfies that GL(A)≃GR(A). Let us see the first result of this work that, looking into the digraph, talks about nilpotency. It is important to point out that the property required to the graph: to be a acyclic digraph, is very useful and commonly used in Graph Theory since it is possible to check whether a given directed graph is acyclic in linear time. Theorem 5. Let Abe an algebra and GR(A)its associated digraph. If GR(A)has no oriented cycles then the lower central series converges to {0}. Particularly, if Ais a Leibniz algebra, then Ais nilpotent. Proof Since GR(A) has no oriented cycle, the topological sorting is possible. Let B={x1,1, x1,2,..., x1,n1, x2,1, x2,2, . . . x2,n2, . . . , xm,1, . . . , xm,nm}be a basis of A which respects the order of the topological sorting (see Figure 5). Figure 5. Topological Ordering The law of the algebra is defined by the products xi,j ∗xp,q = m X r=1 nr X s=1 Cr,s i,j;p,qxr,s!,for all xi,j, xp,q, xr,s ∈ B.Moreover, due to the topological sorting, we assert that Cr,s i,j;p,q = 0,∀r≤i. Therefore xi,j ∗xp,q = m X r=i+1 nr X s=1 Cr,s i,j;p,qxr,s. We claim that for every x∈Asuch that x= m X r=1 nr X s=1 αr,sxr,s!with α1,s 6= 0 for some s∈N, we have x /∈A2.In fact, if x∈A2then ∃y, z ∈Asuch that y∗z=x. Since one may write y, z with respect to the elements of the basis as follow: y= m X r=1 ( nr X s=1 βr,sxr,s) z= m X r=1 ( nr X s=1 γr,sxr,s), 16 R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ Since each edge belongs to one of these configurations, all of them have a defined colour and belong to one subdigraph with the configuration of Figure 11. Therefore, by Theorem 3, there is an algebra whose union digraph is isomorphic to G. Note that could be more than one algebra associated to this digraph G. As we said at the beginning of this section, we want to give one step more and analyse the structure of a graph (without neither orientation or weight) in order to study its associated algebra. That is possible thanks to the previous results and mainly, to the following corollary. Corollary 10. Let Gbe a non-oriented and non-weighted graph. If each edge in Gbelongs to a cycle with length three, then there exists an orientation of edges of G, a weight function, and a colour function such that Gequipped with them is the union digraph of (at least) one algebra. Proof Thanks to the previous corollary, it remains to show that it is possible to orient the edges in a way that each of them belongs to a triple of vertices with the configuration given in Figure 11. We will make a constructive proof. Firstly, select an edge and give it any orientation. Then, keep selecting edges and orienting them following the instructions below: •Select one of the 3-cycles that contains this edge. •If the 3-cycle has one oriented edge, define any orientation to other edge. •If the 3-cycle has two oriented edges, it is always possible define an orientation to the third edge in order to satisfy the allowed configuration (see Figure 11). In this orientation, each edge belongs to at least one cycle with the configuration in Figure 11. In fact, the possibilities to two oriented edges in a cycle with length three are shown in Figure 14 (a) Possibility 1 (b) Possibility 2 (c) Possibility 3 Figure 14. Possibilities to 3-cycles with two edges oriented It is enough to apply the previous corollary to finish the proof. Finally, let us see as some important algebraic properties of an algebra can be studied by analysing the union digraph. Previously, we need to introduce the fine sequence (a sequence of subdigraphs). Definition 5. Let Gbe the union digraph associated to the algebra Aand Ξits colour function. The fine sequence of Gis the sequence of subdigraphs G(1), G(2), ..., G(n)started at G(1) =Gand such that G(i+1) is obtained from G(i)by the following procedure: •Remove all vertices that are not the endpoint of an edge which image by the function π2◦Ξis equal to 1. •Set as zero the coordinates of images of function Ξthat does not satisfy the Lemma 3. •Remove all edges such that the image by function Ξis (0,0,0). •If G(i+1) is not longer a digraph (if there is no more vertices), then make G(i+1) =G(i). Remark 8. The fine sequence is non–increasing with the ordering relation ⊆(to edge set or to vertex set). Since the digraph is finite, the sequence stabilizes. It is worthwhile to point out the importance of the fine sequence, because it allows to describe a method to recognise soluble Leibniz algebras. We formalize this result in the following theorem. CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 17 Theorem 11. Let Gbe the union digraph of an algebra A. If the fine sequence stabilizes in a digraph without edges, then the derivated series of Astabilizes in zero. Proof Let B={e1, . . . , en}be a basis of Aand y=Pn i=1 αieian element of A(k). Firstly, we will prove that for any i∈[1, n] such that αi6= 0, ei∈V(G(k)).This proof will be done by induction on k. For k= 1 the assertion is clear and suppose that it is true for k. On one hand, recall that in step kof the fine sequence, vertices of Bthat do not belong to the linear combinations of elements of A(k) are excluded. On the other hand, the colours of edges generated by these vertices can be removed. From definition of the fine sequence we know that the remove vertices from G(k)to G(k+1) are those whose indegree are zero. By induction, the product of any elements of A(k)is represented in G(k). Hence, the vertices with indegree zero are vertices that are not in the linear combinations of A(k)∗A(k) and then, they do not belong to A(k+1).Therefore A(k+1) ⊂V(G(k+1)). Finally, if the fine sequence stabilizes at a digraph G(n)without edges, there are no products between the elements represented by remaining vertices, therefore A(n+1) ={0}. Remark 9. Even though this algorithm covers all cases of Theorem 7, it is important to note that this algorithm is not a characterization, since there are algebras whose derivated series estabilize in zero but the fine sequence of its associated digraph does not estabilize in a digraph without edges. Let us see that in the following example. Example 9. Let Abe the 4-dimensional algebra given by the following table of multiplication. A:         [x1, x2] = x3+x4, [x2, x1] = −x3−x4, [x3, x4] = x3+x4, [x4, x3] = −x3−x4. The derivated series of Ais given by: A[1] =hx1, x2, x3, x4i; A[2] =hx3, x4i; A[3] ={0}; The union digraph of Aand its fine sequence are showed in Figures 15 and 16. For the sake of simplicity, we will omit the weights of the edges. Figure 15. GU(A)(1) =GU(A) Figure 16. GU(A)(2) =GU(A)(n)∀n≥2. 18 R.M. AQUINO, L.M. CAMACHO, E.M. CA˜ NETE, C. CAVALCANTE, A. M´ ARQUEZ It is clear that the hypothesis of the previous theorem is not obtained, because even the derivated series of Astabilises in zero, the fine sequence stabilises in GU(A)(2), which is a digraph with edges, as we can see in the Figure 16. 6. Concluding remarks and open questions In this paper we have introduced new and natural methods to associate (directed) graphs to algebras. Regarding previous works as that of Carriazo et al. [5], our method has two advantages: on one hand, we have not the restriction of considering only Lie algebras and our graphs can be associated to any algebra. And, on the other hand, by checking very simple properties of graphs (as to be a directed acyclic graph, fact that can be checked algorithmically in linear time) we can deduce important properties of the original algebra (as to be a nilpotent algebra in the case of directed acyclic graph). Additionally, in our weighted digraphs, we store all the information of the original algebra, thus this can be rebuild completely from the discrete structures. Of course, this work must be considered as a first step toward a more general goal. So, we thing that more properties of the algebras can be extracted by examining the associated digraphs. Finally, it is known that the classification of algebras up to isomorphism is a extremely difficult task, so we know only results in some particular cases and in low dimensions (see [3], [4], and [12] for more details). Although, the problem of graph isomorphism is not known to be a polynomial problem, in practice, it is much easier to solve that the equivalent in algebraic structures, so we think that it can be worthy to study what kind of partition is obtained if we group the algebras with the same (or similar under some discrete property) associated digraphs. Acknowledgement: The authors want to thank Antonio Ca˜nete, Isabel Fern´andez, Antonio Gonz´alez, and Juan Vicente Guti´errez for a very fruitful help. References [1] Auslander M., Reiten I., Smalo S.O., Representation theory of Artin algebras Cambridge Studies in Advanced Mathematics, Cambridge University Press, vol. 36, 1995. [2] Barnes D.W., On Levi’s theorem for Leibniz algebras, Bull. Aust. Math. Soc., vol. 86(2), 2012, 184–185 [3] Casas J.M., Ladra M., Omirov B.A., Karimjanov I.A. Classification of solvable Leibniz algebras with null-filiform nilradical, Linear and multilinear Algebra, vol. 61(6), 2013, 758–774. [4] Ca˜nete E.M., Khudoyberdiyev A. Kh. The classification of 4-dimensional Leibniz algebras, Linear Algebra and its Applications, vol. 439, 2013, 273–288. [5] Carriazo A., Fern´andez L.M., N´u˜nez J. Combinatorial structures associated with Lie algebras of finite dimension, Linear Algebra and its Applications, vol. 389, 2004, 43–61. [6] Diestel R., Graph Theory, Springer-Verlag, New York, 1997 (Ed. 2000). [7] Dynkin E. B., The structure of semi-simple algebras, Uspehi Matem. Nauk, (N.S.) (in Russian), vol. 4(20), 1947, 59—127. [8] Godsil C., Royle G., Algebraic Graph Theory, Springer Verlag, 2013. [9] Jacobson N., Lie algebras, Interscience Publishers, Wiley, New York, 1962. [10] Loday J.-L., Cyclic Homology, Cround. math. Wiss, Springer Verlag, vol. 301, 1992. [11] Mal’cev A., On solvable Lie algebras, (in Russian), Bull. Acad. Sci. URSS, Ser. Math. (Izvestia Akad. Nauk SSSR), vol. 9, 1945, 329–356. [12] Rikhsiboev I.M., Rakhimov I.S., Classification of three dimensional complex Leibniz algebras, The 5th International Conference on Research and Education in Mathematics AIP Conf. Proc. 1450, 2012, 358-362 (doi: 10.1063/1.4724168). [13] Sedgewick R., Wayne K. D., Algorithms, Addison-Wesley Professional, 4th ed., 2011. [Aquino R.M.] Dpto. de Matem´ atica. Universidade Federal do Espirito Santo. Vitoria (Brasil) E-mail address:[email protected] [Camacho L.M. — M´ arquez A.] Dpto. Matem´ atica Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes, s/n. 41012 Sevilla. (Spain) E-mail address:[email protected] --- [email protected] [Ca˜ nete E.M.] Instituto Matem´ atica. Universidade Federal de Alagoas. Macei´ o (Brasil) E-mail address:[email protected] CLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 19 [Cavalcante C.] Instituto de Ciˆ encias Matem´ aticas e Computac¸˜ ao. Universidade de S˜ ao Paulo. S˜ ao Paulo (Brasil) E-mail address:[email protected]