Classical properties of algebras using a new graph association
R.M. Aquino, L.M. Camacho, E.M. Cañete, C. Cavalgante, A. Márquez
CCLASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPHASSOCIATION
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 definedfive different weighted digraphs associated to a finite dimensional algebra over a field in order totackle important properties of the associated algebras, mainly the nilpotency and solvability in thecase 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 contri-butions to this topic have made, for instance, in Graph Theory, one may consider matrices associatedto a graph in order to obtain properties of those graphs as a consequence of the study of the spectrumof the adjacency matrix or the Laplacian matrix of such graph. One may also recall the case of Cayley’sgraphs associated to groups. We refer to [8] for further details. It is also well known the classificationof semi-simple Lie algebras by Dynkin diagrams, for the case of finite dimensional Lie algebras overalgebraically closed fields (see [7]). In [1] the studies of quiver algebras and representation theory offinite dimensional associative algebras over a field are shown. Finally it is worthwhile to point outCarriazo, Fern´andez and N´u˜nez’s paper [5], where the authors have developed a method to associate adiscrete structure to a finite dimensional Lie algebra that lead them to presente interesting applicationsconcerning solvability and nilpotency of those algebras. The problem of that work is that the authorsstrongly applied the identity of Jacobi as a fundamental tool to obtain their results, thus their resultsare 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, inorder to obtain algebraic properties of the algebra by looking the properties of the graph. The aimof this work is to answer this question in the affirmative. Specially, in this paper we are going toshow that given a finite dimensional algebra over a field (and a fixed basis), it is possible to associatesome weighted oriented finite graphs, which allow us to recover completely the algebra, i.e.,to recoverthe product structure of the algebra. But the main contribution of this work is that if we drop theinformation about the labels (obtaining thus more simple digraphs), one obtains interesting algebraicproperties of the algebra by analysing the properties of these simple digraphs. Furthermore, we succeedto apply our results to an important family of algebras: the Leibniz algebras, obtaining in this casesignificant 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. Liealgebras are related to Lie groups which are closely related to Yang-Mills’ theory, one of the best reasonto study Lie algebras hence Leibniz algebras.
The authors were supported by MCTI/CNPQ/Universal 10/2014, CAPES, and individually L.M.Camacho byMTM2016-79661-P (European FEDER support included) and A. M´arquez by BFU2016-74975-P. a r X i v : . [ m a t h . C O ] J un 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 byour 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 F and as digraph any oriented finite graph. In Section 1 we recollect the main definitions and resultsuseful to introduce our results. In Section 2 we define two classes of digraphs associated to a finitedimensional algebra A over a field, with a fixed basis and a given list of relations. We also show thatthe digraphs associated to Lie algebras in [5] are a particular case of the digraphs defined in our currentwork, in Subsection 2.1. In Section 3 some basic properties of general algebras and their associateddigraphs are given. In Section 4 interesting relationships between algebraic properties (as stability ofderived series and the lower central series) and digraphs are shown. A complete discussion of nilpotencyand solvability of Leibniz algebras is given in that section, in particular, in Theorem 5 , Proposition 7and Theorem 7. In Section 5 some new digraphs are introduced in order to preserve more informationabout the algebraic and discrete structures and achieve more general results. Finally, some interestingcounterexamples have been presented to provide the strength of our hypotheses on those results.1. Preliminaries
Throughout this paper F will denote an algebraically closed field with characteristic zero.We remark that any graph G throughout this paper is directed in the sense to be a digraph.Particularly, we will denote a (directed) graph G by a pair of sets ( V ( G ) , A ( G )) where V ( G ) is thenon-empty set of vertices and A ( G ) is a set of directed edges or arrows, presented by ordered pairs ofvertices x i and x j , denoted by ( x , x ). We notice that hypothesis fix as two the number of arrowsbetween two fixed vertices, one arrow to each direction. In this context, we will call two arrows betweenthe same pair of vertices by parallel arrows or parallel edges.We recall that the outdegree (respectively the indegree) of a vertex v ∈ G (denoted δ out G ( v ), or,respectively, δ in G ( v )) is the number of arrows in G starting at v (resp. ending at v ).Throughout this work, we consider the definition of cycles in graphs as usual, meaning, a closedpath over that graph (we ignore the orientation of the arrows). We will denote a cycle with length n by n -cycle. When necessary, we will consider cycles over oriented graphs, that is, a closed orientedwalk 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, itadmits an ordering of the vertices such that the starting endpoint of every arrow occurs earlier in theordering 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 H such that V ( H ) ⊆ V ( G ) and A ( H ) ⊆ A ( G ) subdigraphof G . This subdigraph is said induced by X ⊆ V ( G ) if V ( H ) = X and every arrow between elementsof X appearing in A ( G ) appears in A ( H ) as well.Furthermore, we will call two digraphs G and H isomorphic if there is a bijective function f : V ( G ) → V ( H ) satisfying that there exists ( v , v ) ∈ A ( G ) if and only if, there exists ( f ( v ) , f ( v )) ∈ A ( H ) . Wewill denoted G (cid:39) H. From the next definitions we follow [6]. A vertex colouring of a graph G is a map c : V → S suchthat c ( v ) (cid:54) = c ( w ) whenever v and w are adjacent. The elements of the set S are called the availablecolours and c is called colour map. The set of all vertices with one colour is independent and is calleda ”colour class”. An n -colouring of a graph G uses n colours; it thereby partions V into n colourclasses. The chromatic index of G, denoted by χ ( G ), is defined as the minimum n for which G has 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, forinstance, natural, real or complex numbers. This function and the graph associated to it are calledweight function and weighted graph respectively.We recall that an algebra A over a field F is a ring A = ( A, + , ∗ ) such that ( A, +) has a structureas vector space over F . We will denote shortly by A or by ( A, ∗ ) when it is necessary to identify theproduct of that algebra.Let A be an n -dimensional algebra over the field F . Let { x , x , · · · , x n } be a fixed basis of thevector space ( A, +) . Then A is determined, up to isomorphisms, by the multiplication rule for the basis LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 3 elements: x i ∗ x j = n (cid:88) k =1 γ kij x k , for γ kij ∈ F . We will take the same notations as used in Lie algebras bycalling those elements, the γ kij , as the structure constants of such algebra. Therefore, for a fixed basisof ( A, +), we can regard each algebra of dimension n over a field F as a point in the n -dimensionalspace of structure constants endowed with the Zariski topology. Recall that the Zariski topology of analgebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the caseof an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usualtopology, as every algebraic set is closed for the usual topology.As usual in Lie theory, the sequences A = A, A k +1 = A k ∗ A , k ≥ ,A [1] = A, A [ s +1] = A [ s ] ∗ A [ s ] , s ≥ , are said to be the lower central series and the derived series of A, respectively. We remark that thoseconcepts can be applied to any algebra.We recall that an algebra ( A, [ − , − ]) over a field F is called a Leibniz algebra if for any x, y, z ∈ A ,the Leibniz identity
Leib ( x, y, z ) : (cid:2) [ x, y ] , z (cid:3) − (cid:2) [ x, z ] , y (cid:3) − (cid:2) x, [ y, z ] (cid:3) = 0holds. In presence of anti-commutativity, Leibniz identity becomes Jacobi identity and therefore Liealgebras 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 Ann r ( A ), the right annihilator of the algebra A , defined asusual. Let g and h be two Leibniz algebras, a homomorphism of Leibniz algebras is a F -linear map φ : g −→ h such 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 A m = 0 (respectively, A [ t ] = 0). The minimal number m (respectively, t ) such that thisproperty is valid is called the nilindex (respectively, index of solvability) of the algebra A. 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 n with a fixed basis { x , . . . , x n } as follows. Definition 1.
Let the algebra ( A, ∗ ) over a field F and { x , . . . , x n } its basis. • Action to the right digraph: We denoted by G R ( A ) the following weighted oriented graph asdescribed below: (1) The vertices of the digraph are { x , . . . , x n } . (2) For each i ∈ { , . . . , n } , we consider the products x i ∗ x j = n (cid:88) l =1 c lij x l different to zero for j ∈ { , . . . , n } . Hence there exists an edge α il , oriented from x i to x l , if c lij (cid:54) = 0 , for j = 1 , . . . , n. (3) The weight of the edge α il is the vector ( c lij ) j . • We define dually the action to the left digraph, by considering the action on the left and wewill denoted that digraph by G L ( A ) . Remark 1.
We notice that those digraphs are well defined for any finite dimensional F -algebra andeven for locally finite dimensional algebras. Example 1.
Group algebras.
Let G be a finite group of order 2 and the group algebra A = F G associated to G for F being the complex number field. We fix a basis for A, { g , g } , where g and g are the elements of the group G such that g ∗ g j = g and g ∗ g i = g i for any i, j = 1 , . Hence wehave the associated digraphs to A shown in the following figures. Figure1. G R ( A ) Figure2. G L ( A ) R.M. AQUINO, L.M. CAMACHO, E.M. CA˜NETE, C. CAVALCANTE, A. M´ARQUEZ
Example 2.
In this example we will present a complete non-redundant list of digraphs (previosly defined) asso-ciated with a 2-dimensional Leibniz algebra over the field of the complex numbers. In order to achievethat, we need the following two results.
Proposition 1.
Let A be a 2-dimensional Leibniz algebra over C . Then A is isomorphic either theabelian algebra, denoted by A or to one of the non isomorphic algebras: A : (cid:110) [ x , x ] = x , A : (cid:110) [ x , x ] = x , A : (cid:40) [ x , x ] = x , [ x , x ] = − x . As the isomorphism between algebras is not transmitted to isomorphism between digraphs, we aregoing to show all 2-dimensional Leibniz algebras, that can be obtained from the above algebras bychange of bases.
Corollary 1.
Let A be a 2-dimensional Leibniz algebra over C . Then either A is the abelian algebra A or it is one of the following algebras: A : (cid:110) [ x , x ] = x , A : (cid:110) [ x , x ] = x , A : (cid:40) [ x , x ] = x , [ x , x ] = − x .A , : (cid:110) [ x , x ] = x , A , : [ x , x ] = − bx + b a x , [ x , x ] = − ax + bx , [ x , x ] = − ax + bx , [ x , x ] = − a b x + ax .a, b (cid:54) = 0 A , : (cid:40) [ x , x ] = x , [ x , x ] = x .A , : (cid:40) [ x , x ] = x , [ x , x ] = x . A , : (cid:110) [ x , x ] = x . A , : [ x , x ] = − dx + cx , [ x , x ] = − d c x + dx ,c, d (cid:54) = 0 .A , : [ x , x ] = − egf x + ex , [ x , x ] = − eg f x + egf x , [ x , x ] = − gx + fx , [ x , x ] = − g f x + gx ,e, f, g (cid:54) = 0 , g ( f − eg ) (cid:54) = 0 . A , : [ x , x ] = ix − hx , [ x , x ] = hx − h i x ,h, i (cid:54) = 0 . A , : (cid:40) [ x , x ] = x , [ x , x ] = − x .A , : [ x , x ] = − jx − kx , [ x , x ] = jx + kx ,j, k (cid:54) = 0 . We notice that those classifications will follow straightforward from solving the space of solutionsof the system of equations given by the structure constants and the equations provided by the Leibnizidentity.
Remark 2.
Note that the algebra A , and the algebras of the family A , are isomorphic to A , ; thealgebras A , , A , , A , and the algebras of the families A , , A , and A , are isomorphic to A ; and A , and the algebras of the family A , are isomorphic to A . The table below shows the action to the right digraphs associate to the 2-dimensional Leibnizalgebras. 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 wasfounded in [5] which leads us to improve the link between Lie algebras and Graph Theory given by theauthors in that paper. The digraphs defined above in
Definition 1 show us that the link between Liealgebras and Graph Theory can be generalized for any algebraic structure. Furthermore if we considerthe action to the left or to the right digraphs, results concerning solvability and nilpotency have beenobtained for general classes of algebras, by considering the algebraic structure and the properties ofthe digraphs instead of using the structure constants. We refer to Section 3, 4 and 5 to the reader.
LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 5 A A A A A , A , A , A , A , A , A , A , A , A , Table 1.
Classes of the action to the right grahps of 2-dimensional Leibniz algebrasIn fact, if we assume that the algebra A has the same hypothesis given in [5], that is, A is a Liealgebra defined by the law [ x i , x j ] = a i x i + b j x j , for a i , b j ∈ F , and we consider the action to the rightdigraphs G R ( A ) , similar results like the obtained in [5] can be achieved by following theirs proofs. Wenotice that the most important tool used to prove those results is the Jacoby identity, thus we followthe same steps to obtain the same kind of results as given by the authors in [5].Precisely, given a Lie algebra A under the hypotheses considered in [5], both the digraph G R ( A ) and G L ( A ) are a generalization of the graph presented in [5], provides that they represent the relationshipsbetween each operand and the result of the product, the same relationships investigated in [5]. Underthese hypothesis, the operands and the result of product lead us to the same elements, therefore thedigraphs contains the same information.We remark that for any Lie algebra A, G R ( A ) (cid:39) G L ( A ), see Proposition 5. Furthermore, recognizingthat G R ( A ) is equivalent to digraphs in [5] except for loops occurring at the going-in vertex and zerocoordinates in weight of digraphs action, we can see that equivalent results to those found in [5] canbe obtained by considering these digraphs.In order to clear up that the digraphs defined in previous section are the generalization of thegraphs defined in [5], we show as examples the following results, by denoting into parentheses thecorresponding results of [5]. It is worthwhile to note that in the following results of this subsection weare going to consider A as is a Lie algebra defined by the law [ x i , x j ] = a i x i + b j x j , for a i , b j ∈ F . Lemma 1. (Lemma 3.1 in [5] ) Let A be a Lie algebra associated to a digraph G R ( A ) . Then, theconfigurations of Figure 3 are allowed in G R for any three different vertices i, j, k. 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 G R ( A ). Furthermore, double edgesin [5] creates in G R ( A ) two loops, one for each vertex linked by double edge. The weight in [5] graphsis equivalent to the weights in G R ( A ) except for the signal, that is opposite in edges directed from thevertex with the highest index to lowest. Theorem 2. ( Theorem 3.2 in [5] ) Let G R ( A ) be the action to the right digraph associated with a Liealgebra A . If G R ( A ) has no 3-cycles, then G R ( 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 algebrasince that the one single coordinate nonzero of weights provided to each edge be the coordinate thatcorresponds to incoming vertex.
Theorem 3. (Theorem 3.6 in [5] ) Let G R ( A ) be a digraph containing 3-cycles associated with a Liealgebra. Then the following conditions holds: (1) The double edges of G R ( 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 G R ( A ) by removing its double edges and loops attached to itsatisfies 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 notcontain the second configuration of Figure 4 is solvable, but not nilpotent. Algebras and digraphs
In section 2 we have associated two digraphs with an algebra but, as we saw in Example 2, isomorphicalgebras can be associated to non isomorphic digraphs under these definitions. Hence, the following
LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 7 natural question rises up: when a digraph determines an algebraic structure? In this section, we willpresent basic relations between an algebra and their associated digraphs to attempt to answer thisquestion.
Proposition 2.
Let G be a non-weighted digraph. Then, there exists an algebra whose action to theright digraph (or action to the left digraph) is isomorphic to G with an associated weight. Proof
Let us see the proof for the action to the right digraph, for the action to left digraph isanalogous.Given a digraph G , define A as the linear envelope of V ( G ). From the definition of linear envelope, A is a vector space and V ( G ) is a basis of that vector space. Now, for each arrow ( x i , x j ) ∈ A ( G ), wedefine the following multiplication x i ∗ x j = x j in A . It is clear that if we extend that operation bybilinearity we will obtain an algebra such that its associated action to the right digraph is isomorphicto G with the corresponding weight.Some basic properties related to an algebra and its associated digraphs are given by the followingresult. Lemma 2.
Let A be an algebra, and let G R ( A ) , and G L ( A ) be their associated digraphs. Then, thefollowing properties hold: • An element x of the basis of A belongs to Ann ( A ) if and only if δ outG R ( A ) ( x ) = δ outG L ( A ) ( x ) = 0 . • An element x of the basis of A belongs to Ann r ( A ) if and only if δ outG L ( A ) ( x ) = 0 . Proof
The proof follows straightforward from the definitions of the annihilator of A, the right anni-hilator of A, and the considered digraphs. Definition 2.
Let A be an algebra and B = { x , ..., x n } a basis to A . We will use the notation G R ( A ) to indicate the subdigraph of G R ( A ) induced by the vertices that belong to W = { x k ∈ B : ∃ y ∈ A such that y = (cid:80) ni =1 α i x i with α k (cid:54) = 0 } . Respectively, G L ( A ) will be denote the subdigraphof G L ( A ) induced by vertices that belong to W .This definition is equivalent to the subdigraph induced by all the vertices with nonzero indegree in G R ( A ) or, respectively, in G L ( A ) . Example 3.
Let A be the 4-dimensional algebra given by the following table of multiplication. A : [ x , x ] = 2 x − x , [ x , x ] = − x , [ x , x ] = x , [ x , x ] = − x . The action to the left and to the right digraphs and its subdigraphs G L ( A ) and G R ( A ) arepresented in the next figure. (a) G L ( A ) (b) G R ( A )(c) G L ( A ) (d) G R ( A ) R.M. AQUINO, L.M. CAMACHO, E.M. CA˜NETE, C. CAVALCANTE, A. M´ARQUEZ G R ( A ) and G L ( A ) are useful to obtain an upper and a lower bound of the chromatic index of G R ( A ) and G L ( A ) . Proposition 3.
Let A be an algebra. The following statements about its associated digraphs are true: χ ( G R ( A ) ) ≤ χ ( G R ( A )) ≤ χ ( G R ( A ) ) + 1 χ ( G L ( A ) ) ≤ χ ( G L ( A )) ≤ χ ( G L ( A ) ) + 1 Proof
It is enough to prove the first statement, the second one follows analogously.The bound χ ( G R ( A ) ) ≤ χ ( G R ( A )) is clear because G R ( A ) is a subdigraph of G R ( A ) . On the one hand, from digraph colouring, if χ ( G R ( A )) = n , there is a complete subdigraph of G R ( A )with n vertices. On the other hand, note that the vertices that have only output arrows do not belongto G R ( A ) , but it is impossible that neither of two connected vertices have an input arrow. Therefore,we have a complete subdigraph with n − G R ( A ) , i.e., χ ( G R ( A )) ≤ χ ( G R ( A ) ) + 1 .4. Non-associative algebras
The goal of this work is to study some classical properties of algebras by analysing the structure ofthe associated graphs. It is worthwhile to point out that although the relationship between algebraicand discrete structures has been worked in other papers as we have referenced in the Introduction ofthis work, it has not been studied thoroughly the fact of obtaining algebraic properties of an algebraby analysing, mainly, the structure of its graphs. In this section, we will consider the defined digraphsin 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 inAlgebra, particularly, in Leibniz algebras case. It is known that every finite dimensional Lie algebra isdecomposed into a semi-direct sum of a semisimple subalgebra and its solvable radical, we refer Levi’sTheorem in [9] to the reader. Moreover, the study of solvable Lie algebras is reduced to the studyof nilpotent algebras, see Mal’cev’s results in [11]. Recently, Barnes in [2] has proved an analogue ofLevi’s Theorem for Leibniz algebras, namely, a Leibniz algebra is decomposed into a semi-direct sumof its solvable radical and a semisimple Lie algebra.Although Proposition 2 proves that any digraph is associated to a certain algebra, we have verifiedthat there exist digraphs that can not be realised as a digraph associated to some algebra under somespecific conditions. We have the following result.
Proposition 4.
There are digraphs which are not action to the right digraph (respectively, action tothe left digraph) associated to Leibniz algebras. In particular, let G be the digraph with n vertices andarrows given by loops, one for each vertex. Then there is no Leibniz algebra whose associated actionto the right digraph(respectively, action to the left digraph) is G . Proof
Suppose A is a Leibniz algebra and G (as above) its associated action to the right digraph. Wehave: [ x j , x i ] = c jji x j , ≤ i (cid:54) = j ≤ n, where, for each j it must exist i j such that c jj i j (cid:54) = 0 . By applying the Leibniz identity
Leib ( x k , x j , x i j ) , for every j and j + 1 ≤ k ≤ n, we obtain that c kk j = 0 for all k. Therefore we have proved that the digraph G has not a loop in the vertex x k , whichis a contradiction.Considering G the action to the left digraph of A , the proof is analogous. Proposition 5.
Let A be a Lie algebra. Then G L ( A ) ≈ G R ( A ) . Proof
We observe that [ x, x ] = 0 and [ x, y ] = − [ y, x ] for x, y ∈ A since A is a Lie algebra.Therefore, the anty-symmetric property of the product in A tell us that G L ( A ) ≈ G R ( A ) sinceedges leaving x i reach the same set of vertices that the edges leaving x j whenever [ x i , x j ] is a nonzeroelement of A, for any pair ( x i , x j ) in the fixed basis of A . LASSICAL 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 A be a 3-dimensional Leibniz algebra given by the following relations with respectto the basis { x , x , x } : [ x , x ] = αx , [ x , x ] = βx , αβ (cid:54) = 0 , β (cid:54) = − α. We have that A is a non-Lie Leibniz algebra and it satisfies that G L ( A ) (cid:39) G R ( A ) . Let us see the first result of this work that, looking into the digraph, talks about nilpotency. It isimportant to point out that the property required to the graph: to be a acyclic digraph, is very usefuland commonly used in Graph Theory since it is possible to check whether a given directed graph isacyclic in linear time.
Theorem 5.
Let A be an algebra and G R ( A ) its associated digraph. If G R ( A ) has no oriented cyclesthen the lower central series converges to { } . Particularly, if A is a Leibniz algebra, then A isnilpotent. Proof
Since G R ( A ) has no oriented cycle, the topological sorting is possible. Let B = { x , , x , , . . . ,x ,n , x , , x , , . . . x ,n , . . . , x m, , . . . , x m,n m } be a basis of A which respects the order of the topo-logical sorting (see Figure 5). Figure 5.
Topological OrderingThe law of the algebra is defined by the products x i,j ∗ x p,q = m (cid:88) r =1 (cid:32) n r (cid:88) s =1 C r,si,j ; p,q x r,s (cid:33) , for all x i,j , x p,q , x r,s ∈ B . Moreover, due to the topological sorting, we assert that C r,si,j ; p,q = 0 , ∀ r ≤ i. Therefore x i,j ∗ x p,q = m (cid:88) r = i +1 n r (cid:88) s =1 C r,si,j ; p,q x r,s . We claim that for every x ∈ A such that x = m (cid:88) r =1 (cid:32) n r (cid:88) s =1 α r,s x r,s (cid:33) with α ,s (cid:54) = 0 for some s ∈ N , wehave x / ∈ A . In fact, if x ∈ A then ∃ y, z ∈ A such that y ∗ z = x . Since one may write y, z withrespect to the elements of the basis as follow: y = m (cid:88) r =1 ( n r (cid:88) s =1 β r,s x r,s ) z = m (cid:88) r =1 ( n r (cid:88) s =1 γ r,s x r,s ) , hence the element x = y ∗ z can be written as follow: y ∗ z = [ m (cid:88) r =1 ( n r (cid:88) s =1 β r,s x r,s )] ∗ [ m (cid:88) u =1 ( n u (cid:88) v =1 γ u,v x u,v )]= m (cid:88) r =1 n r (cid:88) s =1 [ m (cid:88) u =1 ( n u (cid:88) v =1 β r,s γ u,v ( x r,s ∗ x u,v ))]= m (cid:88) r =1 n r (cid:88) s =1 ( m (cid:88) u =1 [ n u (cid:88) v =1 β r,s γ u,v ( m (cid:88) p = r +1 ( n p (cid:88) q =1 C p,qr,s ; u,v x p,q )))] . Thus every element x ∈ A satisfies that the structural constants associated with x ,s are equal tozero for all s. We will apply induction on k to prove that ∀ x ∈ A k +1 , x = m (cid:88) i = k +1 ( n i (cid:88) j =1 γ i,j x i,j ) . Since x ∈ A k +1 , wehave x = y ∗ z for any y ∈ A k and z ∈ A. Hence we have the following equalities. x = m (cid:88) i = k ( n i (cid:88) j =1 α i,j x i,j ) ∗ ( m (cid:88) p =1 ( n p (cid:88) q =1 β p,q x p,q ))= m (cid:88) i = k n i (cid:88) j =1 [ m (cid:88) p =1 ( n p (cid:88) q =1 α i,j β p,q ( x i,j ∗ x p,q ))]= m (cid:88) i = k n i (cid:88) j =1 ( m (cid:88) p =1 [ n p (cid:88) q =1 α i,j β p,q ( m (cid:88) r = i +1 ( n r (cid:88) s =1 C r,si,j ; p,q x r,s )))] . Therefore we have proved that the vertices of each k -line of the topological sorting of the digraph,that is, the elements that correspond to { x k − , , x k − , , . . . , x k − ,n k − } have coefficients nulls for eachelement of A k represented in chosen basis. In other words, elements of A k are a linear combination ofelements corresponding to lines bellow line k − A is a finite-dimensional algebra, there is a finite number of vertices on G R ( A ) hencea finite number of lines in G R ( A ) which we have denoted by m , in the topological sorting. Thereforethe elements from A m are written as linear combination of the m -th line elements, and these have notoutgoing edges, that is, A m +1 = A m ∗ A = 0 as we wish to prove. Corollary 6.
Let A be a Leibniz algebra and G R ( A ) its associated digraph. If G R ( A ) has no orientedcycles then the longest directed path in G R ( A ) is an upperbound of the nilindex of A . Remark 4.
We observe that in general the problem of finding the longest directed path in a graph isan NP–hard problem. However, if the digraph have not oriented cycles, it is known that one can solvethat problem in linear time. We refer, for instance, [13].Although there are digraphs with oriented cycles associated to algebras whose central lower seriesdo not stabilize, our next result ensures the existence of a basis in which the digraph associated with A has not oriented cycles for any algebra A with this feature. Thereby providing a characterization. Proposition 6.
Let A be an algebra whose lower central series stabilizes at { } . Then there exists abasis of A whose associated action on the right digraph has not oriented cycles. Proof
There are some m ∈ N such that A m = { } due to the lower central series of A stabilizes at { } . Let n be the smallest of these. It is clear that ∀ k ≤ n , A k (cid:40) A k − . Therefore, the dimensions of A s are decreasing.We will now proceed with a constructive proof. LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 11
Let b i − be a basis of A i − /A i , ∀ < i ≤ n, i ∈ N and let B be the basis consisting of union of b i .Since A n − /A n ∪ A n − /A n − ∪ ... ∪ A /A = A , B is a basis of A .Finally, it is possible define a topological ordering to the action to the right digraph associated to A and this basis with the vertices that represents elements in b i in the i -th row. Then this digraphcontains no oriented cycles.Our next result has established a necessary and sufficient condition to have finite dimensionalnilpotent Leibniz algebras with a multiplicative basis by considering the property of having no orientedcycles (of its associated action to the right digraph). Proposition 7.
Let A be an algebra admitting a multiplicative basis and G R ( A ) its action to the rightdigraph. Then, the lower central series of A converges to { } if and only if G R ( A ) has no orientedcycles. Particularly, if A is a Leibniz algebra, A is nilpotent if and only if G R ( A ) has no orientedcycles. Proof
We denote B = { x , x , . . . , x n } a fixed basis for A . Suppose that the lower central seriesconverge to { } . We assume that there exists a oriented cycle C = < x , x , . . . , x t > in G R ( A ) . Hencewithout lost of generality one can write the following equalities. x i ∗ − = β i x i +1 , ≤ i ≤ t − ,x t ∗ − = β t x , where − is some unknown element of B. It follows that x i ∈ A k for any k ≥ ∀ i ∈ { , · · · , t } . Thereforethere are k elements in each element of the lower central series, a contradiction with our assumption.Conversely we suppose that G R ( A ) has no oriented cycles, then the proof follows directly fromTheorem 5.We remark that the hypothesis on A to admit a multiplicative basis is crucial in Proposition 7. Infact, the following example show us a nilpotent Leibniz algebra A whose associated digraph G R ( A )has oriented cycles. Example 5.
Let A be the 4-dimensional Lie algebra given by the following table of multiplication. A : [ x , x ] = x + x , [ x , x ] = − x − x , [ x , x ] = x , [ x , x ] = − x − x , [ x , x ] = − x . One can verify that the nilindex of A is 3 and its associated digraph G R ( A ) has oriented cyclesgiven in Figure 6. Figure 6.
Action to the right digraph of ATheorem 5 proves that if the action to the right digraph of a Leibniz algebra A is an acyclic digraph,then A is nilpotent. In the next Theorem, the main result of this section, we are going to prove thatit is enough to check if either the action to the right digraph or the action to the left digraph of A isacyclic in order to know if A is solvable. Theorem 7.
Let A be an algebra. If either G R or G L has no oriented cycles then the derived seriesof A converges to { } . Particularly, if A is a Leibniz algebra, then A is solvable. Proof
By considering the Theorem 5, it is enough to prove that if G L ( A ) has no oriented cycles then A is solvable. In order to prove this statement, we define the following sequence. A (1) = A,A ( i ) = [ A, A ( i − ] for i > . It is clear that A (1) ⊆ A [1] . We claim that A ( k ) ⊆ A [ k ] ∀ k ≥ . This proof follows analogously to the proof of Theorem 5. Precisely, by using the topological sortingof G L ( A ) and the sequence A ( n ) , one can prove, for all k, that the elements of A ( k ) belonging to each k -line of the topological sorting of G L ( A ) have null coeficients in the linear combination representingthe elements of A ( k +1) in the specified basis. Finally, we consider the fact that G L ( A ) has a finitenumber of vertices. Hence there exists m > A [ m +1] = 0 . Remark 5.
We observe that the reciprocal is not true, in general. The 4-dimensional solvable non-nilpotent Leibniz algebra A given by A : [ x , x ] = x , [ x , x ] = x , [ x , x ] = − x . has associated digraphs with oriented cycles. Figure 7.
Action to theright digraph of A . Figure 8.
Action to the leftdigraph of A .5. The union digraph
The steps followed in this work have been firstly, to create a biunivocal relationship between weigtheddigraphs with finite dimensional algebra with a fixed basis, and secondly, to analise the digraph struc-ture in order to tackle and conclude from them some interesting algebraic properties of our algebra. Inthis section, we are going to give one step more, we are going to be able to obtain algebraic properties ofthe algebra by analysing the structure of the subjacent graph (without neither orientation or weight).In order to achieve that goal, we introduce new digraphs and sequence.
Definition 3.
Let ( A, ∗ ) be an algebra and { x , . . . , x n } a basis of A .The weighted digraph whose vertices are x , ..., x n and with only one arrow oriented from x i to x j to each non-zero product x i ∗ x j = (cid:80) nk =1 c kij x k with weight ( c ij , ..., c nij ) is called operands digraph andrepresented by G O ( A ) . Example 6.
Let A be a 4-dimensional algebra given by the following table of multiplication. A : [ x , x ] = 2 x − x , [ x , x ] = − x , [ x , x ] = x , [ x , x ] = 2 x . The operands digraph of A is shown in Figure 9. LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 13
Figure 9. G O ( A )Thanks to this definition, we can define the main digraph in this section: the union digraph. Definition 4.
Let ( A, ∗ ) be an algebra over a field F and { x , . . . , x n } a basis of A .The union digraph of A (denoted G U ( A ) ) is defined as the digraph with coloured edges whose verticesare { x , . . . , x n } and edges as follows. • The set of edges in G U ( A ) is the union of the sets of edges in G O ( A ) , G R ( A ) and G L ( A ) . • The weight of each edge is a vector with n coordinates, where each n coordinates correspondto the weight of this edge in G O ( A ) , G R ( A ) and G L ( A ) ) respectively. If an edge of the uniondigraph does not appear in one the digraphs G O ( A ) , G R ( A ) or G L ( A ) ) then, the weight of thisedge will have the corresponding n zeros. • The colouring of each edge is represented by a vector in Z where the digit in each coordinateindicates the existence of this edge in G O ( A ) , G R ( A ) or/and G L ( A ) . Example 7.
Let A be the algebra in Example 6, the union digraph of A is shown in Figure 10, wherethe red coloured label represents the image by colour function of each edge. Figure 10. G U ( A ). Remark 6.
We will use the notation π i ( v ) to denote the projection on the i -th coordinate of thevector v .Directly from definition of the union digraph, we have a fundamental result that will be useful forthe main result of this section. In Lemma 3 one can note the fact that the union digraph is related toan algebra makes that the digraph has a specific structure. Lemma 3.
Let G be a weighted digraph with weight function f : A ( G ) → F n and colour function Ξ : A ( G ) → Z . If G is the union digraph of an F -algebra A , then each edge of G belongs to asubdigraph with the configurations of the Figure 11. Figure 11.
General model to cell of the union digraph where (1) π (Ξ( a )) = π (Ξ( b )) = π (Ξ( c )) = 1 . Proof
This result is trivially obtained from the definitions of the digraphs G R ( A ), G L ( A ), G O ( A )and G U ( A ). In fact, each product e i ∗ e j = ... + c kij e k + ... is represented by one edge at G O ( A ) from e i to e j , one edge at G R ( A ) from e i to e k and one edge at G L ( A ) from e j to e k . Therefore, they aremutually dependent in the sense that one can not exist without the others. Remark 7.
If an element of a fixed basis of A plays more than one role in the product e i ∗ e j = ... + c kij e k + ... , we need to rewrite the configuration of Figure 11 as those in Figure 12. (a) Model to cell of theunion digraph when e i = e j (b) Model to cell of theunion digraph when e j = e k (c) Model to cell ofthe union digraph when e i = e k (d) Model to cell ofthe union digraph when e i = e j = e k Figure 12.
Alternative models to singular cases to cell of the union digraphNote that similarly to the general case, these cells have the following restrictions: • In Figure 12(a), π (Ξ( a )) = π (Ξ( b )) = π (Ξ( c )) = 1. • In Figure 12(b), π (Ξ( a )) = π (Ξ( b )) = π (Ξ( b )) = 1. • In Figure 12(c), π (Ξ( a )) = π (Ξ( c )) = π (Ξ( c )) = 1. LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 15 • In Figure 12(d), π (Ξ( a )) = π (Ξ( a )) = π (Ξ( a )) = 1.It is worthwhile to note that not every digraph with these properties and colour function can beassociated with a weight function such that it is a union digraph of some algebra. The next exampleshows this statement. Example 8.
Let G be the digraph coloured by function Ξ : A − > Z represented in Figure 13. Figure 13.
General model to cell of the union digraph.where Ξ( a ) = (0 , ,
0) ; Ξ( b ) = (1 , ,
1) and Ξ( c ) = (1 , , b would represent a non-null product e j ∗ e k since that π (Ξ( b )) = 1, but there are no edges where e k represents this product in G L ( A ) , giving rise to a contradiction.Therefore, the condition of Lemma 3 is not enough to give a characterization. Note that it occursdue to the presence of an extra colour on an edge which only satisfies the condition of Lemma 3when it is playing the role of another colour. Thus, the lemma can be rewritten in order to find acharacterization as follows. Theorem 8.
Let G be a digraph coloured by the function Ξ : A ( G ) − > Z and with n vertices. If thefollowing statements are true to any α ∈ A ( G ) : π (Ξ( α )) = 1 ⇒ exists a subdigraph following the settings in Figure 11 in which α plays the role of c ; π (Ξ( α )) = 1 ⇒ exists a subdigraph following the settings in Figure 11 in which α plays the role of a ; π (Ξ( α )) = 1 ⇒ exists a subdigraph following the settings in Figure 11 in which α plays the role of b ,then there exists a weight function f : A ( G ) → F n that can be associated to G, such that this newdigraph is the union digraph of an algebra A. Proof
The proof is given by construction.Let A be an algebra. For every vertex e i , there must be an element x i of the basis of A . On theother hand, to each edge α oriented from e i to e j playing the role of c in configuration of Figure 11,let { e α , e α , ..., e α p } be the list of vertex playing the role of e k in any subdigraph such that α playsthe role of c . Define the product x i ∗ x j = (cid:80) pt =1 x α t .Since all edges are in this configuration by hypotesis, each edge in A ( G ) occurs at the union digraphof A , and, by construction, it has the same orientation and colour of edge than in G . Corollary 9.
Let G be a digraph with n vertices. If each edge of G belongs to (at least) one subdigraphwith the configuration of Figure 11, then there exists a weight function and a colour function such that G eqquiped with them is the union digraph of an algebra. Proof
Let Ξ : A ( G ) − > Z be a function colour. On the one hand, for every edge playing the roleof c in this configuration, let define π (Ξ( c )) = 1. On the other hand, for every a i and b i living in thisconfiguration with c , let define π (Ξ( a i )) = π (Ξ( b i )) = 1. Finaly, set the remaining coordinates aszero. Since each edge belongs to one of these configurations, all of them have a defined colour and belongto one subdigraph with the configuration of Figure 11. Therefore, by Theorem 3, there is an algebrawhose 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 structureof a graph (without neither orientation or weight) in order to study its associated algebra. That ispossible thanks to the previous results and mainly, to the following corollary.
Corollary 10.
Let G be a non-oriented and non-weighted graph. If each edge in G belongs to acycle with length three, then there exists an orientation of edges of G , a weight function, and a colourfunction such that G equipped 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 ina 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, keepselecting 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 thirdedge 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 orientedIt 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 analysingthe union digraph. Previously, we need to introduce the fine sequence (a sequence of subdigraphs).
Definition 5.
Let G be the union digraph associated to the algebra A and Ξ its colour function.The fine sequence of G is the sequence of subdigraphs G (1) , G (2) , ..., G ( n ) started at G (1) = G andsuch 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 π ◦ Ξ isequal to 1. • Set as zero the coordinates of images of function Ξ that does not satisfy the Lemma 3. • Remove all edges such that the image by function Ξ is (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 vertexset). 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 amethod to recognise soluble Leibniz algebras. We formalize this result in the following theorem. LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 17
Theorem 11.
Let G be the union digraph of an algebra A . If the fine sequence stabilizes in a digraphwithout edges, then the derivated series of A stabilizes in zero. Proof
Let B = { e , . . . , e n } be a basis of A and y = (cid:80) ni =1 α i e i an element of A ( k ) . Firstly, we will prove that for any i ∈ [1 , n ] such that α i (cid:54) = 0 , e i ∈ V ( G ( k ) ) . This proof will be doneby 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 k of the fine sequence, vertices of B that 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 thosewhose 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 betweenthe elements represented by remaining vertices, therefore A ( n +1) = { } . Remark 9.
Even though this algorithm covers all cases of Theorem 7, it is important to note thatthis algorithm is not a characterization, since there are algebras whose derivated series estabilize inzero but the fine sequence of its associated digraph does not estabilize in a digraph without edges. Letus see that in the following example.
Example 9.
Let A be the 4-dimensional algebra given by the following table of multiplication. A : [ x , x ] = x + x , [ x , x ] = − x − x , [ x , x ] = x + x , [ x , x ] = − x − x . The derivated series of A is given by: A [1] = (cid:104) x , x , x , x (cid:105) ; A [2] = (cid:104) x , x (cid:105) ; A [3] = { } ;The union digraph of A and its fine sequence are showed in Figures 15 and 16. For the sake ofsimplicity, we will omit the weights of the edges. Figure 15. G U ( A ) (1) = G U ( A ) Figure 16. G U ( A ) (2) = G U ( A ) ( n ) ∀ n ≥ . It is clear that the hypothesis of the previous theorem is not obtained, because even the derivatedseries of A stabilises in zero, the fine sequence stabilises in G U ( 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 onehand, we have not the restriction of considering only Lie algebras and our graphs can be associatedto any algebra. And, on the other hand, by checking very simple properties of graphs (as to be adirected acyclic graph, fact that can be checked algorithmically in linear time) we can deduce importantproperties 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 thiscan 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 thingthat 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 difficulttask, so we know only results in some particular cases and in low dimensions (see [3], [4], and [12] formore 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 itcan be worthy to study what kind of partition is obtained if we group the algebras with the same (orsimilar under some discrete property) associated digraphs.
Acknowledgement:
The authors want to thank Antonio Ca˜nete, Isabel Fern´andez, AntonioGonz´alez, and Juan Vicente Guti´errez for a very fruitful help.
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E-mail address : [email protected] [Camacho L.M. — M´arquez A.] Dpto. Matem´atica Aplicada I. Universidad de Sevilla. Avda. ReinaMercedes, 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] LASSICAL PROPERTIES OF ALGEBRAS USING A NEW GRAPH ASSOCIATION. 19 [Cavalcante C.] Instituto de Ciˆencias Matem´aticas e Computac¸˜ao. Universidade de S˜ao Paulo. S˜ao Paulo(Brasil)
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