Finite-dimensional Zinbiel algebras and combinatorial structures

Finite-dimensional Zinbiel algebras and combinatorial structures Manuel Ceballos†,1, Juan N´u˜nez‡,´ Angel F. Tenorio§ †Dpto. de Ingenier´ıa. Universidad Loyola Andaluc´ıa. Av. de las Universidades, s/n, 41704 Dos Hermanas, Sevilla (Spain). ‡Departamento de Geometr´ıa y Topolog´ıa. Facultad de Matem´aticas. Universidad de Sevilla. Calle Tarfia, s/n. 41012-Seville (Spain). §Dpto. de Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica. Escuela Polit´ecnica Superior. Universidad Pablo de Olavide. Ctra. Utrera km. 1. 41013-Seville (Spain). [email protected] jnv[email protected] [email protected] Abstract In this paper, we study the link between finite-dimensional Zinbiel algebras and combinatorial structures or (pseudo)digraphs determining which configurations are associated with those algebras. Some properties of Zinbiel algebras that can be read from their associated combinatorial structures are studied. We also analyze the isomorphism classes for each configuration associated with these algebras providing a new method to classify them and we compare our results with the current classifications of 2and 3-dimensional Zinbiel algebras. We also obtain the 3vertices combinatorial structures associated with such algebras. In order to complement the theoretical study, we have designed and performed the implementation of an algorithm which constructs and draws the (pseudo)digraph associated with a given Zinbiel algebra and, conversely, another procedure to test if a given combinatorial structure is associated with some Zinbiel algebra. Keywords: Graph; combinatorial structure; Zinbiel algebra; algorithm; complexity 2010 Mathematics Subject Classification: 17A32; 05C25; 05C85; 05C90; 68W30; 68R10. 1Corresponding author 1 1 Introduction In these days, one of the most important and stimulating research in Science and, particularly, in Mathematics is finding and studying new links between different fields. Alternative techniques and procedures allow researchers to solve many open problems, improve known theories and achieve new results. This paper deals with the relation between Zinbiel algebras and Graph Theory. More concretely, our main goal is to continue with the research line started in [1, 3, 4], where a link between combinatorial structures and Lie or Leibniz algebras was established. Due to this link, several properties on those non-associative algebras can be translated into the field of Graph Theory and vice versa. Now, we want to extend these studies to the case of Zinbiel algebras. Non-associative algebras have been deeply studied due to its own theoretical importance and its many applications to different fields like Physics, Engineering or Applied Mathematics. A particular type of these algebras is formed by Zinbiel algebras. They were introduced by J.-L. Loday [7] in 1995. They are the Koszul dual of Leibniz algebras and, in fact, J.M. Lemaire (see [8]) proposed the name of Zinbiel as a mirror of Leibniz algebras. As happens with any class of non-associative algebras, there exist many general questions to be solved and these questions (as, for example, the classification of Zinbiel algebras) require alternative techniques since the traditional ones are not sufficient. Analogously, Graph Theory is also running at a very high level nowadays. Graphs have been used to deal with a wide range of problems in many different fields including non-associative algebras. For example, they have been really helpful in order to compute degenerations of Zinbiel algebras in [6]. In this way, we believe that graphs and simplicial complexes (its generalization to higher dimensions) might be an useful tool in the study of non-associative algebras, providing new ways to solve many open problems, like the above-mentioned classification problem of Zinbiel algebras. Hence, our main goal is to study the link between combinatoral structures and Zinbiel algebras, giving a generalization for the techniques introduced in [1] and then developed in [3, 4] to the case of Zinbiel algebras. We will also provide a new method to classify this type of algebras. The structure of this paper is the following: Section 2 focuses on reviewing some well-known results on Zinbiel algebras and Graph Theory. Then, Section 3 is devoted to the association between combinatorial structures 2 and Zinbiel algebras. In Section 4, we study some properties of Zinbiel algebras that can be read from their associated combinatorial structure. Next, Section 5 analyzes the structure of (pseudo)digraphs associated with Zinbiel algebras and some of their properties. For each configuration, we study both the solvability and the isomorphism classes for its associated Zinbiel algebra. In Section 6 we study the 3-vertex combinatorial structures including full triangles that are associated with Zinbiel algebras. In addition, we also show how our algorithm may be useful for the classification of Zinbiel algebras. Finally, Section 7 shows the implementation of the two algorithmic procedures used in the previous sections. The first one is devoted to check if a given combinatorial structure (not necessarily a (pseudo)digraph) is associated or not with a Zinbiel algebra and, conversely, the second one computes the (pseudo)digraph, if possible, associated with a given finite-dimensional Zinbiel algebra starting from its law. In addition, we give a brief computational study, showing the complexity order and computing time of the procedures here presented. We believe that the tools and results achieved in this paper might be useful to advance in the research line connecting Zinbiel algebras and Graph Theory. In addition, combinatorial structures may provide us with a new method to classify Zinbiel algebras. 2 Preliminaries For a general overview on Zinbiel algebras and Graph Theory, the reader can consult [7]. We only consider finite-dimensional Zinbiel algebras over the complex number field C. Definition 1. AZinbiel algebra Zis a vector space with a second bilinear inner composition law ([·,·]), called the bracket product or commutator, which satisfies [[A, B], C] = [A, [B, C]] + [A, [C, B]],∀A, B, C ∈ Z. The latter is called the Zinbiel identity. From here on, we use the notation Z(A, B, C) = [[A, B], C]−[A, [B, C]] −[A, [C, B]]. Given a basis {ei}n i=1 of Z, its structure (or Maurer-Cartan) constants are defined by [ei, ej] = Pn h=1 ch i,jeh, for 1≤i<j≤n. Definition 2. Given a Zinbiel algebra Z, its center is defined as Z(Z) = {X∈ Z | [X, Y ]=0,∀Y∈ Z}. 3 Definition 3. Given a finite-dimensional Zinbiel algebra Z, its derived series is Z1=Z,Z2= [Z,Z], . . . , Zk= [Zk−1,Zk−1], . . . Thus, Zis called solvable if there exists m∈Nsuch that Zm={0}. In addition, if Zm−1={0}also holds, then Zis (m−1)-step solvable. Definition 4. Given a finite-dimensional Zinbiel algebra Z, its central series is Z1=Z,Z2= [Z,Z], . . . , Zk= [Zk−1,Z], . . . Thus, Zis called nilpotent if there exists m∈Nsuch that Zm={0}. In addition, if Zm−1={0}also holds, then Zis (m−1)-step nilpotent. Remark 1. Every nilpotent algebra is trivially solvable because Zi⊆ Zi, for all i∈N. Remark 2. The derived algebra of a Zinbiel algebra Zwill be denoted by DZ=Z2=Z2. Although the reader can consult [5] as an introductory reference to Graph Theory, some notions are recalled next. Definition 5. Adigraph consists in an ordered pair G= (V, E), where V is a non-empty set called vertex-set and Eis a set of ordered pairs (edges) of two vertices, called edge-set. Definition 6. Aloop in the digraph Gis an edge that connects a vertex with itself. If the digraph Gcontains loops, then Gis called a pseudodigraph. Throughout the paper, we consider (pseudo)digraphs admitting double edges. Definition 7. Given a digraph G= (V, E), a vertex v∈Vis a sink (resp. asource) if each edge incident with vis oriented towards v(resp. from v). See Figure 1. Definition 8. A vertex vin Gis said to be simple if vhas no loop. Otherwise, it will be non-simple. 4 Figure 1: Example of sink and source, respectively. 3 Associating combinatorial structures with Zinbiel algebras Let Zbe a n-dimensional Zinbiel algebra with basis B={ei}n i=1 and law [ei, ej] = Pn h=1 ch i,jeh. The pair (Z,B) can be associated with a combinatorial structure by following the procedure introduced in [4, Section 3] for Leibniz algebras. Therefore, every Zinbiel algebra with a fixed basis can be associated with a combinatorial structure. This method for Zinbiel algebras provides a generalization of the one described in [1] for Lie algebras, as we prove in the following. Example 1. The 3-dimensional Zinbiel algebra with multiplication law [e1, e1] = e2−e3,[e1, e2]=[e1, e3] = −e2+e3is associated with the combinatorial structure shown in Figure 2. Figure 2: Combinatorial structure associated with a 3-dimensional Zinbiel algebra. 5 4 Reading properties from the combinatorial structure In this section, given a Zinbiel algebra Z, we analyze the properties of Z algebras that can be read from its associated combinatorial structure, G. More concretely, we characterize the case when Zis a Lie algebra and then we study the center, Z(Z), and the derived algebra DZ. Proposition 1. Let Gbe the combinatorial structure associated with a Zinbiel algebra Z. Then Zis a Lie algebra, if Gsatisfies the following conditions 1. There are no loops. 2. The weight for the edge from vertex ito vertex jis given by (cj i,j,−cj i,j). 3. The weight for the edges in a full triangle ijk is given by (ck i,j,−ck i,j), (ci j,k,−ci j,k)and (cj i,k,−cj i,k). Proof. Trivial from the self-annihilation and the skew-symmetry of the commutator. Remark 3. Given an edge in a combinatorial structure associated with a Zinbiel algebra, both coordinates in Conditons 2 and 3 from Proposition 1 are opposite each other and, then, only one coordinate is required for saving the information of the structure constants as happened in [1]. Example 2. The combinatorial structure of Figure 3 is associated with the 3-dimensional Lie algebra with law [e1, e2] = −[e2, e1] = −2e2+e3,[e1, e3] = −[e3, e1] = −e2+e3. Figure 3: Combinatorial structure associated with a 3-dimensional Lie algebra. 6 Proposition 2. Let Gbe the combinatorial structure associated with a Zinbiel algebra Z. Then, Z(Z)⊃span{ei|i∈Γ} ∪ span{ej|j∈Λ},where Γis the set of simple and isolated vertices of Gand Λis the set of simple vertices in full triangles that are adjacent only to ghost edges. Proof. Let us assume that G= (V, E) with V={1, . . . , n}, therefore B= {ek}n k=1 is a basis of Z. First, if the vertex i∈Vis a source then ∃j∈V such that (cj i,j, cj j,i)= (0,0). Consequently, ei/∈Z(Z). In case that i∈V is a simple and isolated vertex then ei∈Z(Z) since [ei, ej] = [ej, ei] = 0, ∀1≤j≤n. Now, we suppose that i∈Vis a simple vertex in a full triangle which is adjacent only to ghost edges. In that case, [ei, ej]=[ej, ei] = 0, ∀1≤j≤n. Therefore, ei∈Z(Z). Remark 4. Notice that the center of a Zinbiel algebra can be given by a linear combination of the basis vectors associated to the type of vertices indicated in Proposition 2. For example, the 3-dimensional Zinbiel algebra Zof Example 1 verifies Z(Z) = span{e2−e3} Proposition 3. Let Gbe the combinatorial structure associated with a Zinbiel algebra Z. Then, the derived algebra of Zis given by DZ= span (X i∈Υ ch i,ieh)∪span ncj i,jej+ck i,jek, cj j,iej+ck j,iek|j∈Πo,where Υis the set of non-simple vertices (vertices with a loop) and Πis the set of vertices that are not sources. Proof. Let us assume that G= (V, E) with V={1, . . . , n}, therefore B={ek}n k=1 is a basis of Z. First, if iis a non-simple vertex, then [ei, ei] = n X h=1 ch i,ieh. Consequently, n X h=1 ch i,ieh∈DZ. From now on, we will only consider simple vertices. In case that iis a source, then ci j,i =ci i,j = 0, ∀1≤j≤n. Therefore, ei/∈DZ. If iis not a source, then ∃jwith 1 ≤j≤n such that (ci i,j, ci j,i)= (0,0). Consequently, ci i,j ei, ci j,i ei∈DZ. Moreover, if iis a vertex in a full triangle with vertices {i, j, k}, then it may happen that (ck i,j, ck j,i)= (0,0), so we would have to consider the terms ci i,j ei+ck i,j ekand ci j,i ei+ck j,i ek. 7 5 Zinbiel algebras and (pseudo)digraphs In this section, we study the structure of (pseudo)digraphs associated with Zinbiel algebras. For each case, we analyze the type of Zinbiel algebra according to its solvability and isomorphism class. Let Zbe a Zinbiel algebra with basis Bsuch that the combinatorial structure Gassociated with (Z,B) consists of a (pseudo)digraph; that is, there are no triangles in G. This assertion is equivalent to affirm that the law of Zwith respect to the basis B={ei}n i=1 is given by [ei, ej] = ci i,jei+cj i,jej,1≤i=j≤n; [ek, ek] = n X h=1 ch k,keh(1) and the rest of products are null. Proposition 4. The unique (pseudo)digraph associated with a 1-dimensional Zinbiel algebra is that formed by an isolated vertex. Remark 5. Clearly, there is only one isomorphism class, Z1, which corresponds to the 1-dimensional abelian Zinbiel algebra. Proposition 5. If Gis a (pseudo)digraph of 2 vertices, then Gis associated with a 2-dimensional Zinbiel algebra Zif and only if Gis isomorphic to configurations a),b)or j)in Figure 4. The rest of configurations cannot be associated with Zinbiel algebras. Figure 4: Pseudodigraphs with two vertices. Proof. Figure 4 includes all the possible (pseudo)digraphs of two vertices. Configuration c) cannot be associated with a Zinbiel algebra since Z(e1, e1, e1) = Z(e1, e1, e2) = Z(e2, e2, e2) = 0 implies that there is no loop on vertex 1. Configurations d), e) and f) cannot be associated with a Zinbiel algebra 8 since Z(e1, e2, e1) = Z(e2, e1, e1) = 0 implies the non-existence of edges from vertex 1 into vertex 2. Finally, configurations g), h) and i) cannot be associated with a Zinbiel algebra since Z(e2, e1, e2) = Z(e1, e2, e2) = 0 implies the non-existence of edges from vertex 2 into vertex 1. Any other configuration in Figure 4 is associated with a Zinbiel algebra if and only if the following restrictions hold for each of them i) Configuration a): No constraints. ii) Configuration b): c1 1,1= 0 ∧c2 1,1= 0. iii) Configuration j): c1 1,1=(c1 2,1)2 c1 2,2∧c2 1,1=−(c1 2,1)3 (c1 2,2)2∧c1 1,2=c1 2,1∧c2 1,2= −(c1 2,1)2 c1 2,2∧c2 2,1=−(c1 2,1)2 c1 2,2∧c2 2,2=−c1 2,1. Proposition 6. Under the assumptions in Proposition 5,  Configuration a)is associated with the abelian 2-dimensional Zinbiel algebra.  Configurations b)and j)are associated with 2-step nilpotent Zinbiel algebras. Proof. Let Zbe the Zinbiel algebra associated with Configuration b). From Proposition 5, Z2=Z2= span(e2) is an abelian ideal and, hence, Z3= Z3={0}. Consequently, Zis 2-step nilpotent. On the other hand, if Z denotes the Zinbiel algebra associated with Configuration j), Proposition 5 implies the following Z2=Z2= span (c1 2,1)2 c1 2,2 e1−(c1 2,1)3 (c1 2,2)2e2, c1 2,1e1−(c1 2,1)2 c1 2,2 e2, c1 2,2e1−c1 2,1e2!=span e1−c1 2,1 c1 2,2 e2! Moreover, "e1, e1−c1 2,1 c1 2,2 e2#=(c1 2,1)2 c1 2,2 e1−(c1 2,1)3 (c1 2,2)2e2−c1 2,1 c1 2,2 c1 2,1e1−(c1 2,1)2 c1 2,2 e2!= 0 and "e2, e1−c1 2,1 c1 2,2 e2#= [e2, e1]−c1 2,1 c1 2,2 [e2, e2] = 0 Therefore, Z3=Z3={0}and Zis a 2-step nilpotent Zinbiel algebra. 9 Conversely, we suppose that Gis a (pseudo)digraph with nvertices (n > 3) and associated with a Zinbiel algebra. We prove that Gmust be as described in (i) or (ii). Let {i, j, k}be three arbitrary vertices of G. According to Proposition 8, we only have 4 possible allowed configurations for these three vertices. Moreover, we cannot have two double edges from three vertices since Configuration 6) from Figure 6 is forbidden. Consequently, and from these considerations, it follows that Gmust be a set of nisolated vertices with a maximum of (n−1) loops or a (pseudo)digraph formed by a double edge with loops plus isolated vertices. Finally, if Gis formed by isolated vertices with no loops, then we obtain an abelian Zinbiel algebra and otherwise, we can use an analogous reasoning to that considered in the proof of Proposition 9 to prove that Zinbiel algebras are 2-step nilpotent. 6 Combinatorial structures of three vertices associated with Zinbiel algebras In this section, we study the combinatorial structures of 3 vertices including full triangles and being associated with Zinbiel algebras. We also analyze the isomorphism classes for those configurations. To do so, we consider a set of three vertices, {i, j, k}, and define a vector space Vendowed with basis {ei, ej, ek}and law given by the following brackets [eh, eh] = ci h,hei+cj h,hej+ck h,hek,for h=i, j, k; [ei, ej] = ci i,jei+cj i,jej+ck i,jek,[ej, ei] = ci j,iei+cj j,iej+ck j,iek, [ei, ek] = ci i,kei+cj i,kej+ck i,kek,[ek, ei] = ci k,iei+cj k,iej+ck k,iek, [ej, ek] = ci j,kei+cj j,kej+ck j,kek,[ek, ej] = ci k,jei+cj k,jej+ck k,jek. where the structure constants may be zero or not. The main difficulty in this study consists in determining under what conditions the previous vector space is a Zinbiel algebra. By imposing the Zinbiel identities, we obtain an equation system which has to be solved. Therefore, we have obtained the following Proposition 13. Let Gbe a combinatorial structures of three vertices containing full triangles. Then, Gis associated with a 3-dimensional Zinbiel algebra Zif and only if Gis isomorphic to configurations shown in Figure 7. Moreover, the restrictions for each configuration are 16 Figure 7: Combinatorial structures of 3 vertices associated with Zinbiel algebras. k) (c1 2,3, c1 3,2)= (0,0) l) c2 2,2= 0,c3 2,2(2c1 2,3−c1 3,2) = 0,(c1 2,3, c1 3,2)= (0,0),(c1 2,2, c3 2,2)= (0,0) m) c2 2,2=c3 2,2=c2 3,3=c3 3,3= 0,c1 2,2= 0,c1 3,3= 0,(c1 2,3, c1 3,2)= (0,0) n) c1 2,2=−c3 2,3(2c3 2,3c1 3,3+ 3c1 3,2c2 3,2) (c2 3,2)2,c2 2,2=−c3 2,3,c3 2,2=−(c3 2,3)2 c2 3,2 ,c1 2,3= c3 2,3c1 3,3+ 2c1 3,2c2 3,2 c2 3,2 ,c2 2,3=c2 3,2,c3 3,2=c3 2,3,c2 3,3=−(c2 3,2)2 c3 2,3 ,c3 3,3= −c2 3,2,(c1 2,3, c1 3,2)= (0,0),c2 2,3= 0,c3 2,3= 0, o) c1 1,1=−(c1 3,1)2c1 2,2 (c1 3,2)2,c2 1,1=(c1 3,1)3c1 2,2 (c1 3,2)3,c3 1,1=−(c1 3,1)3(c1 2,2)2 (c1 3,2)4,c1 1,3= c1 3,1,c2 1,3=−(c1 3,1)2 c1 3,2 ,c3 1,3=(c1 3,1)2c1 2,2 (c1 3,2)2,c2 2,2=−c1 2,2c1 3,1 c1 3,2 ,c3 2,2= (c1 2,2)2c1 3,1 (c1 3,2)2,c1 2,3=c1 3,2,c2 2,3=−c1 3,1,c3 2,3=c1 2,2c1 3,1 c1 3,2 ,c2 3,1=−(c1 3,1)2 c1 3,2 , c3 3,1=(c1 3,1)2c1 2,2 (c1 3,2)2,c2 3,2=−c1 3,1,c3 3,2=c1 2,2c1 3,1 c1 3,2 ,c1 3,2= 0,c1 3,1= 0, c1 2,2= 0; p) c1 1,1=−c3 1,3,c2 1,1=−c3 1,3c2 3,1 c1 3,1 ,c3 1,1=−(c3 1,3)2 c1 3,1 ,c1 1,3=c1 3,1,c2 1,3=c2 3,1, c1 2,2=−c3 1,3c2 3,2c1 3,1 (c2 3,1)2,c2 2,2=−c3 1,3c2 3,2 c2 3,1 ,c3 2,2=−(c3 1,3)2c2 3,2 (c2 3,1)2,c1 2,3= 17 c2 3,2c1 3,1 c2 3,1 ,c2 2,3=c2 3,2,c3 2,3=c3 1,3c2 3,2 c2 3,1 ,c3 3,1=c3 1,3,c1 3,2=c2 3,2c1 3,1 c2 3,1 ,c3 3,2= c3 1,3c2 3,2 c2 3,1 ,c1 3,3=−c1 3,1(c1 3,1+c2 3,2) c3 1,3 ,c2 3,3=−c2 3,1(c1 3,1+c2 3,2) c3 1,3 ,c3 3,3= −c1 3,1−c2 3,2,c1 3,1= 0,c2 3,1= 0,c3 1,3= 0,c2 3,2= 0,c1 3,1=−c2 3,2; q) c1 1,2=−c3 3,2,c2 1,2=c2 2,3c3 3,2 c1 3,2 ,c3 1,2=−(c3 3,2)2 c1 3,2 ,c1 1,3=−c2 2,3,c2 1,3= (c2 2,3)2 c1 3,2 ,c3 1,3=−c2 2,3c3 3,2 c1 3,2 ,c1 2,1=c3 3,2,c2 2,1=−c2 2,3c3 3,2 c1 3,2 ,c3 2,1=(c3 3,2)2 c1 3,2 , c1 2,3=−c1 3,2,c3 2,3=−c3 3,2,c1 3,1=c2 2,3,c2 3,1=−(c2 2,3)2 c1 3,2 ,c3 3,1=c2 2,3c3 3,2 c1 3,2 , c2 3,2=−c2 2,3,c1 3,2= 0,c2 2,3= 0,c3 3,2= 0 r) c1 1,1=−c3 3,1(c1 2,3+c1 3,2) c1 3,2 ,c2 1,1=c1 3,1c3 3,1(c1 2,3+c1 3,2) (c1 3,2)2,c3 1,1=−(c3 3,1)2(c1 2,3+c1 3,2) c1 3,2c1 3,1 , c1 1,2=−c3 3,1c1 3,2 c1 3,1 ,c2 1,2=c3 3,1,c3 1,2=−(c3 3,1)2c1 3,2 (c1 3,1)2,c1 1,3=c1 2,3c1 3,1 c1 3,2 , c2 1,3=−c1 2,3(c1 3,1)2 (c1 3,2)2,c3 1,3=c1 2,3c3 3,1 c1 3,2 ,c1 2,1=−c1 2,3c3 3,1 c1 3,1 ,c2 2,1=c1 2,3c3 3,1 c1 3,2 , c3 2,1=−c1 2,3(c3 3,1)2 (c1 3,1)2,c2 2,3=−c1 2,3c1 3,1 c1 3,2 ,c3 2,3=c1 2,3c3 3,1 c1 3,1 ,c2 3,1=−(c1 3,1)2 c1 3,2 , c2 3,2=−c1 3,1,c3 3,2=c3 3,1c1 3,2 c1 3,1 ,c1 3,2= 0,c1 3,1= 0 s) c1 1,1=(c1 3,1+c2 3,2)2 c1 3,3 ,c2 1,1=−(c1 3,1+c2 3,2)2c2 3,2 c1 2,3c1 3,3 ,c3 1,1=−(c1 3,1+c2 3,2)3 (c1 3,3)2, c1 1,2=−(c1 3,1+c2 3,2)c1 2,3 c1 3,3 ,c2 1,2=c2 3,2(c1 3,1+c2 3,2) c1 3,3 ,c3 1,2=(c1 3,1+c2 3,2)2c1 2,3 (c1 3,3)2, c1 1,3=c1 3,1+2c2 3,2,c2 1,3=−c2 3,2(c1 3,1+ 2c2 3,2) c1 2,3 ,c3 1,3=−(c1 3,1+c2 3,2)(c1 3,1+ 2c2 3,2) c1 3,3 , c1 2,1=(c1 3,1+c2 3,2)c1 2,3 c1 3,3 ,c2 2,1=−c2 3,2(c1 3,1+c2 3,2) c1 3,3 ,c3 2,1=−(c1 3,1+c2 3,2)2c1 2,3 (c1 3,3)2, c2 2,3=−c2 3,2,c3 2,3=−(c1 3,1+c2 3,2)c1 2,3 c1 3,3 ,c2 3,1=−c1 3,1c2 3,2 c1 2,3 ,c3 3,1=−c1 3,1(c1 3,1+c2 3,2) c1 3,3 , c1 3,2=−c1 2,3,c3 3,2=(c1 3,1+c2 3,2)c1 2,3 c1 3,3 ,c2 3,3=−c2 3,2c1 3,3 c1 2,3 ,c3 3,3=−c1 3,1− c2 3,2,c1 3,3= 0,c1 2,3= 0 18 t) c1 1,1=−c3 3,1(c2 2,2+c3 3,2) c3 3,2 ,c2 1,1=c2 2,2(c3 3,1)2 (c3 3,2)2,c1 1,2=−c2 2,2−c3 3,2,c2 1,2= c2 2,2c3 3,1 c3 3,2 ,c3 1,2=c3 1,1c3 3,2 c3 3,1 ,c1 1,3=−(c3 3,1)2(c2 2,2+c3 3,2) c3 1,1c3 3,2 ,c2 1,3=c2 2,2(c3 3,1)3 (c3 3,2)2c3 1,1 , c3 1,3=c3 3,1,c1 2,1=−c2 2,2−c3 3,2,c2 2,1=c2 2,2c3 3,1 c3 3,2 ,c3 2,1=c3 1,1c3 3,2 c3 3,1 ,c1 2,2= −(c2 2,2+c3 3,2)c3 3,2 c3 3,1 ,c3 2,2=c3 1,1(c3 3,2)2 (c3 3,1)2,c1 2,3=−c3 3,1(c2 2,2+c3 3,2) c3 1,1 ,c2 2,3= c2 2,2(c3 3,1)2 c3 3,2c3 1,1 ,c3 2,3=c3 3,2,c1 3,1=−(c3 3,1)2(c2 2,2+c3 3,2) c3 1,1c3 3,2 ,c2 3,1=c2 2,2(c3 3,1)3 (c3 3,2)2c3 1,1 , c1 3,2=−c3 3,1(c2 2,2+c3 3,2) c3 1,1 ,c2 3,2=c2 2,2(c3 3,1)2 c3 3,2c3 1,1 ,c1 3,3=−(c3 3,1)3(c2 2,2+c3 3,2) c3 3,2(c3 1,1)2, c2 3,3=(c3 3,1)4c2 2,2 (c3 3,2)2(c3 1,1)2,c3 3,3=(c3 3,1)2 c3 1,1 ,c3 3,2= 0,c3 3,1= 0,c3 1,1= 0 Proof. Similar to the proof of Proposition 10. Now, we study the isomorphism class for Zinbiel algebras associated with configurations shown in Figure 7. Theorem 2. Let Z3 αbe the Zinbiel algebra associated with Configuration α, where α∈ {k, l, m, n, o, p, q, r, s, t}from Figure 7 and let Z3 xviii)be the Zinbiel algebra associated to Configuration xviii)from Figure 5. Then, we have the following isomorphisms (∼ =) 1) Z3 m∼ =Z3 p∼ =Z3 t∼ =Z3 xviii) 2) Z3 l∼ =Z3 n∼ =Z3 o∼ =Z3 r∼ =Z3 s Proof. We start proving 1) that Z3 m∼ =Z3 xviii). According to Proposition 8, the law of Z3 xviii)is given by [v1, v1] = c3 1,1c2 3,2 c3 2,3 v2+c3 1,1v3,[v2, v2] = −c3 2,3v2−(c3 2,3)2 c2 3,2 v3, [v3, v3] = −(c2 3,2)2 c3 2,3 v2−c2 3,2v3,[v2, v3] = [v3, v2] = c2 3,2v2+c3 2,3v3 By considering the basis change ϕ:Z3 xviii)→ Z3 xviii)given by e1=ϕ(v1) = c2 3,2v2+c3 2,3v3;e2=ϕ(v2) = v2+ 2c3 2,3v1;e3=ϕ(v3) = −v3+1 c3 1,1v1, [e2, e2] = c3 2,3(4c3 1,1−1 c2 3,2 )e1,[e3, e3] = 1 c3 2,3 (1 c3 1,1 −c2 3,2)e1,[e2, e3] = [e3, e2] = e1 19 With a trivial scale on the coefficients, we obtain the law of Z3 m. In order to prove 2), we bear in mind that, from Proposition 13, the law of Z3 nis [v2, v2] = −c3 2,3(2c3 2,3c1 3,3+ 3c1 3,2c2 3,2) (c2 3,2)2v1−c3 2,3v2−(c3 2,3)2 c2 3,2 v3,[v3, v3] = c1 3,3v1−(c2 3,2)2 c3 2,3 v2−c2 3,2v3 [v2, v3] = c3 2,3c1 3,3+ 2c1 3,2c2 3,2 c2 3,2 v1+c2 3,2v2+c3 2,3v3,[v3, v2] = c1 3,2v1+c2 3,2v2+c3 2,3v3 Next, we consider the basis change ϕ:Z3 n→ Z3 ngiven by e1=ϕ(v1) = v1; e2=ϕ(v2) = c2 3,2v2+c3 2,3v3;e3=ϕ(v3) = v3, we obtain the law [e3, e3] = c1 3,3e1−c2 3,2 c3 2,3 e2,[e2, e3] = 2[e3, e2] = 2(c3 2,3c1 3,3+c1 3,2c2 3,2)e1 Relabeling the vertices, we get the law of Z3 l. We continue proving that Z3 o∼ =Z3 l. According to Proposition 13, the non-zero brackets of Z3 oare [v1, v1] = −(c1 3,1)2c1 2,2 (c1 3,2)2v1+(c1 3,1)3c1 2,2 (c1 3,2)3v2−(c1 3,1)3(c1 2,2)2 (c1 3,2)4v3, [v1, v3]=[v3, v1] = c1 3,1v1−(c1 3,1)2 c1 3,2 v2+(c1 3,1)2c1 2,2 (c1 3,2)2v3, [v2, v2] = c1 2,2v1−c1 2,2c1 3,1 c1 3,2 v2+(c1 2,2)2c1 3,1 (c1 3,2)2v3,[v2, v3]=[v3, v2] = c1 3,2v1−c1 3,1v2+c1 2,2c1 3,1 c1 3,2 v3 Now, we consider the basis change ϕ:Z3 o→ Z3 ogiven by e1=ϕ(v1) = v3; e2=ϕ(v2) = c1 2,2v1−c1 3,1c1 2,2 c1 3,2v2+c1 3,1(c1 2,2)2 (c1 3,2)2v3;e3=ϕ(v3) = v1, we obtain the law [e3, e3] = −(c1 3,1)2 (c1 3,2)2e2,[e1, e3] = [e3, e1] = c1 3,1 c1 3,2 e2 After exchanging subindexes, we obtain the law of Z3 l. By using the same reasoning, one can prove the isomorphism of statement 3). Next, we prove statement 4). In order to do so, we consider the law of Z3 robtained from Proposition 13: [v1, v1] = −c3 3,1(c1 2,3+c1 3,2) c1 3,2 v1+c1 3,1c3 3,1(c1 2,3+c1 3,2) (c1 3,2)2v2−(c3 3,1)2(c1 2,3+c1 3,2) c1 3,2c1 3,1 v3, [v1, v2] = −c3 3,1c1 3,2 c1 3,1 v1+c3 3,1v2−(c3 3,1)2c1 3,2 (c1 3,1)2v3,[v1, v3] = c1 2,3c1 3,1 c1 3,2 v1−c1 2,3(c1 3,1)2 (c1 3,2)2v2+c1 2,3c3 3,1 c1 3,2 v3, 20 [v2, v1] = −c1 2,3c3 3,1 c1 3,1 v1+c1 2,3c3 3,1 c1 3,2 v2−c1 2,3(c3 3,1)2 (c1 3,1)2v3,[v2, v3] = c1 2,3v1−c1 2,3c1 3,1 c1 3,2 v2+c1 2,3c3 3,1 c1 3,1 v3, [v3, v1] = c1 3,1v1−(c1 3,1)2 c1 3,2 v2+c3 3,1v3,[v3, v2] = c1 3,2v1−c1 3,1v2+c3 3,1c1 3,2 c1 3,1 v3 Let us consider v=v1−c1 3,1 c1 3,2v2+c3 3,1 c1 3,1v3. Then, [v1, v1] = −c3 3,1(c1 2,3+c1 3,2) c! 3,2 v, [v1, v2] = −c3 3,1c1 3,2 c1 3,1 v, [v1, v3] = c1 2,3c1 3,1 c1 3,2 v, [v2, v1] = −c1 2,3c3 3,1 c1 3,1 v, [v2, v3] = c1 2,3v, [v3, v1] = c1 3,1v, [v3, v2] = c1 3,2v Now, we consider the basis change ϕ:Z3 r→ Z3 rgiven by e1=ϕ(v1) = v1; e2=ϕ(v2) = v;e3=ϕ(v3) = v3, we obtain the law [e1, e1] = −c3 3,1(c1 2,3+c1 3,2) c1 3,2 e2,[e1, e3] = c1 2,3 c1 3,2 [e3, e1] = c1 2,3c1 3,1 c1 3,2 e2 Relabeling the vertices 1 and 2, the law of the algebra Z3 lis obtained. Finally, with a similar reasoning, it is possible to prove that Z3 s∼ =Z3 land Z3 t∼ =Z3 m. Finally, we clarify how all the isomorphism classes obtained in Sections 5 and 6 correspond to those in the well-known classification of 2and 3dimensional Zinbiel algebras. We will relate the notation given in [2, Theorem 1.7] with that used in this paper for the isomorphism classes. Z1Z2 aZ2 bZ3 i)Z3 ii)Z3 iii)Z3 xviii)Z3 kZ3 l A Q(0) Q(1) R(0,0,0,0) R(1,0,0,0) R(1,0,0,1) R(α, 1,1,1 α) α=−c2 3,2 c3 2,3 R(0, β, γ, 0) W(3) Table 2: Comparison with the classification of 2and 3-dimensional Zinbiel algebras. Remark 6. We have not included configuration q)from Figure 7 in Table 2, since it corresponds to a Lie algebra with the following law: [e1, e2] = −[e2, e1] = −c3 3,2e1+c2 2,3c3 3,2 c1 3,2 e2−(c3 3,2)2 c1 3,2 e3, [e1, e3] = −[e3, e1] = −c2 2,3e1+(c2 2,3)2 c1 3,2 e2−c2 2,3c3 3,2 c1 3,2 e3, [e2, e3] = −[e3, e2] = −c1 3,2e1+c2 2,3e2−c3 3,2e3 21 7 Algorithmic procedures This section is devoted to introduce two algorithmic procedures: The first one checks if a given combinatorial structure is associated or not with a Zinbiel algebra; the second one, conversely, computes the (pseudo)digraph associated with a given finite-dimensional Zinbiel algebra starting from its law when this is not providing full triangles. Let us note that the procedure presented in Subsection 7.1 is developed starting from the techniques to prove the existence or non-existence of Zinbiel algebras associated with combinatorial (pseudo)digraphs in previous sections; moreover, this procedure has been run later to check the theoretical results for dimensions 2 and 3. 7.1 Checking if a given combinatorial structure is associated with a Zinbiel algebra We have implemented this algorithmic procedure by using the symbolic computation package Maple, working the implementation in version 12 or higher. To do this, we have used the libraries linalg and combinat to activate commands related to Linear and Combinatorial Algebra. This algorithmic procedure consists of the following three steps: a) Defining the values of the structure constants according to the combinatorial structure. b) Generating the law which should be satisfied by the Zinbiel algebra, starting from the structure constants. c) Checking if the Zinbiel identities are satisfied for this law. In order to develop the implementation, we use three subprocedures for the two first steps and one main procedure for the last one. Before running the procedure, we must restart all the variables and delete all the computations saved in the kernel by using the command restart. The first step of this algorithm is executed by the subprocedure assignment, which allows us to define the dimension and the value of the structure constants of the vector space associated with the combinatorial structure and to determine the candidate for the bracket product. To do so, assignment receives the following two inputs: The list Vwith the vertices of the combinatorial structure as natural numbers, and the set Ewith its weighted, directed edges. The elements of the set Eare inserted as [[i, j, k], l], denoting ck i,j =l. As output, we 22 obtain the value of the variable dim with the dimension of the combinatorial structure and also the value of all the non-zero structure constants. > restart: > assignment:=proc(V,E) > local B,L; > B:=[];L:=[]; > for x from 1 to nops(V) do > B:=[op(B),e[x]]; > end do; > assign(dim,nops(V)); > for i from 1 to nops(E) do > assign(c[E[i][1][1],E[i][1][2],E[i][1][3]],E[i][2]); > end do > end proc: Now, we can run the second subprocedure, named law, which receives two natural numbers as inputs. These numbers represent the subindexes of two vectors in the endowed vector space or, equivalently, two vertices from the combinatorial structure. The subroutine computes the bracket of these two vectors. In the implementation, we use a local variable, v, to save the value of the bracket, which is computed by using the structure constants defined in the previous subprocedure. > law:=proc(i,j) > local v; v:=0; > for k from 1 to dim do > if type(c[i,j,k],numeric)=true then > v:=v+c[i,j,k]*e[k]; > end if > end do; > return v; > end proc: Next, we implement the subprocedure called bracket to compute the product between two arbitrary vectors expressed as linear combinations of the basis vectors used in the previous subprocedure. > bracket:=proc(u,v,n) > local exp; exp:=0; > for i from 1 to n do > for j from 1 to n do > exp:=exp + coeff(u,e[i])*coeff(v,e[j])*law(i,j); > end do; > end do; > exp; > end proc: 23 Finally, we show the implementation of the main procedure called Zinbiel, which checks if the vector space is or is not a Zinbiel algebra. This procedure receives as input the dimension nof the vector space Zand returns a message which will be “True” in case that the vector space Zis a Zinbiel algebra and “False” otherwise. >Zinbiel:=proc(n) > local L,M,N,P; > L:=[];M:=[];N:=[];P:=[]; > for i from 1 to n do > L:=[op(L),i,i]; > end do; > M:=permute(L,3); > for j from 1 to nops(M) do > eq[j]:=bracket(bracket(e[M[j][1]],e[M[j][2]],n),e[M[j][3]],n) -bracket(e[M[j][1]],bracket(e[M[j][2]],e[M[j][3]],n),n) -bracket(e[M[j][1]],bracket(e[M[j][3]],e[M[j][2]],n),n); > end do; > N:=[seq(eq[k], k=1..nops(M))]; > for i from 1 to nops(N) do > if N[i]<>0 then > P:=[op(P),N[i]]; > end if; > end do; > if P=[] then return "True" > else return "False"; > end if; > end proc: Example 3. To illustrate the algorithmic procedure, we show an example by considering the combinatorial structure of Figure 2. According to the notation that we are using in this algorithm, we have to consider > V=[1,2,3]; > E={[[1,1,2],1],[[1,1,3],-1],[[1,2,2],-1],[[1,2,3],1],[[1,3,2],-1],[[1,3,3],1]}; Now, we run all the procedures obtaining > assignment(V,E); > Zinbiel(dim); > "True" Therefore, the combinatorial structure is associated with a 3-dimensional Zinbiel algebra. 24 7.2 Obtaining (pseudo)digraphs associated with Zinbiel algebra In this subsection, we show an algorithmic procedure that computes the (pseudo)digraph associated with a given Zinbiel algebra with finite dimension. According to the notation previously used, we consider a n-dimensional Zinbiel algebra Zwith basis Band whose non-zero brackets are the ones given in (1). In this way, there will be no full triangles in the configuration. To implement the algorithm, we have used the symbolic computation package MAPLE 12, loading the libraries linalg,combinat,GraphTheory and Maplets[Elements]. The first three libraries allow us to apply commands of Linear Algebra, Combinatorics and Graph Theory, respectively; whereas the last is used to display a message so that the user introduces the required input in the first subprocedure, corresponding to the definition of the law of the algebra Z. Our algorithm is based on the following four steps 1. Obtaining the bracket between basis vectors of B. The implementation of this step is carried out by the subprocedure law2, which receives as input the subindexes of two basis vectors in B. The output is the bracket between these vectors. Moreover, conditional sentences are necessary for the non-zero brackets. Since the subprocedure must be completed by the user, according to the law of Z, we have added a sentence at the beginning recalling this fact. We also have to restart all the variables and delete the previous calculations before updating the value of dim, which is the variable saving the dimension of Z. > restart: > maplet:=Maplet(AlertDialog("Don’t forget to introduce non-zero brackets of the algebra and its dimension in subprocedure law", ’onapprove’=Shutdown("Continue"),’oncancel’=Shutdown("Aborted"))): > Maplets[Display](maplet): > assign(dim,...): > law2:=proc(i,j) > if (i,j)=... then ...; end if; > if .... > else 0; end if; > end proc; The ellipsis in command assign are for dim(Z). The other suspension points correspond to the introduction of the non-zero brackets of Z. 25 [6] I. Kaygorodov, Y. Popov, A. Pozhidaev and Y. Volkov. Degenerations of Zinbiel and nilpotent Leibniz algebras. Linear and Multilinear Algebra 66:4 (2018). [7] J.L. Loday: Cup-product for Leibniz cohomology and dual Leibniz algebras. Math. Scand. 77:2 (1995), 189–196. https://doi.org/10.7146/math.scand.a-12560 [8] J.L. Loday: Dialgebras. In J.L. Loday, F. Chapoton, A. Frabetti and F. Goichot: Dialgebras and Related Operads. Lecture Notes in Mathematics, vol. 1763. Springer, Berlin, Heidelberg, pp. 7–66. https://doi.org/10.1007/3-540-45328-8 2 [9] H.S. Wilf: Algorithms and Complexity. 2nd edition. A K Peters, Natick, 2002. https://doi.org/10.1201/b10621 32