Finite-dimensional flexible algebras associated with directed and weighted CW complexes

Finite-dimensional flexible algebras associated with directed and weighted CW complexes M. Ceballos1 1Dpto. de Ingenier´ıa. Universidad Loyola Andaluc´ıa. Av. de las Universidades, s/n, 41704 Dos Hermanas, Sevilla (Spain). [email protected] Abstract In this paper, we study a link between directed and weighted CW complexes (also called configurations) and flexible algebras determining which configurations are associated with those algebras. Some important elements that can be read from the (pseudo)digraph that is associated with a flexible algebra are studied. Moreover, the isomorphism classes of each 2-dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: the first one checks if a given directed and weighted CW complex is associated or not with a flexible algebra, while the second one constructs and draws the (pseudo)digraph associated with a given flexible algebra. Keywords: Digraph, Pseudodigraph, CW complex, Flexible algebra. 2010 Mathematics Subject Classification: 17A20 05C25, 05C20, 68W30, 68R10. 1 Introduction Nowadays, one of the most important and stimulating research in Mathematics is finding and studying new links between different fields. Alternative techniques and procedures allow researchers to solve many unsolved problems, improve known theories and achieve new results. This paper deals with the relation between Graph Theory and flexible algebras. More concretely, the main goal is to make progress on the research line started in [2, 5], where a mapping between Lie algebras and directed CW complexes was introduced in order to translate properties of Lie algebras into the language of Graph Theory and vice versa. Now, the main goal is to obtain an analogous mapping for flexible algebras. 1 Non-associative algebras have been deeply studied due to their own theoretical importance and their many applications to different fields like Physics, Engineering or Applied Mathematics [11, 16]. A particular type of these algebras is formed by flexible algebras. There exists a close relation between flexible algebras and other types of algebras. In this sense, every Lie, Jordan, associative or alternative algebra is flexible. The first papers dealing with flexible algebras were written by Oehmke and Schafer in 1954 and 1958. In [22], Schafer studied the algebras generated by the CayleyDickson process over a field and proved that they satisfy the flexible identity. In [19], Oehmke studied several properties on flexible algebras. More recently, Pumpl¨un [21] has analyzed algebraic constructions that yield to flexible quadratic algebras and Behn Correa and Hentzel in [3] have studied flexible algebras satisfying the polynomial identity x(yz) = y(zx). Flexible algebras have also been investigated in terms of degree of algebras [14, 17]. A very important example of flexible algebras is formed by the octonions, which are a normed division algebra over the real number field. They are non-commutative, non-associative and flexible. Octonions have applications in many different fields such as string theory, special relativity and quantum logic (see [1]). Other characterizations and applications of these algebras can be found in [13, 15] and references therein. Currently, Graph Theory has become a very useful tool to deal with a wide range of problems in many research fields. This theory may be used to study non-associative algebras in general and flexible algebras in particular. Concerning flexible algebras, there is no reference in the literature about the study of the link among graph theory and these algebras. However, this theory has been essential in order to study other non-associative algebras such as Lie, Leibniz, Malcev, Zinbiel and evolution algebras. For instance, in case of Lie algebras, trees perform an important role to determine the Dynkin diagrams associated to such algebras [23] and graphs are used to represent Lie algebras [20]. Leibniz algebras have also been studied and classified starting from their associated graphs [6]. A similar study was done for Malcev and Zinbiel algebras [7, 9]. Another example is the use of finite connected bipartite graph to construct finite-dimensional indecomposable semisimple Leibniz algebras [25]. One of the most important types of nonassociative algebras are evolution algebras. There are several papers dealing with the link between these algebras and graphs [8, 10, 24, 4]. In those papers graphs are used in order to study several properties of their associated evolution algebra. Finally, in [18], the authors considered to solve some open 2 problems related to graphicable algebras. This paper is organized as follows: in Section 2, some well-known concepts on Graph Theory and flexible algebras are recalled. An algorithmic procedure to associate directed and weighted CW complexes with flexible algebras and vice versa is developed in Section 3. Next, Section 4 shows some properties flexible algebras that can be read from its associated directed and weighted CW complex such that the center and derived algebra. Section 5 studies the structure of (pseudo)digraphs associated with flexible algebras and some of their properties. For each configuration, the type of flexible algebra considering solvability and nilpotency is analyzed. Section 6 is devoted to determining the isomorphism classes of the 2-dimensional algebras obtained in the previous section. Section 7 shows the implementation of the two algorithmic methods used in the previous sections. The first one is designed to check if a given directed and weighted CW complex is associated or not with a flexible algebra and the second one draws the (pseudo)digraph, if possible, associated with a given finite-dimensional flexible algebra. Moreover, a brief computational study, showing the complexity order and computing time of the routines of the algorithm is given. At the end of the paper, there is a conclusion section, acknowledgements and references. Finally, and in order to make the paper more legible, an appendix section in included, which contains several lists of restrictions from some results of Sections 4 and 6. 2 Preliminaries This section recalls some preliminary concepts, results and notations about flexible algebras, Graph Theory and CW complexes. Concerning the former, the reader can consult [19]. Regarding the latters, [12] is an introductory reference to Graph Theory and CW complexes were introduced by J.H.C. Whitehead [27]. 2.1 Flexible algebras Let Kbe a field. A flexible algebra Fis a vector space over Kwith a second bilinear inner composition law ([·,·]), called the bracket product or commutator, which satisfies [X, [Y, X]] = [[X, Y ], X] 3 This is known as the flexible identity and we will use the following notation: F(X, Y ) = [X, [Y, X]] −[[X, Y ], X]. Given a basis {ei}n i=1 of F, its structure (or Maurer-Cartan) constants are defined by the coefficients ch i,j which determines the law of the algebra: [ei, ej] = Pn h=1 ch i,jeh, for 1 ≤i<j≤n. Given a flexible algebra F, its center is defined as Z(F)={X∈ F | [X, Y ]=0,∀Y∈ F}. The flexible algebra Fis abelian if Z(F) and Fare isomorphic. In that case, Fis called zero or trivial algebra. The derived series of a given finite-dimensional flexible algebra Fis C1(F) = F,C2(F) = [F,F], . . . , Ck(F)=[Ck−1(F),Ck−1(F)], . . . We say that Fis solvable if there exists m∈N,m > 1 such that Cm(F) = {0}. In addition, if Cm−1(F)={0}also holds, then mis known as the solvability index or solvindex of Fand it is said that Fis (m−1)-step solvable. The central series of a given finite-dimensional flexible algebra Fis C1(F) = F,C2(F)=[F,F], . . . , Ck(F)=[Ck−1(F),F], . . . We say that Fis nilpotent if there exists m∈N,m > 1 such that Cm(F) = {0}. In addition, if Cm−1(F)={0}also holds, then mis known as the nilpotency index or nilindex of Fand it is said that Fis (m−1)-step nilpotent. Notice that every nilpotent algebra is trivially solvable because Ci(F)⊆ Ci(F) for all i∈N. The derived algebra of a flexible algebra Fwill be denoted by DF= C2(F) = C2(F). A flexible algebra Fis perfect if Fand DFare isomorphic. 2.2 Graph Theory and CW complexes Adigraph consists of an ordered pair G= (V, E), where Vis a non-empty set called vertex-set and Eis a set of ordered pairs (edges) of two vertices, called edge-set. It is possible to associate a weight to each edge. In that case Gwill be a weighted digraph. Aloop in the digraph G= (V, E) is an edge that connects a vertex with itself. If the digraph Gcontains loops, then Gis called a pseudodigraph. A vertex v∈Vis called simple if there is no loop on this vertex. Throughout the paper, weighted (pseudo)digraphs will be considered. 4 Given a (pseudo)digraph G= (V, E), a sub(pseudo)digraph G′= (V′, E′) of Gis a (pseudo)digraph verifying V′⊆Vand E′⊆E. A sub(pseudo)digraph His said to be induced by a vertex-subset V(H) in Gif the edge-set of H consists of all the edges of Gbetween two vertices in V(H). Two vertices u, v ∈Vare adjacent if there is an edge from vertex uto v or viceversa. In that case, we will say that those vertices are incident with that edge. A vertex v∈Vis a sink (resp. a source) if each edge incident with vis oriented towards v(resp. from v). See Figure 1. Figure 1: Example of sink and source, respectively. A sequence of consecutive vertices and edges in a digraph is known as a walk. A (pseudo)digraph is connected if there is a walk between any pair of vertices. Otherwise, we will say that the digraph is non-connected. ACW complex is a topologycal space built out of smaller spaces iteratively by a process called attaching cells, where a k-cell is a k-dimensional disc Dk={x∈Rk:|x| ≤ 1} A CW complex is directed when we establish a direction in the elements that formed the complex. In case that we have some weight over the edges of that complex, we call it a directed and weighted CW complex. This class of spaces generalizes simplicial complexes and retains a combinatorial structure nature. Discrete points, digraphs and directed full triangles (in the sense that will be seen in Section 3) are examples of 0,1 and 2-dimensional directed CW complexes, respectively. 3 Associating flexible algebras with directed and weighted CW complexes Let Fbe an n-dimensional flexible algebra with basis B={ei}n i=1. The structure constants are given by [ei, ej] = Pn h=1 ch i,jehand, hence, the pair 5 (F,B) is associated with an directed and weighted CW complex by using the following procedure. a) Draw a vertex ifor each vector ei∈ B. b) For every vertex iverifying [ei, ei]= 0, draw a loop such that its weight is the n-tuple (c1 i,i, c2 i,i, . . . , cn i,i). See Figure 2. c) Given two positive integers i<j≤nverifying (cj i,j, cj j,i)= (0,0), draw a directed edge from vertex ito jwhose weight is given by the pair (cj i,j, cj j,i). Draw another directed edge with weight (ci i,j, ci j,i), but now from jto i, in case that (ci i,j, ci j,i)= (0,0). See Figure 3. Figure 2: Loop over vertex i.Figure 3: Directed edge. d) Given three positive integers i < j < k ≤nsuch that (ck i,j, ck j,i, ci j,k, ci k,j, cj i,k, cj k,i)= (0,0,0,0,0,0), draw a full triangle ijk such that the edges ij,jk and ik have weights (ck i,j, ck j,i), (ci j,k, ci k,j) and (cj i,k, cj k,i), respectively; see Figure 4. Moreover, d1) a discontinuous line (named ghost edge) will be used for edges with weight (0,0). d2) If two triangles ijk and ijl satisfy that (ck i,j, ck j,i)=(cl i,j, cl j,i), only one edge between vertices iand jshared by both triangles will be drawn. See Figure 5. Figure 4: Full triangle. Figure 5: Two triangles sharing an edge. Therefore, every flexible algebra with a given basis can be associated with a directed and weighted CW complex. 6 Example 1. The 3-dimensional flexible algebra with non-zero brackets [e1, e2] = e1= [e2, e1],[e2, e2] = e2,[e1, e3]=[e3, e1] = e2,[e2, e3]=[e3, e2]=2e2and [e3, e3] = e3is associated with the directed and weighted CW complex shown in Figure 6. Figure 6: Directed and weighted CW complex associated with a 3dimensional flexible algebra. Now, we see how to define the flexible algebra associated with a fixed pseudodigraph (directed and weighted 1-dimensional CW complex). Let G= (V, E) be a pseudodigraph with V={1, . . . , n}. Then Gcan be associated with a flexible algebra Fwith basis Bas follows: a) Define the vector space W={e1, . . . , en}from the set of vertices V. b) In case that i(1 ≤i≤n) is an isolated vertex, we define [ei, ei] = 0. c) If two vertices iand j(1 ≤i < j ≤n) are not adjacent, then [ei, ej]=[ej, ei]=0 d) In case that i(1 ≤i≤n) is a non-simple vertex, then we define [ei, ei] = Pn h=1 ch i,iei. e) Given two vertices iand jwith 1 ≤i < j ≤n, if there is a directed edge from ito jand there is no directed edge from jto i, then we define [ei, ej] = cj i,jej, [ej, ei] = cj j,iejwith (cj i,j, cj j,i)= (0,0). f) Given two vertices iand jwith 1 ≤i < j ≤n, in case that there is a directed edge between iand jand also a directed edge between jand i, then we define [ei, ej] = ci i,jei+cj i,jej, [ej, ei] = ci j,iei+cj j,iejwith (ci i,j, ci j,i),(cj i,j, cj j,i)= (0,0). 7 The above definitions and linear extension provide a product on V. Finally, we have to impose the flexible identity in order to obtain a flexible algebra. 4 Reading properties from the directed and weighted CW complexes In this section, we analyze the properties that can be read from the directed and weighted CW complex associated with a flexible algebra: being a Lie algebra, the center and the derived algebra. The method shown in Section 3 for flexible algebras provides a generalization of the one described in [2] for Lie algebras, as it is proved in the following Proposition 1. Given a flexible algebra which is also a Lie algebra, its associated directed and weighted CW complex satisfies 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 1. Given an edge in a directed and weighted CW complex associated with an flexible 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 [2]. Lemma 1. Let us denote by Gthe directed and weighted CW complex associated with a flexible algebra F. Then, Z(F)⊇span{ei|iis an isolated vertex} Proof. It follows from the fact that isolated vertices without loops on this structure correspond to basis vectors in the center of the flexible algebra. 8 Remark 2. If a directed and weighted CW complex is formed only by isolated vertices (trivial graph), then it will be associated with an abelian flexible algebra. From here on, only non-abelian flexible algebras will be considered. Regarding Lemma 1, let us note that the center of a flexible algebra may contain basis vectors which do not correspond to isolated vertex. Example 2. Let Fbe the 3-dimensional flexible algebra with basis B= {e1, e2, e3}and law [e1, e2] = e2,[e3, e2] = −e2. This algebra is associated with digraph a)of Figure 10 and Z(F) = span(e1+e3). Lemma 2. Let Gthe connected (pseudo)digraph associated with a flexible algebra F. Then it is verified that DF= span n X h=1 ch i,j eh|his not a simple source vertex!. Proof. First, it is trivial that DF=span({[ei, ej]|1≤i, j ≤n}) = span n X h=1 ch i,j eh!. Let us note there is no edge directed to a simple source vertex. Therefore, we conclude that DF= span n X h=1 ch i,j eh|his not a simple source vertex!. Corollary 1. Let Gthe connected (pseudo)digraph associated with a flexible algebra F. If Gcontains a simple source vertex, then Fis not perfect. Proof. If Gcontains a source vertex i, then there is no edge directed to i. Therefore, we can affirm that ei/∈DFand, hence, Fis not perfect. 5 Flexible algebras associated to pseudodigraphs This section studies the structure of (pseudo)digraphs associated with flexible algebras. For each case, the condition on the structure constants and the type of flexible algebra is analyzed according to its solvability. Let Fbe a non-trivial or non-zero flexible algebra with basis Bwhose directed and weighted CW complex Gconsists of a (pseudo)digraph; that is, there are no triangles in G. This assertion is equivalent to affirm that the law of Fwith 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 brackets are null. 9  Configurations d)and e)are associated with a 3-step solvable nonnilpotent flexible algebra if c1 1,2=−c1 2,1,c2 1,2=−c2 2,1. Otherwise, it is non-solvable and non-perfect. The same happens for Configurations i) and k)with conditions c2 3,2=−c2 2,3,c3 3,2=−c3 2,3.  Configuration f)is associated with a non-solvable flexible algebra.  Configurations g)and ℓ)are associated with a perfect flexible algebra.  Configuration j)is associated with a 3-step solvable flexible algebra if c2 2,3= 0. Otherwise, it is a perfect flexible algebra.  Configuration m)is associated with a 2-step solvable flexible algebra if c1 1,2=−c3 2,3,c2 1,2=c3 1,3,c1 1,3=−c2 3,2,c1 2,1=c3 2,3,c2 2,1=−c3 1,3, c2 2,3=−c2 3,2,c1 3,1=c2 3,2,c3 3,1=−c3 1,3,c3 3,2=−c3 2,3. Otherwise, it is non-solvable. Proof. Let us denote by Fxthe flexible algebra associated with Configuration x) and we will consider restrictions indicated in the appendix section for every configuration. In case of Configurations a) and c), it is satisfied that C2(Fa) = span(e2) and C2(Fc) = span(e1, e3) are abelian ideals. For Configurations b) and h), we have that C2(Fb) = span(e2, e3), C3(Fb) = span(e3), C2(Fh) = span(e2, e3), C3(Fh) = span(e2) and Ck(Fb) = Ck(Fh) = {0}, for k≥4. In case of Fd, if c1 1,2=−c1 2,1and c2 1,2=−c2 2,1, then [e2, e1] = −[e1, e2], C2(Fd) = span([e1, e2], e3), C3(Fd) = span(e3) and Cp(Fd) = {0}, for p≥4. Otherwise, Cq(Fd) = span([e1, e2], e3), for q≥2. The proof is similar for Configurations e), i) and k). Now, we consider Ff. This algebra is not perfect and could be solvable if c1 2,1=−c1 1,2, c2 2,1=−c2 1,2,c2 3,2=−c2 2,3,c3 3,2=−c3 2,3. In such a case, [e2, e1] = −[e1, e2] and [e3, e2] = −[e2, e3]. Consequently, C2(Ff) = span([e1, e2],[e2, e3]) and [[e1, e2],[e1, e2]] = [[e2, e3],[e2, e3]] = 0. However, Ck(Ff) = span([[e1, e2],[e2, e3]]), for k≥3. Therefore, Ffis non-solvable. Next, Fgand C2(Fg) are isomorphic, so Fgis a perfect flexible algebra. The same happens for Fℓ. If we consider Fj, then C2(Fj) = span(e1, e3, c2 2,3e2+c3 2,3e3). In case that c2 2,3= 0, then C2(Fj) = span(e1, e3) and C3(Fj) = span(e3) is an abelian ideal. Consequently, Fjis 3-step solvable. Otherwise, C2(Fj) = span(e1, e2, e3) = Fj. Finally, we consider the algebra Fm. According to the restrictions in appendix, we have that [e2, e1] = c1 2,1 c1 1,2 [e1, e2], [e3, e2] = c2 3,2 c2 2,3 [e2, e3] and [e3, e1] = c1 3,1 c1 1,3 [e1, e3]. In case that c1 1,2=−c3 2,3,c2 1,2=c3 1,3,c1 1,3=−c2 3,2,c1 2,1=c3 2,3, c2 2,1=−c3 1,3,c2 2,3=−c2 3,2,c1 3,1=c2 3,2,c3 3,1=−c3 1,3,c3 3,2=−c3 2,3, then 16 [e2, e1] = −[e1, e2], [e3, e2] = −[e2, e3] and [e3, e1] = −[e1, e3]. Therefore C2(Fm) = span([e1, e2],[e2, e3],[e1, e3]), where [e1, e2] = −c3 2,3e1+c3 1,3e2,[e1, e3] = −c2 3,2e1+c3 1,3e3,[e2, e3] = −c2 3,2e2+c3 2,3e3 Moreover,  ci,i ci,j ci,k cj,i 0cj,k ck,i ck,j ck,k  = 0 and C3(Fm) = {0}, so Fmis 2-step solvable. Proposition 9. Let Fbe a 3-dimensional non-abelian flexible algebra associated to a connected pseudodigraph G. Then, Gis isomorphic to one configuration in Figure 11. Proof. Figure 11 includes all the possible 3-vertices connected pseudodigraphs. Following the procedure of Section 3, we can construct the algebra associated with each configuration. For each of them, flexible identity is imposed. Therefore, every configuration in Figure 11 is associated with a flexible algebra if and only if the restrictions indicated in Section 9 hold for each of them. 6 Classification of flexible algebras In this section, the isomorphism class for each configuration with two vertices from Section 5 associated with non-abelian flexible algebras is analyzed. In this way, we provide a new method to classify these algebras is provided. Proposition 10. Flexible algebras associated with Configuration a)from Figure 2 belong to the isomorphism class F2 0) = span(e1, e2)defined by the law [e1, e1] = e1or F2 i)= span(e1, e2)given by [e1, e1] = e2. Proof. Let Fbe the flexible algebra associated with Configuration a) from Figure 7. Let {v1, v2}be the basis of F. If c1 1,1= 0, then we consider the basis change ϕ:F → F given by e1=ϕ(v1) = 1 c1 1,1 v1+c2 1,1 (c1 1,1)2v2; e2=ϕ(v2) = v2and the law [e1, e1] = e1is obtained. In case that c1 1,1= 0, then c2 1,1= 0 and the basis change ϕ:F → F given by e1=ϕ(v1) = 1 c2 1,1 v1; e2=ϕ(v2) = 1 c2 1,1 v2leads to the law [e1, e1] = e2. 17 Figure 11: Connected pseudodigraphs with three vertices. 18 Proposition 11. Flexible algebras associated with Configuration b)from Figure 2 belong to the isomorphism class F2 ii)= span(e1, e2)defined by the law [e1, e1] = e1,[e2, e2] = e2. Proof. If Fis the flexible algebra associated with Configuration b) from Figure 7, then its law is given by [v1, v1] = c1 1,1v1, [v2, v2] = c2 2,2v2. Now, with the basis change ϕ:F → F given by ei=ϕ(vi) = 1 ci i,i vifor i= 1,2, the law [e1, e1] = e1, [e2, e2] = e2is obtained. Proposition 12. Flexible algebras associated with Configuration c)from Figure 8 belong to the isomorphism class F2 iii)= span(e1, e2)defined by the law [e1, e2] = e2,[e2, e1] = αe2, with α∈C. Proof. Let Fbe the flexible algebra associated with Configuration c) from Figure 8, then its law is given by [v1, v2] = c2 1,2v2, [v2, v1] = c2 2,1v2, where (c2 1,2, c2 2,1)= (0,0). If c2 1,2= 0 (the other case is similar) then the basis change ϕ:F → F given by e1=ϕ(v1) = 1 c2 1,2 v1,e2=v2leads to the law [e1, e2] = e2, [e2, e1] = c2 2,1 c2 1,2 e2. Proposition 13. Flexible algebras associated with Configuration d)from Figure 8 belong to the isomorphism class F2 iv)= span(e1, e2)defined by the law [e1, e2] = e1+e2,[e2, e1] = α(e1+e2), with α∈C. Proof. Let Fbe the flexible algebra associated with Configuration d) from Figure 8, then its law is given by [v1, v2] = c1 1,2v1+c2 1,2v2, [v2, v1] = c1 2,1v1+ c2 2,1v2, where c1 1,2c2 2,1=c2 1,2c1 2,1and (c1 1,2, c1 2,1),(c2 1,2, c2 2,1)= (0,0). Now, with the basis change ϕ:F → F given by e1=ϕ(v1) = 1 c2 1,2 v1,e2=1 c1 1,2 v2leads to the law [e1, e2] = e1+e2, [e2, e1] = c2 2,1 c2 1,2 (e1+e2). Proposition 14. Flexible algebras associated with Configuration e)from Figure 8 belong to the isomorphism class F2 v)= span(e1, e2)defined by the law [e1, e1] = αe1+βe2,[e1, e2]=[e2, e1] = e2, where α∈Cand β∈ {0,1}. Proof. According to the proof of Proposition 4 and the restrictions indicated in Appendix, there exist two different families of flexible algebras associated with Configuration e) from Figure 8. Let us denote by F1and F2those algebras and let {v1, v2}be a basis for them. The law of F1is given by [v1, v1] = c1 1,1v1+c2 1,1v2, [v1, v2]=[v2, v1] = c2 1,2v2, while the non-zero brackets of F2are [v1, v1] = c2 1,1v2, [v1, v2] = c2 1,2v2, [v2, v1] = c2 2,1v2. The basis changes ϕ:F1→ F1given by e1=ϕ(v1) = 1 c2 1,2 v1;e2=ϕ(v2) = c2 1,1 (c2 1,2)2v2 19 and ϕ′:F2→ F2given by e1=ϕ′(v1) = 1 c2 1,2 v1;e2=ϕ′(v2) = v2lead to the law [e1, e1] = c1 1,1 c2 1,2 e1+βe2, [e1, e2] = e2, where β= 0 for F2and β= 1 for F1. Proposition 15. Flexible algebras associated with Configuration f)from Figure 8 belong to the isomorphism class F2 vi)= span(e1, e2)defined by the law [e1, e1] = αe1+e2,[e1, e2]=[e2, e1] = e2,[e2, e2] = βe1+e2, where α, β ∈C. Proof. Let us denote by Fthe flexible algebra associated with Configuration f). If {v1, v2}is a basis of F, according to the proof of Proposition 4 and the restrictions indicated in Appendix, the law of Fis given by [v1, v1] = c1 1,1v1+c2 1,1v2, [v1, v2] = [v2, v1] = c2 1,2v2, [v2, v2] = c1 2,2v1+c2 2,2v2. The basis change φ:F → F given by e1=φ(v1) = 1 c2 1,2 v1+λ c2 1,2 v2;e2=φ(v2) = µv2, where aand bare the solution of the system c2 2,2λ2+c2 1,2λ+c2 1,1= 0, µ2c2 2,2c2 1,2=λ), leads to the law [e1, e1] = c1 1,1+λ2c1 2,2 (c2 1,2)2e1+e2, [e1, e2]=[e2, e1] = e2, [e2, e2] = µ2c1 2,2c2 1,2e1+e2. Proposition 16. Flexible algebras associated with Configuration g)from Figure 8 belong to the isomorphism class F2 vii)= span(e1, e2)defined by the law [e1, e1] = e1+αe2,[e1, e2]=[e2, e1] = e1, where α∈C. Proof. Let us denote by Fthe flexible algebra associated with Configuration g). If {v1, v2}is a basis of F, according to the proof of Proposition 4 and the restrictions indicated in Appendix, the law of Fis given by [v1, v1] = c1 1,1v1+c2 1,1v2,[v1, v2]=[v2, v1] = c1 1,2v1. Now, we consider the basis changes ϕ:F → F given by e1=ϕ(v1) = 1 c1 1,1 v1=e1,e2=ϕ(v2) = 1 c1 1,2 v2=e2. Applying ϕ, we obtain the law [e1, e1] = e1+αe2, [e1, e2] = [e2, e1] = e1, where α=c2 1,1c1 1,2 (c1 1,1)2. Proposition 17. Flexible algebras associated with Configuration h)from Figure 8 belong to the isomorphism class F2 viii)= span(e1, e2)defined by the law [e1, e1] = e1+αe2,[e1, e2]=[e2, e1] = e1, where α∈C. 20 Proof. We denote by Fthe flexible algebra associated with Configuration h). Let {v1, v2}be a basis of F. According to the proof of Proposition 4 and the restrictions indicated in Appendix, the law of Fis given by [v1, v1] = c1 1,1v1+c2 1,1v2,[v1, v2] = [v2, v1] = c1 1,2v1+c2 1,2v2. Now, we consider the basis changes ϕ:F → F given by e1=ϕ(v1) = 1 2c2 1,2+c1 1,1 v1+c2 1,2 (2c2 1,2+c1 1,1)c1 1,2 v2=e1, e2=ϕ(v2) = 1 c1 1,2 v2=e2. Applying ϕ, we obtain the law [e1, e1] = e1+αe2, [e1, e2] = [e2, e1] = e1, where α=c1 1,2c2 1,1−c1 1,1c2 1,2 (2c2 1,2+c1 1,1)2. Proposition 18. Flexible algebras associated with Configuration i)from Figure 8 belong to the isomorphism class F2 ix)= span(e1, e2)defined by the law [e1, e1] = e1+αe2,[e2, e2] = βe1+e2,[e1, e2]=[e2, e1] = e1, where α, β ∈Cor F2 x)= span(e1, e2)defined by the law [e1, e1] = e1,[e2, e2] = e2, [e1, e2] = (1 −λ)e1+λe2,[e2, e1] = (1 −µ)e1+µe2, where λ, µ ∈C Proof. According to the proof of Proposition 4 and the restrictions indicated in Appendix, there exist three different families of flexible algebras associated with Configuration i) from Figure 8. Let us denote by F1,F2 and F3those algebras and let {v1, v2}be a basis for them. The law of F1is given by [v1, v1] = c1 1,1v1+c2 1,1v2, [v1, v2] = [v2, v1] = c1 1,2v1+c2 1,2v2, [v2, v2] = c1 2,2v1+c2 2,2v2, while the non-zero brackets of F2are [v1, v1] = c1 1,1v1+c2 1,1v2, [v1, v2]=[v2, v1] = c1 1,2v1+c2 1,2v2, [v2, v2] = c1 2,2v1+c2 2,2v2and the law of F3is [v1, v1] = c1 1,1v1+c2 1,1v2, [v1, v2]=[v2, v1] = c1 1,2v1+c2 1,2v2, [v2, v2] = c1 2,2v1+c2 2,2v2. Considering a basis change similar to the one used in the proof of Proposition 17, we obtain the law [e1, e1] = e1+αe2, [e2, e2] = βe1+e2, [e1, e2] = [e2, e1] = e1for F1. For flexible algebras F2and F3, the basis change ϕ:Fi→ Figiven by e1=ϕ(v1) = 1 c1 1,1 v1, e2=ϕ(v2) = 1 c2 2,2 v2, for i= 2,3 leads to the law [e1, e1] = e1, [e2, e2] = e2, [e1, e2] = (1 −c2 1,2 c1 1,1 e1+c2 1,2 c1 1,1 e2, [e2, e1] = (1 −c2 2,1 c1 1,1 e1+c2 2,1 c1 1,1 e2. 7 Algorithmic methods In this section two algorithmic methods are introduced. The first one checks if a fixed directed and weighted CW complex is associated or not with a flexible algebra. The second procedure obtains all the (pseudo)digraphs associated with a parametric family of flexible algebras starting from its law when contains no full triangles. It also draws all those (pseudo)digraphs. 21 Notice that these algorithms have been used in order to achieve all the results of Section 5 and 6. 7.1 Checking if a fixed directed and weighted CW complex is associated with a flexible algebra This algorithmic procedure has been implemented by using the symbolic computation package Maple, working the implementation in version 18. To do this, the libraries linalg and combinat have to be used in order to activate commands related to Linear and Combinatorial Algebra. This algorithmic procedure consists of the following three steps: a) Obtaining the values of the structure constants according to the directed and weighted CW complex. b) Defining the law which should be fulfilled by the flexible algebra, starting from the structure constants. c) Checking if the flexible identity is satisfied for this law. In order to develop the implementation, three subprocedures for the two first steps and one main procedure for the last one are required. Before running the procedure, one need the command restart to reset all the variables and delete all the computations saved in the kernel. The first step of this algorithm is executed by the subprocedure assignment, which allows to define the dimension and the value of the structure constants of the vector space associated with the directed and weighted CW complex 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 directed and weighted CW complex 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. The output is the value of the variable dim with the dimension of the directed and weighted CW complex 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]]; > od; 22 > 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]); > od; > end proc: Next, one 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 directed and weighted CW complex. The subroutine computes the bracket of these two vectors. In the implementation, a local variable, v, is used 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]; > fi; > od; > return v; > end proc: Now, the implementation of the subprocedure called bracket is shown. This subroutine is devoted to computing the bracket 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); > od; > od; > exp; > end proc: Finally, let us proceed with the implementation of the main procedure called flexible, which checks if the vector space is or is not a flexible algebra. This procedure receives as input the dimension nof the vector space Fand returns the message “True” in case that the vector space Fis a flexible algebra and “False” otherwise. 23 > flexible:=proc(n) > local L,M,N,P; > L:=[];M:=[];N:=[];P:=[]; > for i from 1 to n do > L:=[op(L),i,i]; > od; > M:=permute(L,2); > for j from 1 to nops(M) do > eq[j]:=bracket(e[M[j][1]],bracket(e[M[j][2]],e[M[j][1]],n),n)- bracket(bracket(e[M[j][1]],e[M[j][2]],n),e[M[j][1]],n); > od; > 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]]; > fi; > od; > if P=[] then return "True" > else return "False"; > fi; > end proc: Example 3. The following example is shown in order to illustrate the algorithmic procedure. It corresponds to the directed and weighted CW complex of Figure 6. Figure 12: Example. According to the notation followed in this algorithm, one has to consider > V=[1,2,3]; > E={[[1,2,1],1],[[2,1,1],1],[[2,2,2],1],[[1,3,2],1],[[3,1,2],1], [[2,3,2],2],[[3,2,2],2],[[3,3,3],1]}; Now, by running the remaining procedure, the following is obtained 24 > assignment(V,E); > flexible(dim); > "True" Therefore, the directed and weighted CW complex is associated with a 3-dimensional flexible algebra. 7.2 Obtaining the (pseudo)digraph associated with a flexible algebra In this subsection, a parametric family of flexible algebras is considered. The algorithmic method computes and draws all the (pseudo)digraphs associated with flexible algebras obtained from the previous family. Under the same notation of the previous section, let Fbe an n-dimensional flexible algebra with basis Band law given in (1) in order to avoid the presence of full triangles. The algorithm is structured in four steps 1. Obtaining the bracket product between two arbitrary basis vectors in B. 2. Computing the bracket between two vectors expressed as a linear combination of vectors from basis B. 3. Imposing flexible identity and solving the corresponding system of equations. 4. Drawing the (pseudo)digraph associated with the flexible algebra F. This algorithm is implemented by using the symbolic computation package MAPLE 18 loading the libraries linalg,combinat,GraphTheory and Maplets[Elements]. The first three libraries provide 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 F. The first subprocedure, named law2, receives two natural numbers as inputs. These numbers represent the subindexes of two basis vectors in B. The subprocedure returns the result of the bracket between these two vectors. In addition, conditional sentences are inserted to determine the nonzero brackets. 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Whitehead, Combinatorial homotopy I, Bulletin of the American Mathematical Society 55:5 (1949), 213–245. 9 Appendix In this section, we show some lists of restrictions from Propositions 2, 3 Restrictions for Propositions 2 and 3:  Configuration a): (c1 1,1, c2 1,1)= (0,0).  Configuration b): (c1 j,j, c2 j,j)= (0,0), for j= 1,2.  Configuration c): (c2 1,2, c2 2,1)= (0,0).  Configuration d): c1 1,2c2 2,1−c2 1,2c1 2,1= 0 ∧(cj 1,2, cj 2,1)= (0,0), for j= 1,2.  Configuration e): c2 1,1(c2 1,2−c2 2,1)=0∧(c1 1,1, c2 1,1),(c2 1,2, c2 2,1)= (0,0). 34  Configuration f): c1 1,2=c1 2,1= 0, ∧(c1 1,1, c2 1,1),(c1 2,2, c2 2,2)= (0,0).  Configuration g): c1 1,2=c1 2,1= 0, ∧(c1 1,1, c2 1,1)= (0,0).  Configuration h): c2 1,1(cj 1,2−cj 2,1) = 0, for j= 1,2, c1 1,1(c1 2,1−c1 1,2) = 0, c1 1,2c2 2,1−c2 1,2c1 2,1= 0, ∧(c1 1,1, c2 1,1), (c1 1,2, c1 2,1),(c2 1,2, c2 2,1)= (0,0).  Configuration i): c2 1,1(cj 1,2−cj 2,1) = 0, c1 2,2(cj 1,2−cj 2,1) = 0, cj j,j(c1 2,1− c1 1,2) + c1 1,2c2 2,1−c2 1,2c1 2,1= 0, for j= 1,2, ∧(c1 j,j, c2 j,j), (cj 1,2, cj 2,1)= (0,0) for j= 1,2. Restrictions for Proposition 5:  Configuration i): (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration ii): (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,3.  Configuration iii): (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2,3.  Configuration iv): (c3 2,3, c3 3,2)= (0,0).  Configuration v): (c3 2,3, c3 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration vi): c3 2,2(c3 2,3−c3 3,2) = 0, (c3 2,3, c3 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration vii): c3 2,3=c3 3,2, (c3 2,3, c3 3,2)= (0,0), (c1 3,3, c2 3,3, c3 3,3)= (0,0,0).  Configuration viii): c3 2,2(c3 2,3−c3 3,2) = 0, (c3 2,3, c3 3,2)= (0,0), (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2.  Configuration ix): c3 2,3=c3 3,2, (c3 2,3, c3 3,2)= (0,0), (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,3.  Configuration x): c3 2,3=c3 3,2, (c3 2,3, c3 3,2)= (0,0), (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 2,3.  Configuration xi): c3 2,3=c3 3,2, (c3 2,3, c3 3,2)= (0,0), (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2,3.  Configuration xii): c2 2,3c3 3,2=c3 2,3c2 3,2, (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0).  Configuration xiii): c2 2,3c3 3,2=c3 2,3c2 3,2, (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0). 35  Configuration xiv): c3 2,2(c2 2,3−c2 3,2) = c3 2,2(c3 2,3−c3 3,2) = cj 2,2(c2 2,3− c2 3,2) = c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for j= 1,2, (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration xv): c3 2,2(c2 2,3−c2 3,2) = c3 2,2(c3 2,3−c3 3,2) = cj 2,2(c2 2,3−c2 3,2) = c3 3,2c2 2,3−c3 2,3c2 3,2= 0, (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2.  Configuration xvi): c3 2,2(c3 2,3−c3 3,2) = cj 2,2(c2 2,3−c2 3,2) = ck 3,3(c3 2,3− c3 3,2) = c2 3,3(c2 2,3−c2 3,2) = −c2 2,3c2 2,2+c2 3,2c2 2,2+c3 3,2c2 2,3−c3 2,3c2 3,2= c3 3,2c2 2,3−c3 2,3c2 3,2+c3 2,3c3 3,3−c3 3,2c3 3,3= 0, for j= 1,3, k= 1,2, (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 2,3.  Configuration xvii): c3 2,2(c3 2,3−c3 3,2) = c3 2,2(c2 2,3−c2 3,2) = cj j+1,j+1(c2 2,3− c2 3,2) = cj 3,3(c3 2,3−c3 3,2) = −c2 2,3c2 2,2+c2 3,2c2 2,2+c3 3,2c2 2,3−c3 2,3c2 3,2= c3 3,2c2 2,3−c3 2,3c2 3,2+c3 2,3c3 3,3−c3 3,2c3 3,3= 0, for j= 1,2 (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2,3. Restrictions for Proposition 7:  Configuration a): (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0).  Configuration b): (c2 1,2, c2 2,1),(c3 2,3, c3 3,2)= (0,0).  Configuration c): (c1 1,2, c1 2,1),(c3 2,3, c3 3,2)= (0,0).  Configuration d): c1 1,2c2 2,1=c2 1,2c1 2,1, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c3 2,3, c3 3,2)= (0,0).  Configuration e): c1 1,2c2 2,1=c2 1,2c1 2,1, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0).  Configuration f): c1 1,2c2 2,1=c2 1,2c1 2,1, c2 2,3c3 3,2=c3 2,3c2 3,2, (c1 1,2, c1 2,1), (c2 1,2, c2 2,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0).  Configuration g): (c1 1,2, c1 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2)= (0,0).  Configuration h): (c2 1,2, c2 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2)= (0,0).  Configuration i): c2 2,3c3 3,2=c3 2,3c2 3,2, (c2 1,2, c2 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0).  Configuration j): c2 2,3c3 3,2=c3 2,3c2 3,2, (c1 1,2, c1 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0). 36  Configuration k): c2 2,3c3 3,2=c3 2,3c2 3,2, (c1 1,2, c1 2,1), (c1 1,3, c1 3,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0).  Configuration l): c1 1,2c2 2,1=c2 1,2c1 2,1,c2 2,3c3 3,2=c3 2,3c2 3,2, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0).  Configuration m): c1 1,2c2 2,1=c2 1,2c1 2,1,c1 1,3c3 3,1=c3 1,3c1 3,1,c2 2,3c3 3,2= c3 2,3c2 3,2, (c1 1,2, c1 2,1), (c2 1,2, c2 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0). Restrictions for Proposition 9:  Configuration 1): c2 1,1c2 1,2−c2 1,1c2 2,1= 0, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 2): c2 1,2=c2 2,1, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 3): c2 1,2=c2 2,1,c2 2,3=c2 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2.  Configuration 4): c2 1,1(c2 1,2−c2 2,1) = c2 3,3(c2 3,2−c2 2,3) = 0, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,3.  Configuration 5): c2 1,2=c2 2,1, c2 2,3=c2 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2,3.  Configuration 6): c2 1,1(c2 1,2−c2 2,1) = 0, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 7): c1 2,2(c2 1,2−c2 2,1) = c2 2,2(c2 1,2−c2 2,1) = c3 2,2(c2 1,2−c2 2,1) = c3 2,2(c3 2,3−c3 3,2) = 0, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 8): c3 2,3=c3 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 3,3, c2 3,3, c3 3,3)= (0,0,0).  Configuration 9): c2 1,2=c2 2,1, c3 2,3=c3 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2.  Configuration 10): c2 1,2=c2 2,1, c3 2,3=c3 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,3.  Configuration 11): c2 1,2=c2 2,1, c3 2,3=c3 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 2,3. 37  Configuration 12): c2 1,2=c2 2,1, c3 2,3=c3 3,2, (c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2,3.  Configuration 13): c1 1,2=c1 2,1, (c1 1,2, c1 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 14): c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = 0, (c1 1,2, c1 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 15): c1 1,1(c1 1,2−c1 2,1) = c2 1,1(c1 1,2−c1 2,1) = c3 1,1(c1 1,2−c1 2,1) = c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = 0, (c1 1,2, c1 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2.  Configuration 16): c1 1,2=c1 2,1,c3 2,3=c3 3,2, (c1 1,2, c1 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,3.  Configuration 17): c1 1,2=c1 2,1,c3 2,3=c3 3,2, (c1 1,2, c1 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2,3.  Configuration 18): ci 1,1(c1 1,2−c1 2,1) = c2 1,1(c2 1,2−c2 2,1) = c2 2,1c1 1,2− c2 1,2c1 2,1= 0, for i= 1,2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 19): c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2−c2 1,2c1 2,1+c2 1,2c2 2,2− c2 2,1c2 2,2=c3 2,2(c2 1,2−c2 2,1) = c1 2,2(c1 1,2−c1 2,1) = c1 2,2(c2 1,2−c2 2,1) = c3 2,2(c3 2,3− c3 3,2) = 0, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c2 2,3, c2 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 20): c1 1,2c2 2,1=c2 1,2c1 2,1, c3 2,3=c3 3,2, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c2 2,3, c2 3,2)= (0,0), (c1 3,3, c2 3,3, c3 3,3)= (0,0,0).  Configuration 21): −c1 1,2c1 1,1+c1 2,1c1 1,1+c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2− c2 1,2c1 2,1+c2 1,2c2 2,2−c2 2,1c2 2,2=ci+1 i,i (c2 1,2−c2 2,1) = cj 1,1(c1 1,2−c1 2,1) = c1 2,2(c2 1,2−c2 2,1) = c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = 0, for i= 1,2, j= 2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2.  Configuration 22): c2 2,1c1 1,2−c2 1,2c1 2,1=ci 1,1(c1 1,2−c1 2,1) = c2 1,1(c2 1,2− c2 2,1) = ci 3,3(c3 2,3−c3 3,2) = 0, for i= 1,2,3 (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,3.  Configuration 23): c1 1,2=c1 2,1, c2 1,2=c2 2,1, c3 2,3=c3 3,2, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c2 2,3, c2 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 2,3. 38  Configuration 24): −c1 1,2c1 1,1+c1 2,1c1 1,1+c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2− c2 1,2c1 2,1+c2 1,2c2 2,2−c2 2,1c2 2,2=c3 2,2(c2 1,2−c2 2,1) = c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = c2 1,1(c2 1,2−c2 2,1) = c1 2,2(c2 1,2−c2 2,1) = cj i,i(c1 1,2−c1 2,1) = ck 3,3(c3 2,3−c3 3,2) =, for i= 1, j= 2,3, k= 1,2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c2 2,3, c2 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2,3.  Configuration 25): ci 1,1(c1 1,2−c1 2,1) = c2 1,1(c2 1,2−c2 2,1) = c2 2,1c1 1,2− c2 1,2c1 2,1= 0, for i= 1,2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 26): c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2−c2 1,2c1 2,1+c2 1,2c2 2,2− c2 2,1c2 2,2=c3 2,2(c2 1,2−c2 2,1) = c1 2,2(c1 1,2−c1 2,1) = c1 2,2(c2 1,2−c2 2,1) = c3 2,2(c3 2,3− c3 3,2) = 0, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c3 2,3, c3 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 27): c1 1,2c2 2,1=c2 1,2c1 2,1, c3 2,3=c3 3,2, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c3 2,3, c3 3,2)= (0,0), (c1 3,3, c2 3,3, c3 3,3)= (0,0,0).  Configuration 28): −c1 1,2c1 1,1+c1 2,1c1 1,1+c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2− c2 1,2c1 2,1+c2 1,2c2 2,2−c2 2,1c2 2,2=ci+1 i,i (c2 1,2−c2 2,1) = cj 1,1(c1 1,2−c1 2,1) = c1 2,2(c2 1,2−c2 2,1) = c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = 0, for i= 1,2, j= 2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2.  Configuration 29): c2 2,1c1 1,2−c2 1,2c1 2,1=ci 1,1(c1 1,2−c1 2,1) = c2 1,1(c2 1,2− c2 2,1) = ci 3,3(c3 2,3−c3 3,2) = 0, for i= 1,2,3 (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,3.  Configuration 30): c1 1,2=c1 2,1, c2 1,2=c2 2,1, c3 2,3=c3 3,2, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1), (c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 2,3.  Configuration 31): −c1 1,2c1 1,1+c1 2,1c1 1,1+c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2− c2 1,2c1 2,1+c2 1,2c2 2,2−c2 2,1c2 2,2=c3 2,2(c2 1,2−c2 2,1) = c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = c2 1,1(c2 1,2−c2 2,1) = c1 2,2(c2 1,2−c2 2,1) = cj i,i(c1 1,2−c1 2,1) = ck 3,3(c3 2,3−c3 3,2) =, for i= 1, j= 2,3, k= 1,2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2,3.  Configuration 32): ci 1,1(c1 1,2−c1 2,1) = c2 1,1(c2 1,2−c2 2,1) = c2 2,1c1 1,2− c2 1,2c1 2,1=c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for i= 1,2,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0). 39  Configuration 33): ci 2,2(c2 1,2−c2 2,1) = ci 2,2(c2 2,3−c2 3,2) = c2 2,1c1 1,2− c2 1,2c1 2,1=c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = c3 3,2c2 2,3−c3 2,3c2 3,2= −c2 1,2c1 2,2+c2 2,1c1 2,2+c3 2,2c2 2,3−c3 2,2c2 3,2= 0, for i= 1,2,3, (c1 1,2, c1 2,1), (c2 1,2, c2 2,1), (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 34): ci 1,1(c1 1,2−c1 2,1) = c1 2,2(c1 1,2−c1 2,1) = cj 2,2(c2 2,3− c2 3,2) = ck 2,2(c2 1,2−c2 2,1) = c2 1,1(c2 1,2−c2 2,1) = c3 2,2(c3 2,3−c3 3,2) = c1 2,2(c2 2,3− c2 3,2) = −c1 1,2c1 1,1+c1 2,1c1 1,1+c2 2,1c1 1,2−c2 1,2c1 2,1=c2 2,1c1 1,2−c2 1,2c1 2,1+ c2 1,2c2 2,2−c2 2,1c2 2,2=c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for i= 2,3, j= 2,3, k= 1,3, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2.  Configuration 35): −c1 1,2c1 1,1+c1 2,1c1 1,1+c2 2,1c1 1,2−c2 1,2c1 2,1=−c2 1,2c1 2,2+ c2 2,1c1 2,2+c3 2,2c2 2,3−c3 2,2c2 3,2=−c2 2,3c2 2,2+c2 3,2c2 2,2+c3 3,2c2 2,3−c3 2,3c2 3,2= c2 2,1c1 1,2−c2 1,2c1 2,1+c2 1,2c2 2,2−c2 2,1c2 2,2=ci 1,1(c1 1,2−c1 2,1) = c1 2,2(c1 1,2−c1 2,1) = cj 2,2(c2 2,3−c2 3,2) = c2 3,3(c2 2,3−c2 3,2) = ck 3,3(c3 2,3−c3 3,2) = c3 2,2(c3 2,3−c3 3,2) = cj 2,2(c2 1,2−c2 2,1) = c2 1,1(c2 1,2−c2 2,1) = c3 3,2c2 2,3−c3 2,3c2 3,2+c3 2,3c3 3,3−c3 3,2c3 3,3= 0, for i= 2,3, j= 1,3, k= 1,2, (c1 1,2, c1 2,1),(c2 1,2, c2 2,1),(c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2,3.  Configuration 36): c3 1,1(c3 1,3−c3 3,1), ci 1,1(c1 2,1−c1 1,2) = 0, for i= 1,2,3, (c1 1,2, c1 2,1), (c3 1,3, c3 3,1),(c2 2,3, c2 3,2)= (0,0),(c1 1,1, c2 1,1, c3 1,1)= (0,0,0)  Configuration 37): c3 1,1(c3 1,3−c3 3,1), ci 1,1(c1 2,1−c1 1,2) = ci 2,2(c2 3,2−c2 2,3) = c1 2,2(c1 2,1−c1 1,2) = 0, for i= 1,2,3, (c1 1,2, c1 2,1),(c3 1,3, c3 3,1),(c2 2,3, c2 3,2)= (0,0),(c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2.  Configuration 38): c1 1,2=c1 2,1, c3 1,3=c3 3,1, c2 2,3=c2 3,2, (c1 1,2, c1 2,1),(c3 1,3, c3 3,1), (c2 2,3, c2 3,2)= (0,0),(c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2,3.  Configuration 39): ci 1,1(c2 1,2−c2 2,1) = 0, for i= 2,3, (c2 1,2, c2 2,1),(c3 1,3, c3 3,1), (c2 2,3, c2 3,2)= (0,0),(c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 40): c2 1,2=c2 2,1,c2 2,3=c2 3,2, (c2 1,2, c2 2,1),(c3 1,3, c3 3,1), (c2 2,3, c2 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 41): −c2 1,2c1 2,2+c2 2,1c1 2,2+c2 2,3c3 2,2−c2 3,2c3 2,2=ci 2,2(c2 1,2− c2 2,1) = ci 2,2(c2 3,2−c2 2,3) = cj 1,1(cj 1,j −cj j,1) = 0, for i= 1,2,3, j= 2,3, (c2 1,2, c2 2,1),(c3 1,3, c3 3,1),(c2 2,3, c2 3,2)= (0,0),(c1 k,k, c2 k,k, c3 k,k)= (0,0,0), for k= 1,2.  Configuration 42): c2 1,2=c2 2,1, c3 1,3=c3 3,1, c2 2,3=c2 3,2, (c2 1,2, c2 2,1),(c3 1,3, c3 3,1), (c2 2,3, c2 3,2)= (0,0),(c1 j,j, c2 j,j, c3 j,j)= (0,0,0), for j= 1,2,3. 40  Configuration 43): c1 1,2=c1 2,1, c1 1,3=c1 3,1, (c1 1,2, c1 2,1), (c1 1,3, c1 3,1), (c2 2,3, c2 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 44): c1 2,2(c1 1,2−c1 2,1) = ci 2,2(c2 3,2−c2 2,3) = 0, for i= 1,2,3, (c1 1,2, c1 2,1), (c1 1,3, c1 3,1), (c2 2,3, c2 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 45): c1 1,2=c1 2,1, c1 1,3=c1 3,1, c2 2,3=c2 3,2, (c1 1,2, c1 2,1),(c1 1,3, c1 3,1), (c2 2,3, c2 3,2)= (0,0),(c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2.  Configuration 46): c1 1,2=c1 2,1, c1 1,3=c1 3,1, c2 2,3=c2 3,2, (c1 1,2, c1 2,1),(c1 1,3, c1 3,1), (c2 2,3, c2 3,2)= (0,0),(c1 i,i, c2 i,i, c3 i,i)= (0,0,0), for i= 1,2,3.  Configuration 47): ci 1,1(c2 1,2−c2 2,1) = c2 2,3c3 3,2−c3 2,3c2 3,2= 0, for i= 2,3, (c2 1,2, c2 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0),(c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 48): ci 2,2(c2 1,2−c2 2,1) = ck 2,2(c2 2,3−c2 3,2) = c2 1,2c1 2,2+ c2 2,1c1 2,2=−c2 2,3c2 2,2+c2 3,2c2 2,2+c3 3,2c2 2,3−c3 2,3c2 3,2=c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for i= 1,2,3, k= 1,3 (c2 1,2, c2 2,1), (c3 1,3, c3 3,1),(c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0),(c1 2,2, c2 2,2, c3 2,2)= (0,0,0).  Configuration 49): ci 2,2(c2 1,2−c2 2,1) = cj 1,1(cj 1,j −cj j,1) = ck 2,2(c2 2,3−c2 3,2) = c3 2,2(c3 2,3−c3 3,2) = c2 1,2c1 2,2+c2 2,1c1 2,2=−c2 2,3c2 2,2+c2 3,2c2 2,2+c3 3,2c2 2,3− c3 2,3c2 3,2=c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for i= 1,2,3, j= 2,3, k= 1,3, (c2 1,2, c2 2,1),(c3 1,3, c3 3,1),(c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2.  Configuration 50): ci 2,2(c2 1,2−c2 2,1) = ci 3,3(c3 1,3−c3 3,1) = cj 3,3(c3 2,3−c3 3,2) = ck 2,2(c2 2,3−c2 3,2) = c2 1,1(c2 1,2−c2 2,1) = c2 1,2c1 2,2+c2 2,1c1 2,2=−c2 2,3c2 2,2+ c2 3,2c2 2,2+c3 3,2c2 2,3−c3 2,3c2 3,2=c3 2,2(c3 2,3−c3 3,2) = c2 3,3(c2 2,3−c2 3,2) = c3 1,1(c3 1,3−c3 3,1) = c3 3,2c2 2,3−c3 2,3c2 3,2+c3 2,3c3 3,3−c3 3,2c3 3,3=−c3 1,3c1 3,3− c3 2,3c2 3,3+c3 3,1c1 3,3+c3 3,2c2 3,3= 0, for i= 1,2,3, j= 1,2, k= 1,3, c1 1,2= c1 2,1, c1 1,3=c1 3,1, c2 2,3=c2 3,2, (c2 1,2, c2 2,1),(c3 1,3, c3 3,1),(c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0),(c1 ℓ,ℓ, c2 ℓ,ℓ, c3 ℓ,ℓ)= (0,0,0), for ℓ= 1,2,3.  Configuration 51): c3 1,1(c3 1,3−c3 3,1) = ci 1,1(−c1 2,1+c1 1,2) = c3 3,2c2 2,3− c3 2,3c2 3,2=c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for i= 1,2,3, (c1 1,2, c1 2,1),(c3 1,3, c3 3,1), (c2 2,3, c2 3,2),(c3 2,3, c3 3,2)= (0,0), (c1 1,1, c2 1,1, c3 1,1)= (0,0,0).  Configuration 52): c1 2,2(c1 1,2−c1 2,1) = c3 2,2(c3 2,3−c3 3,2) = ci 2,2(c2 2,3−c2 3,2) = −c2 2,3c2 2,2+c2 3,2c2 2,2+c3 3,2c2 2,3−c3 2,3c2 3,2=c3 3,2c2 2,3−c3 2,3c2 3,2= 0, for i= 1,3, (c1 1,2, c1 2,1), (c3 1,3, c3 3,1), (c2 2,3, c2 3,2), (c3 2,3, c3 3,2)= (0,0), (c1 2,2, c2 2,2, c3 2,2)= (0,0,0). 41