The computation of Abelian subalgebras in low-dimensional solvable Lie algebras

The Computation of Abelian Subalgebras in Low-Dimensional Solvable Lie Algebras MANUEL CEBALLOS, JUAN N ´ U˜ NEZ University of Seville Geometry and Topology Aptdo. 1160. 41080-Seville. SPAIN [email protected], jnv[email protected] ´ ANGEL F. TENORIO Pablo de Olavide University Economics, Quantitative Methods and Ec. History Ctra. Utrera km. 1. 41013-Seville SPAIN [email protected] Abstract: The main goal of this paper is to compute the maximal abelian dimension of each solvable nondecomposable Lie algebra of dimension less than 7. To do it, we apply an algorithmic method which goes ruling out non-valid maximal abelian dimensions until obtaining its exact value. Based on Mubarakzyanov and Turkowsky’s classical classifications of solvable Lie algebras (see [13] and [19]) and the classification of 6-dimensional nilpotent Lie algebras by Goze and Khakimdjanov [7], we have explicitly computed the maximal abelian dimension for the algebras given in those classifications. Key–Words: Solvable Lie algebra, maximal abelian dimension. 1 Introduction The main topic of this paper is the maximal abelian dimension of a given finite-dimensional Lie algebra g. Let us recall that it is the maximum among the dimensions of the abelian subalgebras of g. Most papers about this topic (like [4, 18]) work with abelian ideals instead of abelian subalgebras, which implies more restrictive hypotheses. In our work, such assumptions and restrictions are not used because we are considering every abelian subalgebra of the Lie algebra g. In this sense, we are working in the line of [3, 6]. Previously, we have already studied this concept. The reader can consult [17] for some properties of Lie algebras using the maximal abelian dimension. The computation of the maximal abelian dimension was done for the Lie algebra gn, of n×nstrictly uppertriangular matrices in [1, 2] and for the Lie algebra hn, of n×nupper-triangular ones in [5]. These papers proved properties of these algebras determining the maximal abelian dimension depending on the order n. To continue our research, the maximal abelian dimension is computed for non-decomposable solvable Lie algebras of dimension up to 7, applying and adjusting the algorithmic method of [2], which uses the main and non-main vectors already introduced in those papers. Before we get down to work with the topic of this paper, we would like to comment, as examples, some applications of Lie Theory to several fields of research. First of all, Lie groups and algebras have been profusely used as tools in Theoretical and Mathematical Physics. In fact, a very classical use corresponds to the study of symmetries in problems involving differential equations as can be seen in Senashov et al. [15] or the classical reference by Olver [14]. In this way, Lie Theory can be applied to dynamical systems and, more specifically, control problems. Examples in this last field are given in Kirillova et al. [10] or Lantos [11]. Let us note that there exist applications of Lie Theory, by means of optimal control problems, related to fields like Medicine as can be seen in [12]. Finally, we would like to indicate that Social Sciences are also applying Lie Theory for dealing with their subjects. In this sense, we would like to cite the papers by Sumedrea and Sangeorzan [16] and Hern´ andez et al. [9] in order to exemplify this assertion. 2 Preliminaries This section is devoted to recall some preliminary concepts and results on Lie algebras. For a general overview, the reader can consult [20]. From here on, only finite-dimensional Lie algebras over the complex number field are considered. The upper central series of a given Lie algebra g is defined by C1(g) = g, C2(g) = [g, g],C3(g) = [C2(g),C2(g)], . . . , Ck(g) = [Ck−1(g),Ck−1(g)], . . . (1) When there exists m∈Nsuch that Cm(g)≡0, the Lie algebra gis called solvable. WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 22 Issue 1, Volume 9, January 2010 Aspecial class of solvable Lie algebras is formed by abelian algebras. A Lie algebra his abelian if [v,w] = 0, for all v,w∈h. A very important abelian subalgebra of a Lie algebra gis its center, which is defined as: cen(g) = {X∈g|[X, Y ] = 0,∀Y∈g}. Another useful ideal of gis its nilradical, which corresponds to the sum of all the nilpotent ideals of the algebra g. Finally, the maximal abelian dimension of g, which will be denoted by M(g), is the maximum among the dimensions of its abelian subalgebras. The invariant M(g)is monotone and additive: for a subalgebra hof gwe have M(h)≤ M(g), and for two Lie algebras aand bwe have M(a⊕b) = M(a) + M(b). By applying these properties, the maximal abelian dimension can be computed for all decomposable solvable Lie algebras of dimension less than 7. Now we explain the method used in this paper to compute the maximal abelian dimension of a given Lie algebra gof dimension d. Let Bd={Xi}d i=1 be a basis of gand let B={vh}r h=1 be a basis of an arbitrary r-dimensional (abelian) subalgebra h(with r≤d). Each vector vh∈ B is expressible as a linear combination vh=Pd i=1 ah,iXiof the vectors in Bd. Hence, Bcan be translated to a matrix in which the hth row saves the coordinates of vhwith respect to Bd    a1,1a1,2· · · a1,d . . .. . ..... . . ar,1ar,2· · · ar,d   (2) Since the rank of the previous matrix is equal to r, its echelon form, obtained by using elementary row and column transformations, is the following       b1,10· · · 0b1,r+1 · · · b1,d 0b2,2· · · 0b2,r+1 · · · b2,d . . .. . ..... . .. . ..... . . 0 0 · · · br,r br,r+1 · · · br,d       (3) Note that the vector Xi∈ Bdcan be associated with a different row depending on Expressions (2) or (3). Consequently, we can suppose that every basis B of his expressible by a matrix similar to (3) and each vector in Bis a linear combination of two different types of vectors Xi: the ones coming from the pivots and the remaining ones. The vectors Xicorresponding to pivots are called main vectors of Bwith respect to Bd, whereas the rest are called non-main vectors. 3 Solvable Lie algebras of dimension less than 5 In this section, the maximal abelian dimension is studied for solvable non-decomposable Lie algebras of dimension less than 5, following Mubarakzyanov’s classification [13]. Obviously, the unique 1-dimensional Lie algebra is the abelian Lie algebra of dimension 1and its maximal abelian dimension is equal to its dimension. Since the maximal abelian dimension of abelian algebras is exactly their dimension, abelian algebras are not considered in this paper. 3.1 Solvable Lie algebras of dimension 2 We compute the maximal abelian dimension of the Lie algebra g2,1. Proposition 1 It is verified that M(g2,1) = 1. Proof: The Lie algebra g2,1is generated by the vectors {Z1, Z2}and there is a unique nonzero bracket: [Z1, Z2] = Z1. Hence, g2,1is non-abelian and M(g2,1)<2. Since 1-dimensional Lie algebras are abelian, both hZ1iand hZ2iare abelian subalgebras of g2,1and M(g2,1) = 1.ut 3.2 Solvable Lie algebras of dimension 3 Now, let us compute the maximal abelian dimension of 3-dimensional solvable Lie algebras. Proposition 2 The maximal abelian dimension of the Lie algebras g3,i, where i∈ {1, . . . , 5}, is given by M(g3,i) = 2. Proof: Fixed and given i∈ {1,2,3,4,5}, let us prove that M(g3,i) = 2. We have to find a 2-dimensional abelian subalgebra in g3,i, in addition to determine the nonexistence of 3-dimensional abelian subalgebras. First, M(g3,i)≤2because the Lie algebra g3,i is non-abelian. Hence, the problem is now reduced to prove the existence of 2-dimensional abelian subalgebras of g3,i. Obviously, the subalgebra hZ1, Z2iof g3,i is abelian for i∈ {1,2,3,4,5}. So, M(g3,i) = 2.ut 3.3 Solvable Lie algebras of dimension 4 Next, we study the maximal abelian dimension of solvable Lie algebra of dimension 4. Proposition 3 It is verified that M(g4,j) = (3,if j∈ {1,4,5,6,7,8}, 2,if j∈ {2,3,9,10,11}.(4) WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 23 Issue 1, Volume 9, January 2010 Proof: Fixed and given i∈ {1,4,5,6,7,8}, let us prove that M(g4,i)=3. So, we have to find a 3dimensional abelian subalgebra in g4,i, in addition to determine the nonexistence of 4-dimensional abelian subalgebras. Since the Lie algebra g4,i is non-abelian, we can set that M(g4,i)≤3. In this way, we only have to prove the existence of 3-dimensional abelian subalgebras. Let us consider the subalgebras h1=hZ1, Z2, Z3i,h2=hZ1, Z2, Z4iand h3=hZ2, Z3, Z4i. It is easy to prove that h1,h2 and h3are abelian subalgebras of g4,5,g4,6and g4,j (for j∈ {1,4,7,8}), respectively. Consequently, M(g4,i) = 3, for i∈ {1,4,5,6,7,8}. Now, let us prove that 2is the maximal abelian dimension of the Lie algebras g4,k, for k∈ {2,3,9,10,11}. Since the Lie algebra g4,k is nonabelian, its maximal abelian dimension is less than 4. We define the subalgebras al=h{Zi+λiZl|1≤i≤4∧i6=l}i,(5) where Zlis the non-main vector. It can be proved that the subalgebras alare not abelian for l= 1,2,3,4 as subalgebras of g4,k. Hence, the maximal abelian dimension of g4,k is less than 3. Since the subalgebra hZ3, Z4iof g4,k is abelian, we can assert that M(h4,k) = 2.ut 4 Solvable Lie algebras of dimension 5 Due to the complexity of the problem, we will study solvable non-decomposable Lie algebras of dimension 5by considering Mubarakzyanov’s classification [13] and distinguishing the following cases: 4.1 Nilpotent Lie algebras and solvable non-nilpotent ones containing a 4dimensional abelian subalgebra We deal now with the maximal abelian dimension of 5-dimensional nilpotent Lie algebras and solvable ones containing an abelian subalgebra of dimension 4. Proposition 4 Given i∈ {1, . . . , 4}, it is verified that M(g5,i) = 3. Proof: For i∈ {1, . . . , 4}, we have to compute a 3dimensional abelian subalgebra of g5,i and to prove the nonexistence of abelian subalgebras of dimension greater than 3. Obviously, the maximal abelian dimension of g5,i is less than 5, due to not being abelian. Moreover, the subalgebra hZ1, Z2, Z3iis abelian. Consequently, it is sufficient to prove that it is not possible to obtain 4-dimensional abelian subalgebras of g5,i, for i∈ {1, . . . , 4}. For reason of length, only one of these algebras is studied explicitly. The same reasoning can be applied for the rest of these algebras. Let us consider the Lie algebra g5,3whose law with respect to a certain basis {Zi}5 i=1 is given by: [Z2, Z4] = Z3,[Z2, Z5] = Z1,[Z4, Z5] = Z2.(6) Let us suppose the existence of a 4-dimensional abelian subalgebra. We can suppose that each vector in a basis of such subalgebra is expressed as a linear combination of a main vector and the non-main one. So 4-dimensional subalgebras would be expressed as ak=h{Zi+λiZk|1≤i≤5∧i6=k}i,(7) where Zkis the non-main vector. Proving that akis non-abelian is equivalent to finding a nonzero bracket in its law. •For k∈ {1,3,5}:[Z2+λ2Zk, Z4+λ4Zk] = Z3+vis nonzero, because v∈ hZ1, Z2, Z4, Z5i. •For k= 2:[Z4+λ4Z2, Z5+λ5Z2] = Z2+vis nonzero, because v∈ hZ1, Z3, Z4, Z5i. •For k= 4:[Z2+λ2Z4, Z5+λ5Z4] = Z1+vis nonzero, because v∈ h{Zi}i=5 i=2i. Hence, there do not exist 4-dimensional abelian subalgebras of g5,3and M(g5,3) = 3.ut Proposition 5 Fixed and given j∈ {5, . . . , 18}the maximal abelian dimension of g5,j is M(g5,j) = 4. Proof: Let us note that the 4-dimensional subalgebra of g5,j hZ1, Z2, Z3, Z4iis abelian for j∈ {5, . . . , 18}. Since g5,j is not abelian, its maximal abelian dimension is exactly 4and the abelian subalgebra previously expounded is a maximal abelian subalgebra. ut 4.2 Solvable, non-nilpotent Lie algebras of dimension 5 containing a 3-dimensional abelian subalgebra Next, we study the maximal abelian dimension of the solvable Lie algebras of dimension 5that contain a 3-dimensional abelian subalgebra. Proposition 6 For j∈ {19, . . . , 38}, the maximal abelian dimension of g5,j is M(g5,j) = 3. Proof: The subalgebras hZ1, Z3, Z4iand hZ1, Z2, Z3i are abelian for j∈ {19, . . . , 29}and for j∈ {30, . . . , 38}, respectively. So, we can set that M(g5,j)≥3for j∈ {19, . . . 38}. WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 24 Issue 1, Volume 9, January 2010 Consequently,it is sufficient to prove the nonexistence of 4-dimensional abelian subalgebras of g5,j. Once more, the reasoning is analogous for all these algebras. So we only study explicitly the algebra g5,22, whose law is with respect to a certain basis {Zi}5 i=1: [Z2, Z3] = Z1,[Z2, Z5] = Z3,[Z4, Z5] = Z4.(8) By applying the reasonings and notations used in Proposition 4, let us suppose the existence of a 4-dimensional abelian subalgebra and find a nonzero bracket in its law. •For k∈ {1,4,5}:[Z2+λ2Zk, Z3+λ3Zk] = Z1+vis nonzero, because v∈ h{Zi}5 i=2i. •For k∈ {2,3}:[Z4+λ4Zk, Z5+λ5Zk] = Z4+w is nonzero, because w∈ hZ1, Z2, Z3, Z5i. Hence, there do not exist 4-dimensional abelian subalgebras of g5,22 and M(g5,22) = 3.ut 4.3 Solvable, non-nilpotent Lie algebras of dimension 5 containing a 2-dimensional abelian subalgebra. Finally, we compute the maximal abelian dimension of 5-dimensional Lie algebras containing a 2dimensional abelian subalgebra. Proposition 7 It is verified that M(g5,39) = 2. Proof: The subalgebra hZ1, Z2iof g5,39 is abelian. Hence, M(g5,39)≥2and we only have to prove the nonexistence of abelian subalgebras of dimension 3. Let us suppose the existence of a 3-dimensional abelian subalgebra. We can suppose that each vector in a basis of such subalgebra is expressed as a linear combination of a main vector and two non-main ones. So 3-dimensional subalgebras would be expressed as aj,k =h{Zi+λiZj+µiZk|1≤i≤5∧j, k 6=i}i,(9) where Zjand Zkare the two non-main vectors. Proving that akis non-abelian is equivalent to finding a nonzero bracket in its law. •For (j, k)∈ {(1,2),(1,4)}: the bracket [Z3+λ3Zj+µ3Zk, Z5+λ5Zj+µ5Zk] = Z2+v is nonzero, because v∈ hZ1, Z3, Z4, Z5i. •For (j, k)∈ {(1,3),(1,5)}: the bracket [Z2+λ2Zj+µ2Zk, Z4+λ4Zj+µ4Zk] = Z2+v is nonzero, because v∈ hZ1, Z3, Z4, Z5i. •For (j, k)∈ {(2,5),(3,5)}: the bracket [Z1+λ1Zj+µ1Zk, Z4+λ4Zj+µ4Zk] = 2Z1+w is nonzero, because w∈ hZii5 i=2. •For (j, k) = (4,5): the bracket [Z2+λ2Zj+µ2Zk, Z3+λ3Zj+µ3Zk] = Z1+w is nonzero, because w∈ hZii5 i=2. •For (j, k) = (2,3): we consider the bracket [Z1+λ1Z2+µ1Z3, Z4+λ4Z2+µ4Z3]= (2 + λ1µ4−µ1λ4)Z1+λ1Z2+µ1Z3. If the subalgebra a2,3is abelian, this bracket would have to be zero, obtaining the system {2 + λ1µ4−µ1λ4= 0, λ1= 0, µ1= 0}, which has no solution. •For (j, k) = (2,4): we have the brackets [Z3+λ3Z2+µ3Z4, Z5+λ5Z2+µ5Z4] = −λ5Z1 +(1 + λ3µ5−µ3λ5)Z2+ (µ5−λ3)Z3,(10) [Z1+λ1Z2+µ1Z4, Z5+λ5Z2+µ5Z4] = 2µ5Z1−λ1Z3+ (λ1µ5−µ1λ5)Z2.(11) If the subalgebra a2,4is abelian, these brackets would have to be zero and the following system without solutions would be obtained: {1 + λ3µ5−µ3λ5= 0, λ5= 0, µ5−λ3= 0, 2µ5= 0, λ1= 0, λ1µ5−µ1λ5= 0}. •For (j, k) = (3,4): we consider the brackets [Z2+λ2Z3+µ2Z4, Z5+λ5Z3+µ5Z4] = λ5Z1 +(λ2µ5−µ2λ5−1)Z3+ (µ5+λ2)Z2,(12) [Z1+λ1Z3+µ1Z4, Z2+λ2Z3+µ2Z4] = −µ1Z2 +(2µ2−λ1)Z1+ (λ1µ2−µ1λ2)Z3,(13) [Z1+λ1Z3+µ1Z4, Z5+λ5Z3+µ5Z4] = 2µ5Z1+λ1Z2+ (λ1µ5−µ1λ5)Z3,(14) If the subalgebra a3,4is abelian, these brackets would have to be zero, obtaining the following system without solutions {λ2µ5−µ2λ5−1= 0, λ5= 0, µ5+λ2= 0,2µ2−λ1= 0, µ1= 0, λ1µ2−µ1λ2= 0, µ5= 0, λ1= 0, λ1µ5−µ1λ5= 0}. Hence, there do not exist 3-dimensional abelian subalgebras of g5,39 and M(g5,39) = 2.ut 5 Solvable Lie algebras of dimension 6 This section is devoted to compute the maximal abelian dimension of each 6-dimensional solvable, non-decomposable Lie algebra g. WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 25 Issue 1, Volume 9, January 2010 5.1 Nilpotent Lie algebras of dimension 6. We deal now with the maximal abelian dimension of 6-dimensional nilpotent Lie algebras. The classification of these algebras is the one given by Goze and Khakimdjanov [7], but including the corrections given later by Goze and Remm [8]. Proposition 8 It is verified that M(g6,j) =      5,if j∈ {1,2}, 4,if j∈ {3,4, . . . , 18}, 3,if j∈ {19,20}. (15) Proof: Fixed and given i∈ {1,2}, let us prove that M(g6,i) = 5. We have to find a 5-dimensional abelian subalgebra in g6,i, and determine the nonexistence of 6-dimensional abelian subalgebras. Since the Lie algebra g6,i is non-abelian, we can set that M(g6,i)≤5. In this way, we only have to prove the existence of 5-dimensional abelian subalgebras. Since the Lie algebra hZ2, Z3, Z4, Z5, Z6iis a 5-dimensional abelian subalgebra of g6,i, we can affirm that M(g6,i)=5, for i∈ {1,2}. For i∈ {3, . . . , 18}, we have to compute a 4dimensional abelian subalgebra of g6,i and to prove the nonexistence of abelian subalgebras of dimension greater than 4. Obviously, the maximal abelian dimension of g6,i is less than 6, due to not being abelian. For reason of length, only one of these algebras is studied explicitly. The same reasoning can be applied for the rest of these algebras. We consider the Lie algebra g6,5whose law with respect to a certain basis {Zi}5 i=1 is given by: [Z1, Z2]= Z3,[Z1, Z4]= Z5,[Z2, Z4]= −Z6.(16) Let us note that hZ3, Z4, Z5, Z6iis a 4-dimensional abelian subalgebra of g6,5. Let us suppose the existence of a 5-dimensional abelian subalgebra. We can suppose that each vector in a basis of such subalgebra is expressed as a linear combination of a main vector and the non-main one. So 5-dimensional subalgebras would be expressed as ak=h{Zi+λiZk|1≤i≤6∧i6=k}i,(17) where Zkis the non-main vector. Proving that akis non-abelian is equivalent to finding a nonzero bracket in its law. •For k∈ {3,4,5,6}: [Z1+λ1Zk, Z2+λ2Zk] = Z3+vis nonzero, because v∈ hZ1, Z2, Z4, Z5, Z6i. •For k= 2:[Z1+λ1Z2, Z4+λ4Z2] = Z5+vis nonzero, because v∈ hZ3, Z6i. •For k= 1:[Z2+λ2Z1, Z4+λ4Z1] = −Z6+v is nonzero, because v∈ hZ3, Z5i. Hence, there do not exist 5-dimensional abelian subalgebras of g6,5and M(g6,5) = 4. Now, let us prove that 3is the maximal abelian dimension of the Lie algebras g6,19 and g6,20. Since these Lie algebra are non-abelian, their maximal abelian dimension is less than 6. For both Lie algebras, hZ4, Z5, Z6iis a 3-dimensional abelian subalgebra. We have to prove that it is not possible to find a4-dimensional abelian subalgebra of g6,19 or g6,20. To do it, we express a generic 4-dimensional subalgebra in g6,19 (an analogous reasoning can be applied to g6,20) as aj,k =h{Zi+λiZj+µiZk|1≤i≤6∧i6=j, k}i (18) •For (j, k)∈ {(3,4),(3,5), . . . , (4,6),(5,6)}, [Z1+λ1Zj+µ1Zk, Z2+λ2Zj+µ2Zk] = Z3+v is a nonzero bracket, because v∈ {Zh}h6=3. •For (j, k)∈ {(2,4),(2,5),(2,6)}: the bracket [Z1+λ1Zj+µ1Zk, Z3+λ3Zj+µ3Zk] = Z4+v is nonzero, because v∈ hZ1, Z3, Z4, Z5i. •For (j, k)∈ {(1,2),(1,5),(1,6)}, the bracket [Z3+λ3Zj+µ3Zk, Z4+λ4Zj+µ4Zk] = −Z6+v is nonzero, because v∈ {Zh}h≤5. •For (j, k)∈ {(1,3),(1,4)}, the bracket [Z2+λ2Zj+µ2Zk, Z5+λ5Zj+µ5Zk] = Z6+v is nonzero, because v∈ {Zh}h≤5. •For (j, k) = (2,3), the following bracket: [Z1+λ1Zj+µ1Zk, Z4+λ4Zj+µ4Zk] = Z5+v is nonzero, because v∈ hZ1, Z2, Z3, Z4, Z6i. Therefore, there do not exist 4-dimensional abelian subalgebras of g6,19 and M(g6,19) = 3. Analogously, the same conclusion can be obtained for the Lie algebra g6,20.ut 5.2 Solvable, non-nilpotent Lie algebras of dimension 6. We show the propositions where the invariant M(g) has been computed for solvable, non-nilpotent Lie algebras of dimension 6. Let us note that the proof of these results will be done in the same way as in the previous subsection. Here we have considered Mubarakzyanov and Turkowski’s classification (see [13] and [19]). Proposition 9 For j∈ {21,22 . . . , 47}, it is satisfied that M(g6,j) = 4. WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 26 Issue 1, Volume 9, January 2010 Proof: To prove this proposition, we have to find a 4dimensional abelian subalgebra of the Lie algebra g6,j (for j∈ {21,22 . . . , 47}) and prove that there do not exist any abelian subalgebras of dimension 5. Let us note that no 6-dimensional abelian subalgebra can be found in these algebras, because they are not abelian. Indeed, the subalgebra hZ3, Z4, Z5, Z6iof g6,j is abelian for j= 21, . . . , 47. So, the maximal abelian dimension of g6,j is, at least, 4. In this way, we only need to see if there exists some abelian subalgebra of dimension 5. Due to reasons of length, we only study one of the Lie algebras shown in the statement of the proposition. The rest can be studied using an analogous reasoning. We consider the algebra g6,39 whose law with respect to a certain basis {Zi}6 i=1 is given by the following nonzero brackets: [Z1, Z3] = Z4+Z5,[Z1, Z4] = −Z3+Z6, [Z1, Z5]= Z6,[Z1, Z6]= −Z5,[Z2, Z3]= Z3,(19) [Z2, Z4] = Z4,[Z2, Z5] = Z5,[Z2, Z6] = Z6. Let us suppose that there exists a 5-dimensional abelian subalgebra of g6,19. Then, we can assume that each vector in the basis of such a subalgebra are expressed as a linear combination of one main vector and the non-main one. So any 5-dimensional abelian subalgebra is expressed by ak=h{Zi+λiZk|1≤i≤6∧i6=k}i (20) where Zkis the nonmain vector. Now, we are proving that we can find a nonzero bracket in its law. •For k∈ {1,3,4,5}: a nonzero bracket is [Z2+λ2Zk, Z6+λ6Zk] = Z6+v, because v∈ hZ1, Z2, Z3, Z4, Z5i. •For k∈ {2,6}, a nonzero bracket is [Z1+λ2Zk, Z5+λ6Zk] = Z6+v, because v∈ hZ1, Z2, Z3, Z4, Z5i. Hence, there do not exist any 5-dimensional abelian subalgebras of g6,39 and M(g6,39) = 4.ut Proposition 10 For j∈ {48,49,...,60}, it is verified that M(g6,j) = 3. Proof: Let us note that hZ3, Z4, Z5iand hZ3, Z4, Z6i are abelian subalgebras of g6,48 and g6,p, for p= {49,...,60}, respectively. Hence, the maximal abelian dimension of these algbras is greater than or equal to 3. So we are now going to prove that no 4-dimensional subalgebra of g6,p is abelian for p= 48, . . . , 60. We are going to prove this fact for one of the algebras and an analogous reasoning can be applied to the rest. Let us consider the algebra g6,51, whose law with respect to a certain basis {Zi}6 i=1 is given by the following nonzero brackets [Z1, Z4] = Z4,[Z1, Z5] = −Z5, [Z2, Z3] = Z3,[Z2, Z5] = Z5,(21) [Z2, Z6] = Z1+Z6,[Z4, Z5] = Z1. Now, we prove that it is not possible to obtain a 4-dimensional abelian subalgebra of g6,51 by arguing analogously to Proposition 7. In this way, we are assuming that the subalgebras are expressed as follows aj,k =h{Zi+λiZj+µiZk|1≤i≤6∧j, k 6=i}i (22) where Zjand Zkare the two non-main vectors. To prove the nonexistence of abelian subalgebras of dimension 4, we only need to find a nonzero bracket in each of these subalgebras: •For (j, k)∈ {(1,2),(1,3),(1,6),(2,3)}, we consider the nonzero bracket given by [Z4+λ4Zj+µ4Zk, Z5+λ5Zj+µ5Zk]= Z1+v, because v∈ hZ3, Z4, Z5i. •For (j, k)∈ {(1,4),(1,5),(3,5)}, this bracket [Z2+λ2Zj+µ2Zk, Z6+λ6Zj+µ6Zk] = Z1+v is nonzero, because v∈Z3, Z4, Z5i. •For (j, k)∈ {(2,5),(2,6)}, the bracket [Z1+λ1Zj+µ1Zk, Z4+λ4Zj+µ4Zk] = Z4+v is nonzero, because v∈ hZ1, Z3, Z5, Z6i. •For (j, k)∈ {(3,4),(3,6)}, the bracket [Z2+λ2Zj+µ2Zk, Z5+λ5Zj+µ5Zk] = Z5+v is nonzero, because v∈ hZ1, Z3, Z4, Z6i. •For (j, k)∈ {(4,5),(4,6),(5,6)}, the bracket [Z2+λ2Zj+µ2Zk, Z3+λ3Zj+µ3Zk] = Z3+v is nonzero, because v∈ hZ1, Z4, Z5, Z6i. •For (j, k) = (2,4), we consider the brackets [Z1+λ1Z2+µ1Z4, Z3+λ3Z2+µ3Z4] = µ3Z4+λ1Z3 [Z1+λ1Z2+µ1Z4, Z5+λ5Z2+µ5Z4] = µ1Z1(23) +µ5Z4+ (λ1−1)Z5 If the subalgebra a2,4is abelian, these brackets would have to be zero, obtaining the following system without solutions {λ1= 0, λ1−1 = 0}. Consequently, there are not abelian subalgebras of dimension 4 in g6,51 and M(g6,51) = 3.ut WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 27 Issue 1, Volume 9, January 2010 6Conclusions In this paper a new method for computing abelian subalgebras and, in particular, the maximum among the dimension of all the abelian subalgebras of a Lie algebra has been proposed. We hope to continue with this research in the future in order to provide the classification of nilpotent and solvable Lie algebras. References: [1] J.C. Benjumea, F.J. Echarte, J. N´ u˜ nez and A.F. Tenorio: An obstruction to represent abelian Lie algebras by unipotent matrices. Extracta Math. 19, 2004, pp. 269–277. [2] J.C. Benjumea, J. N´ u˜ nez and A.F. Tenorio: The maximal abelian dimension of linear algebras formed by strictly-upper triangular matrices. Theor. Math. Phys. 152, 2007, pp. 1225– 1233. [3] R. Campoamor-Stursberg: Number of missing label operators and upper bounds for dimensions of maximal Lie subalgebras. Acta Phys. Polon. B37, 2006, pp. 2745–2760. [4] C. Caranti, S. Mattarei and F. Newman: Graded Lie algebras of maximal class. Trans. Amer. Math. Soc. 349, 1997, pp. 4021–4051. [5] M. Ceballos, J. N´ u˜ nez and A.F. Tenorio: The computation of abelian subalgebras in the Lie algebra of upper-triangular matrices. An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 16:1, 2008, pp. 59–66. [6] D. Gorenstein, R. Lyons and R. Solomon: The classification of the finite simple groups, vol. 3. Mathematical surveys and monographs, American Mathematical Society, Providence, 1997. [7] M. Goze and Y. Khakimdjanov: Nilpotent and solvable Lie algebras. In M. Hazewinkel (ed.): Handbook of Algebra Vol 2. Elsevier, Amsterdam, 2000, pp. 615–664 [8] M. Goze and E. Remm: Classifications of complex nilpotent Lie algebras. Available from World Wide Web (http://www.math.uha.fr/ algebre/goze/LIE/Lie.html), 2004. [9] I. Hern´ andez, C. Mateos, J. N´ u˜ nez and A.F. Tenorio: Lie Theory: Applications to problems in Mathematical Finance and Economics. Applied Mathematics and Computation 208:2, 2009, pp. 446–452. [10] E. Kirillova, T. Hoch and K. Spindler: Optimal control of a spin system acting on a single quantum bit. WSEAS Transactions on Mathematics 7:12, 2008, pp. 687–697. [11] B. Lantos: Path design and control for collision avoidance system of cars. In M.I. Garc´ ıa-Planas and C. Gonz´ alez-Concepci´ on (eds.): Proceedings of 5th WSEAS International Conference on System Science and Simulation in Engineering (ICOSSE’06), WSEAS, Tenerife, 2006, pp. 337– 345. [12] U. Ledzewicz and H. Sch¨ attler: Singular controls in systems describing tumor antiangiogenesis. In M.I. Garc´ ıa-Planas and C. Gonz´ alez-Concepci´ on (eds.): Proceedings of 5th WSEAS International Conference on System Science and Simulation in Engineering (ICOSSE’06), WSEAS, Tenerife, 2006, pp. 156– 161. [13] G.M. Mubarakzyanov: Classification of real structures of Lie algebras of fifth order. Izv. Vysˇs. Uˇcebn. Zaved. Matematika 3:34, 1963, pp. 99– 106. [14] P.J. Olver: Applications of Lie groups to differential equations. Springer, New York, 1986. [15] S.I. Senashov, A. Yakhno and L. Yakhno: LieBacklund symmetries of homogeneus system of bi-dimensional equations. WSEAS Transactions on Mathematics 6:1, 2007, pp. 16–21. [16] A.G. Sumedrea and L. Sangeorzan: A Mathematical Theory of Psychological Dynamics. WSEAS Transactions on Mathematics 8:10, 2009, pp. 604–613. [17] A.F. Tenorio: Solvable Lie algebras and maximal abelian dimensions. Acta Math. Univ. Comenian. (N.S.) 77, 2008, pp. 141–145. [18] J.-L. Thiffeault and P.J. Morrison: Classification and Casimir invariants of Lie-Poisson brackets. Phys. D 136, 2000, pp. 205–244. [19] P. Turkowski: Solvable Lie algebras of dimension six. J. Math. Phys. 31, 1990, pp. 1344– 1350. [20] V.S. Varadarajan: Lie groups, Lie algebras and their representations. Springer, New York, 1984. WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 28 Issue 1, Volume 9, January 2010 ATables Table 1: Non-decomposable Solvable Lie algebras of dimension less than 5. Algebra Nonzerobrackets M(g) g2,1[Z1,Z2]=Z11 g3,1[Z1,Z3]=Z22 g3,2[Z1,Z3]=Z1,[Z2, Z3] = Z22 g3,3[Z1,Z3]=Z2,[Z2, Z3] = −Z12 g3,4[Z1,Z3]=−Z1,[Z2, Z3] = −Z1−Z22 g3,5[Z1,Z3]=−Z12 g4,1[Z1,Z3]=Z2,[Z1, Z4] = Z33 g4,2[Z1,Z3]=Z3,[Z1, Z4] = Z4, [Z2, Z3] = Z42 g4,3[Z1,Z3]=Z3,[Z1, Z4] = Z4, [Z2, Z3] = −Z4,[Z2, Z4] = Z32 g4,4[Z1,Z3]=Z3,[Z1, Z2] = Z43 gα,β 4,5 [Z4,Z1]=Z1,[Z4, Z2] = αZ2, [Z4, Z3] = βZ33 gα 4,6 [Z3,Z1]=αZ1,[Z3, Z2] = Z2, [Z3, Z4] = Z2+Z43 g4,7[Z1,Z2]=Z2+Z3, [Z1, Z3] = Z3+Z4,[Z1, Z4] = Z43 gα,β 4,8 [Z1,Z4]=αZ4,[Z1, Z2] = βZ2−Z3, [Z1, Z3] = Z2+βZ33 gα 4,9 [Z2,Z3]=Z4,[Z1, Z2] = (α−1)Z2, [Z1, Z3] = Z3,[Z1, Z4] = αZ42 g4,10 [Z2,Z3]=Z4,[Z1, Z2] = Z2+Z3, [Z1, Z3] = Z3,[Z1, Z4] = 2Z42 gα 4,11 [Z2,Z3]=Z4,[Z1, Z2] = αZ2−Z3, [Z1, Z4] = 2αZ4,[Z1, Z3] = Z2+αZ32 Table 2: Non-decomposable Solvable Lie algebras of dimension 5(I). Algebra Nonzerobrackets M(g) g5,1[Z2,Z4]=Z3,[Z2, Z5] = Z1, [Z4, Z5] = Z23 g5,2[Z2,Z4]=Z1,[Z3, Z5] = Z13 g5,3[Z3,Z4]=Z1,[Z2, Z5] = Z1, [Z3, Z5] = Z23 g5,4[Z3,Z4]=Z1,[Z2, Z5] = Z1, [Z3, Z5] = Z2,[Z4, Z5] = Z33 g5,5[Z3,Z5]=Z1,[Z4, Z5] = Z24 Table 3: Non-decomposable Solvable Lie algebras of dimension 5(I). Algebra Nonzerobrackets M(g) g5,6[Z2,Z5]=Z1,[Z3, Z5] = Z2, [Z4, Z5] = Z34 gα,β,γ 5,7 [Z1,Z5]=Z1,[Z2, Z5] = αZ2, [Z3, Z5] = βZ3,[Z4, Z5] = γZ44 gγ 5,8 [Z2,Z5]=Z1,[Z3, Z5] = Z3, [Z4, Z5] = γZ4,4 gβ,γ 5,9 [Z1,Z5]=Z1,[Z2, Z5] = Z1+Z3, [Z3, Z5] = βZ3,[Z4, Z5] = γZ44 g5,10 [Z2,Z5]=Z1,[Z3, Z5] = Z2, [Z4, Z5] = Z44 gγ 5,11 [Z1,Z5]=Z1,[Z2, Z5] = Z1+Z2, [Z3, Z5] = Z2+Z3,[Z4, Z5] = γZ44 g5,12 [Z1,Z5]=Z1,[Z2, Z5] = Z1+Z2, [Z3, Z5] = Z2+Z3,[Z4, Z5] = Z3+Z44 gp,s,γ 5,13 [Z1,Z5]=Z1,[Z2, Z5] = γZ2, [Z3, Z5] = pZ3−sZ4, [Z4, Z5] = sZ3+pZ4 4 g5,14 [Z2,Z5]=Z1,[Z3, Z5] = pZ3−Z4, [Z4, Z5] = Z3+pZ44 gγ 5,15 [Z1,Z5]=Z1,[Z3, Z5] = γZ3, [Z2, Z5] = Z1+Z2, [Z4, Z5] = Z3+γZ4 4 gp,s 5,16 [Z1,Z5]=Z1,[Z2, Z5] = Z1+Z2, [Z3, Z5] = pZ3−sZ4, [Z4, Z5] = sZ3+pZ4 4 gp,q,s 5,17 [Z1,Z5]=pZ1−Z2, [Z2, Z5] = Z1+pZ2, [Z3, Z5] = qZ3−sZ4, [Z4, Z5] = sZ3+qZ4 4 gp 5,18 [Z3,Z5]=Z1+pZ3−Z4, [Z2, Z5] = Z1+pZ2, [Z1, Z5] = pZ1−Z2, [Z4, Z5] = Z2+Z3−pZ4 4 gα,β 5,19 [Z2,Z3]=Z1,[Z1, Z5] = (1 + α)Z1, [Z2, Z5] = Z2,[Z3, Z5] = αZ3, [Z4, Z5] = βZ4 3 gα 5,20 [Z2,Z3]=Z1,[Z1, Z5] = (1 + α)Z2, [Z2, Z5] = Z2,[Z3, Z5] = αZ3, [Z4, Z5] = Z1+ (1 + α)Z4 3 g5,21 [Z2,Z3]=Z1,[Z1, Z5]=2Z1, [Z2, Z5] = Z2+Z3,[Z4, Z5] = Z4, [Z3, Z5] = Z3+Z4 3 g5,22 [Z2,Z3]=Z1,[Z2, Z5] = Z3, [Z4, Z5] = Z43 gβ 5,23 [Z2,Z3]=Z1,[Z1, Z5]=2Z1, [Z2, Z5] = Z2+Z3, [Z3, Z5] = Z3,[Z4, Z5] = βZ4 3 g² 5,24 [Z2,Z3]=Z1,[Z1, Z5]=2Z1, [Z2, Z5] = Z2+Z3,[Z3, Z5] = Z3, [Z4, Z5] = ²Z1+ 2Z4 3 gβ,² 5,25 [Z2,Z3]=Z1,[Z1, Z5]=2Z1, [Z2, Z5] = Z2+Z3,[Z4, Z5] = Z4, [Z3, Z5] = Z3+Z4 3 gp,² 5,26 [Z2,Z5]=pZ2+Z3,[Z1, Z5]=2pZ1, [Z2, Z3] = Z1,[Z3, Z5] = −Z2+pZ3, [Z4, Z5] = ²Z1+ 2pZ4 3 g5,27 [Z2,Z3]=Z1,[Z3, Z5] = Z3+Z4, [Z1, Z5] = Z1,[Z4, Z5] = Z1+Z43 WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 29 Issue 1, Volume 9, January 2010 Table 4: Non-decomposable Solvable Lie algebras of dimension 5(II). Algebra Nonzerobrackets M(g) gα 5,28 [Z2,Z3]=Z1,[Z2, Z5] = αZ2, [Z1, Z5] = (1 + α)Z1,[Z4, Z5] = Z4, [Z3, Z5] = Z3+Z4 3 g5,29 [Z2,Z3]=Z1,[Z1, Z5] = Z1, [Z2, Z5] = Z2,[Z3, Z5] = Z43 gh 5,30 [Z2,Z4]=Z1,[Z3, Z4] = Z2, [Z1, Z5] = (2 + h)Z1,[Z4, Z5] = Z4, [Z2, Z5] = (1 + h)Z2,[Z3, Z5] = hZ3 3 g5,31 [Z2,Z4]=Z1,[Z3, Z4] = Z2, [Z1, Z5] = 3Z1,[Z3, Z5] = Z3, [Z2, Z5]=2Z2,[Z4, Z5] = Z3+Z4 3 gh 5,32 [Z2,Z4]=Z1,[Z3, Z4] = Z2, [Z1, Z5] = Z1,[Z2, Z5] = Z2, [Z3, Z5] = hZ1+Z3 3 gβ,γ 5,33 [Z1,Z4]=Z1,[Z3, Z4] = βZ3, [Z2, Z5] = Z2,[Z3, Z5] = γZ33 gα 5,34 [Z1,Z4]=αZ1,[Z2, Z4] = Z2, [Z3, Z4] = Z3,[Z1, Z5] = Z1, [Z3, Z5] = Z2 3 gh,α 5,35 [Z1,Z4]=hZ1,[Z2, Z4] = Z2, [Z3, Z4] = Z3,[Z2, Z5] = −Z3, [Z1, Z5] = αZ1,[Z3, Z5] = Z2 3 g5,36 [Z2,Z3]=Z1,[Z1, Z4] = Z1, [Z2, Z4] = Z2,[Z3, Z5] = Z3, [Z2, Z5] = −Z2 3 g5,37 [Z1,Z4]=Z1,[Z2, Z4] = Z2, [Z1, Z5] = −Z2,[Z2, Z5] = Z1, [Z4, Z5] = Z3 3 g5,38 [Z1,Z4]=Z1,[Z2, Z5] = Z2, [Z4, Z5] = Z33 g5,39 [Z2,Z3]=Z1,[Z1, Z4]=2Z1, [Z2, Z4] = Z2,[Z3, Z4] = Z3, [Z2, Z5] = −Z3,[Z3, Z5] = Z2 2 Table 5: Non-decomposable Solvable Lie algebras of dimension 6(I). Algebra Nonzerobrackets M(g) g6,1[Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z1, Z5] = Z6.5 g6,2[Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6.5 g6,3[Z1,Z2]=Z3,[Z1, Z4] = Z5, [Z2, Z6] = −Z5.4 g6,4 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z1, Z5] = Z6, [Z2, Z3] = Z5,[Z2, Z4] = Z6. 4 g6,5[Z1,Z2]=Z3,[Z1, Z4] = Z5, [Z2, Z4] = −Z6.4 g6,6 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z2, Z6] = Z4, [Z3, Z6] = Z5. 4 g6,7 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z2, Z6] = Z5, [Z2, Z3] = Z5. 4 g6,8[Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z2, Z6] = Z5.4 g6,9[Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z2, Z3] = Z6.4 Table 6: Non-decomposable Solvable Lie algebras of dimension 6(II). Algebra Nonzerobrackets M(g) g6,10 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z2, Z3] = Z6, [Z2, Z6] = Z5. 4 g6,11 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6,[Z3, Z5] = −Z4, [Z2, Z6] = Z4. 4 g6,12 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6,[Z5, Z6] = Z4.4 g6,13 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6,[Z2, Z3] = Z4.4 g6,14 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6,[Z2, Z3] = Z6.4 g6,15 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6,[Z2, Z3] = Z6, [Z2, Z5] = Z6. 4 g6,16 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z5] = Z6,[Z2, Z5] = Z4.4 g6,17 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z1, Z5] = Z6, [Z2, Z3] = Z6. 4 g6,18 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z5, Z6] = −Z4.4 g6,19 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z1, Z5] = Z6, [Z2, Z5] = Z6,[Z3, Z4] = −Z6. 3 g6,20 [Z1,Z2]=Z3,[Z1, Z3] = Z4, [Z1, Z4] = Z5,[Z1, Z5] = Z6, [Z2, Z3] = Z5−Z6,[Z2, Z4] = Z6, [Z2, Z5] = Z6,[Z3, Z4] = −Z6. 3 gα,β,γ,δ 6,21 [Z1,Z3]=αZ3,[Z1, Z4] = γZ4, [Z1, Z6] = Z6,[Z2, Z3] = βZ3, [Z2, Z4] = δZ4,[Z2, Z5] = Z5 4 gα,β,γ 6,22 [Z1,Z3]=αZ3,[Z1, Z4] = Z4, [Z1, Z5] = Z6,[Z2, Z6] = Z6, [Z2, Z3] = βZ3,[Z2, Z4] = γZ4, [Z2, Z5] = Z5 4 gα 6,23 [Z1,Z3]=Z3,[Z2, Z5] = Z5, [Z1, Z4] = Z4,[Z1, Z5] = Z6, [Z2, Z4] = αZ4,[Z2, Z6] = Z6, [Z2, Z3] = αZ3+Z4 4 gα,β 6,24 [Z1,Z3]=Z3,[Z1, Z4] = Z4, [Z1, Z5] = Z6,[Z2, Z6] = αZ6, [Z2, Z3] = Z4,[Z2, Z4] = −Z3, [Z2, Z5] = αZ5+βZ6 4 gα,β 6,25 [Z1,Z3]=αZ3,[Z1, Z6] = Z6, [Z2, Z3] = βZ3,[Z2, Z4] = Z4, [Z1, Z5] = Z5+Z6 4 gα,β 6,26 [Z1,Z3]=αZ3,[Z1, Z4] = αZ4, [Z1, Z5] = Z5+Z6,[Z1, Z6] = Z6, [Z2, Z3] = Z3+Z4,[Z2, Z4] = Z4 4 gα,β,γ 6,27 [Z1,Z3]=αZ3,[Z1, Z4] = αZ4, [Z2, Z5] = βZ6,[Z2, Z3] = γZ3+Z4, [Z1, Z6] = Z6,[Z2, Z4] = −Z3+γZ4, [Z1, Z5] = Z5+Z6 4 g6,28 [Z1,Z3]=Z1,[Z1, Z4] = Z6, [Z2, Z4] = Z2, [Z2, Z5] = Z5+Z6,[Z2, Z6] = Z6 4 g6,29 [Z1,Z3]=Z3,[Z1, Z4] = Z6, [Z2, Z6] = Z6,[Z2, Z4] = Z4+Z5, [Z2, Z5] = Z5+αZ6 4 gα,β 6,30 [Z1,Z3]=αZ3,[Z1, Z4] = Z3+βZ6, [Z1, Z5] = Z5,[Z2, Z5] = Z6, [Z1, Z6] = Z6,[Z2, Z3] = Z3, [Z2, Z4] = Z5 4 WSEAS TRANSACTIONS on MATHEMATICS Manuel Ceballos, Juan Nunez, Angel F. Tenorio ISSN: 1109-2769 30 Issue 1, Volume 9, January 2010