Algorithm to Compute Minimal Matrix Representation of Nilpotent Lie Algebras
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Algorithm to Compute Minimal Matrix Representation of Nilpotent Lie Algebras M. Ceballos 1 , J. Núñez 2 , Á. F. Tenorio 3 1 Dpto. de Matemáticas e Ingeniería. Universidad Loyola Andalucía. Campus Palmas Altas, C/ Energía Solar 1, Ed. E, 41014-Seville (Spain). 2 Departamento de Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Calle Tara, s/n. 41012-Seville (Spain). 3 Dpto. de Economía, Métodos Cuantitativos e Historia Económica. Escuela Politécnica Superior. Universidad Pablo de Olavide. Ctra. Utrera km. 1. 41013-Seville (Spain). [email protected] jnv[email protected] [email protected] Abstract. As it is well-known there exist matrix representations of any given nite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in such representations. In this way, our main goal is to revisit, debug and implement an algorithm which provides the minimal order for matrix representations of any nite-dimensional nilpotent Lie algebra from its law, as well as returning a matrix representative of such an algebra by using the minimal order previously computed. In order to show the applicability of this procedure, we have computed minimal representative for each nilpotent Lie algebra of dimensions 6 and 7 and we have also obtained the representation of some families with an arbitrary dimension. Key words and phrases: nilpotent Lie algebra, faithful matrix representation, minimal representation, symbolic computation, non-numerical algorithm. 2000 Mathematics Subject Classication: 17B30, 17B05, 1708, 68W30, 68W05. 1 Introduction Lie algebra Representation Theory has broad applications to the analysis of continuous symmetries in Mathematics and Physics. More concretely, in Mathematics, the classication of Lie groups reveals symmetries in dierential equations. With respect to Physics, representation theory yields natural connections between representation of Lie algebras and the properties of elementary particles. 1
Ado's Theorem states that given a nite-dimensional complex Lie algebra g , there exists a matrix algebra isomorphic to g (see [12] for the classical proof and [14] for a very short alternative). In this way, every nite-dimensional complex Lie algebra can be represented as a Lie subalgebra of the complex general linear algebra gl(n;C) , of complex n×n matrices, for some n∈N . Moreover, according to Theorem 3.6.6 in [16], it is well-known that a given nite-dimensional nilpotent Lie algebra g is isomorphic to a subalgebra of Lie algebra gn , of n×n strictly upper-triangular matrices, for some n∈N . Therefore, the following interesting question arises for nite-dimensional nilpotent Lie algebras: determining the minimal n∈N such that gn contains g as a subalgebra (i.e. obtaining a minimal faithful representation of g using strictly upper-triangular matrices). Firstly, several authors studied the minimal faithful representations of a given Lie algebra g . However, they all considered faithful g -modules instead of the particular subclass consisting of representations in Lie algebras gn . Regarding this matter, matrix representations were computed by Ghanam et al. [10] for low-dimensional nilpotent Lie algebras, but not studying the minimality and giving non-minimal representations. Then, Benjumea et al. [1, 2] obtained minimal faithful representations by strictly upper-triangular matrices of nilpotent Lie algebras with dimension less than 6 . After that, Ceballos et al. [6] studied minimal faithful matrix representations of liform Lie algebras given explicit representatives for dimension less than 9 . Finally, the same authors in [7] computed the minimal faithful triangular representation of solvable Lie algebras up to dimension 5 providing an algorithmic procedure. In this paper, we give an algorithmic method to compute the minimal faithful unitriangular matrix representation of any nilpotent Lie algebra with an arbitrary dimension. Moreover, we provide an implementation of this algorithm by using the symbolic computation package Maple 12 . This algorithm improves the previous one introduced in [7] obtaining new representatives and reducing the number of routines and operations. Finally, as application of our method, we give a complete list with the representative for all the nilpotent Lie algebras of dimensions 6 and 7 . In order to do so, we use the classications given by de Graaf and Magnin (see [8, 13]). This paper is structured as follows: Section 2 reviews some well-known results on Lie Theory. Section 3 introduces the general method used to compute a minimal faithful representation for nilpotent Lie algebras by using strictly upper-triangular matrices. Then, we show a theoretical example in Subsection 3.1 . We also provide an implementation by using the symbolic computation package Maple 12 in Subsection 3.2 . After that, we give a practical example in Subsection 3.3 . Next, Section 4 gives an application of the algorithm studying the minimal faithful representation of 6 and 7 -dimensional nilpotent Lie algebras and we also study two families of arbitrary dimension. Although our method is applicable to every nite-dimensional nilpotent Lie algebras, their laws are not easy to compute in high dimensions. In fact they have been only classied up to dimension 7 . Finally, Section 5 shows a computational analysis of the algorithm with a complexity study. 2
2 Preliminaries An overall review on Lie algebras can be consulted in [16]. The present section only recalls some denitions and results about Lie algebras which we are applying later. From here on, we only consider nite-dimensional Lie algebras over the complex number eld C . Given a Lie algebra g , its lower central series is dened as follows C1(g) = g,C2(g)=[g,g],C3(g)=[C2(g),g], . . . , Ck(g)=[Ck−1(g),g], . . . (1) Additionally, the Lie algebra g is said to be nilpotent if there exists a natural integer m such that Cm(g)≡0 . The center of a Lie algebra g is given by Z(g) = {v∈g|[v, w] = 0,∀w∈g} . A Lie algebra g is abelian if Z(g) = g . Note that abelian Lie algebras constitute a particular family of nilpotent Lie algebras. The relation between the lower central series of the Lie algebra g and one of its subalgebras is given by the following Proposition 1. If h is a Lie subalgebra of the Lie algebra g , then Ck(h)⊆ Ck(g) , ∀k∈N . Denote by gn the nilpotent matrix algebra of n×n strictly upper-triangular matrices, where n∈N\ {1} . The vectors in gn are expressed as follows gn(xr,s) = 0x1,2· · · x1,n−1x1,n 0 0 · · · x2,n−1x2,n . . .. . ..... . .. . . 0 0 · · · 0xn−1,n 0 0 · · · 0 0 (xi,j ∈C). (2) The dimension of gn is n(n−1) 2 and its law is given by the brackets [Xi,j, Xj,k] = Xi,k,for 1 ≤i < j < k ≤n, (3) with respect to the basis Bn={Xi,j =gn(xr,s)}1≤i<j≤n , where xr,s =(1,if (r, s)=(i, j); 0,if (r, s)= (i, j). (4) According to the law of gn , Z(gn) =< X1,n > and its lower central series is determined by Ci+1(gn) = gn−i , ∀0≤i≤n−2 . Consequently, the dimension sequence of this series is (dim(gn),dim(gn−1),dim(gn−2),..., dim(g2),0) (5) Given a Lie algebra g , a representation of g in Cn is a Lie-algebra homomorphism ϕ:g→gl(Cn) = gl(C, n) . The natural integer n is the dimension of this representation. 3
According to Ado's theorem, every nite-dimensional Lie algebra over a eld of characteristic zero (e.g. C ) admits a linear injective (i.e. faithful ) representation into gl(Cn) . In general, representations can be dened by considering the general Lie algebra gl(V) of endomorphisms on some vector space V (e.g. [9]). Then, representations are Lie-algebra homomorphisms from g to gl(V) , usually called g -modules . Previously, Burde [3] studied minimal representations of Lie algebras introducing the invariant µ(g) for a Lie algebra g as follows: µ(g) is the minimum among the dimensions of faithful g -modules (see [9, p. 481] to consult the notion of g -module). In the theory of the ane crystallographic groups and nitely-generated torsion-free nilpotent groups, the invariant µ(g) plays an important role (for details and reference see [4]). There is also a very strong relation between the existence of a left-invariant ane structure in g and the value of µ(g) . We have for example the following result Proposition 2. [4, Propositions 2.31 and 3.8] Let G be an Lie group of dimension n with g=Lie(G) the Lie algebra of G . If G admits a leftinvariant ane structure, then µ(g)≤n+ 1 . On the other hand, this problem is also related to the representation theory of nitely generated nilpotent groups, since it is important to have methods for nding integer matrix representations of small dimension for this class of groups (see [11, 15]). Since we are interested in computing faithful matrix representations of nilpotent Lie algebras, we are taking into consideration an additional restriction when studying the minimality of such representations: the representation must be embedded as a Lie subalgebra in gn for some n∈N . Consequently, we want to determine which is the minimum n∈N such that there exists a Lie subalgebra of gn being isomorphic to a given nilpotent Lie algebra g . This minimum is denoted by ¯µ(g) and is an invariant of g . Obviously, the values of µ(g) and ¯µ(g) may dier from each other. 3 Computing Minimal Matrix Representations The main goal of this section is to introduce an algorithmic method to compute a minimal faithful representation by strictly upper-triangular matrices for any nite-dimensional nilpotent Lie algebra. This representation is given by a representative which can be expressed in terms of the dimension of the algebra. After giving a step-by-step explanation of the algorithm and some examples, we implement it with the symbolic computation package Maple 12 . Finally, we show as application the computation of these minimal faithful representations for nilpotent Lie algebras of dimensions 6 and 7 . First, we introduce and expound the algorithm which we have designed to compute a minimal faithful matrix representations for a given n -dimensional nilpotent Lie algebra g by using the Lie algebra gk for some k∈N . We consider minimality as the existence of 4
a subalgebra in gk being isomorphic to g , but not in gk−1 . Now, we show a step-by-step explanation of the algorithm: Step 1. We compute the rst k∈N such that the lower central series of gk and g satisfy Ci(g)⊆ Ci(gk) , for all i∈N (see Proposition 1). Step 2. We write the vectors in basis {ei}n i=1 of g as linear combinations of the vectors of basis Bk bearing in mind the central series of both algebras. We obtain expressions as follows eh=X 1≤i<j≤k λh i,jXi,j,for 1 ≤h≤n. (6) Step 3. Bracket [ei, ej] is computed for 1≤i≤j≤n . When imposing the law of g , a system of non-linear equations is obtained by comparing coordinate to coordinate with respect to basis Bk . Step 4. We compute the solutions of the system obtained in the previous step. One of those solutions provides a minimal faithful representation for Lie algebra g if the solution corresponds to a set of vectors being linearly independent. When no solution is obtained, g cannot be represented as a Lie subalgebra of gk ; hence, we go back to Step 2 and consider Lie algebra gk+1 . In order to compute a simple and particular solution from the previous system, we will apply Noether's normalization lemma searching for one having as many coecients λh i,j being equal to 0 as possible. Therefore, a coecient is assumed to be zero if that coecient is not involved in the constrains that dene the set of solutions. Moreover, it is well-known that every nilpotent Lie algebra has a non-trivial center, therefore we will express the basis vectors of Z(g) as a linear combination of Z(gk) =< X1,k > . A solution verifying the previous conditions will be called a natural representative of the Lie algebra g . Although we only compute natural representatives as application of this method, any representative from the same or other equivalence class can be obtained by means of its use. 3.1 Preliminary example Next, we show an example from a theoretical point of view of the previous algorithm in which minimal faithful representations by strictly upper-triangular matrices are computed for nilpotent Lie algebras. Example 1. Let us consider the nilpotent Lie algebra g with basis {ei}4 i=1 and law [e1, e4] = e3,[e1, e3] = e2 . The lower central series of g is given by C1(g) = {e1, e2, e3, e4},C2(g) = {e2, e3},C3(g) = {e2},C4(g) = {0} . According to Proposition 1, the minimal possible dimension for the matrix representation of g is 4 . 5
Next, we apply Step 2 and express the vectors in the basis of g as a linear combination of the vectors from the basis of g4 bearing in mind the lower central series of both algebras e1=λ1 1,2X1,2+λ1 1,3X1,3+λ1 1,4X1,4+λ1 2,3X2,3+λ1 2,4X2,4+λ1 3,4X3,4 e2=λ2 1,4X1,4, e3=λ3 1,3X1,3+λ3 1,4X1,4+λ3 2,4X2,4 e4=λ4 1,2X1,2+λ4 1,3X1,3+λ4 1,4X1,4+λ4 2,3X2,3+λ4 2,4X2,4+λ4 3,4X3,4 Now, we apply Step 3 computing the brackets [ei, ej],∀1≤i<j≤3 and imposing the laws in g and g4 . In this way, we obtain the following equations λ2 1,4=λ1 1,2λ3 2,4−λ1 3,4λ3 1,3, λ3 1,3=λ1 1,2λ4 2,3−λ1 2,3λ4 1,2, λ3 2,4= 0 λ3 1,4=λ1 1,2λ4 2,4λ1 1,3λ4 3,4−λ1 2,4λ4 1,2−λ1 3,4λ4 1,3, λ3 1,3λ4 3,4=λ3 2,4λ4 1,2. (7) Finally, in the Step 4 , we solve the previous system. Each solution of this system denes a Lie subalgebra of g4 isomorphic to g ; i.e. a minimal faithful matrix representation. Now, we can check the cardinality of the solution set in order to know if the number of minimal faithful representation is nite or not. This checking is done by using the test is_finite . In this example, the test returns false and there are innite number of solutions for the system and, hence, of possible minimal faithful representations in g4 for g . To reduce the number of representatives, we apply Noether's normalization lemma to the solution set of our system (which is an algebraic variety). Its intersection with the algebraic variety determined by the implicit equations {λ2 1,4= 1, λ3 1,3=λ3 2,4= 0} is a nite set. By solving the system in these conditions, we can obtained the following natural representative e1=X1,2+X2,3, e2=X1,4, e3=X2,4, e4=X3,4 3.2 Implementation Now, we show the implementation of the dierent routines in order to apply the previous algorithmic method. In order to do so, we have used the symbolic computation package MAPLE 12 , loading the libraries DifferentialGeometry, LieAlgebras to activate commands related to Lie algebras. Notice that an algorithm for the case of solvable Lie algebras was implemented in [7]. Regarding this, we have achieved some improvements in the implementation of several routines. For example, we have been able to reduce the number of routines. This number was eight for the case of solvable Lie algebras and we have obtained a simplied implementation with only six routines. Due to reasons of length, we will only show the implementation of those routines that are dierent from the ones in [7]. In the rst place, the routine law_gn is implemented to compute the law of the nilpotent Lie algebra gn . This routine receives as input the value of n and returns the list of brackets 6
expressing the law of gn with respect to the basis {e1, e2, . . . , en(n−1) 2 } , which corresponds to {Xi,j : 1 ≤i<j≤n} . For the implementation, a list B saves the basis of gn and S keeps all the (non-zero) brackets involved in the law. In order to carry out the computations, three dierent loops are programmed to nd and save the non-zero brackets in Eq. (3). Finally, the law of the algebra is saved in the variable Ext1 to be loaded in a later routine. > law_gn:=proc(n) > local B, S; > B:=[]; S:=[]; > for i from 1 to n-1 do (*Constructing the basis*) > for j from i+1 to n do > B:=[op(B),X[i,j]]; > end do; > end do; > for i from 1 to n-2 do (*Expressing brackets in Eq. (3)*) > for j from i+1 to n-1 do > for k from j+1 to n do > S:=[op(S),[X[i,j],X[j,k]]=X[i,k]]; > end do; > end do; > end do; > return LieAlgebraData(S,B,Ext1, "LieAlgebraData"); (*Defining the algebra*) > end proc: Now, we introduce the nilpotent Lie algebra g according to the following notation > L:= _DG([["LieAlgebra", g, [n]], [A]]); > DGsetup(L); where n is the dimension of g (a value that must be inserted by the user) and A is a list containing information about the structure constants of the law of g . Elements in A must be of the form [[i, j, k], cijk] where cijk is the structure constant λk i,j corresponding to the coecient of ek in the bracket [ei, ej] . Once these data are loaded, we execute the order DGsetup(L) to operate over the Lie algebra g . Next, we compute lower central series of the Lie algebra g with the following command > LS_g:=Series("Lower"); After this, we complete the rst Step of the algorithm. In this way, the routine DimRepresentation computes the minimal dimension for a matrix representation of g by using Lie algebras gn . To implement this routine, we compare the dimension sequence of the derived series of both g and gn . The output is the minimal k ∈N such that Step 1 is veried. The implementation of this routine is the same as the one given in [7]. Now, once we have computed the minimal possible dimension for the matrix representation, k , we introduce the nilpotent Lie algebra gk with the sentence g > DGsetup(law_gn(k),[x],[a]); g > LS_gn:=Series("Lower"); 7
In this expression, [x] is used to denote the basis vectors in Bk as {xi} k(k−1) 2 i=1 instead of {ei} k(k−1) 2 i=1 , since we need dierent notations for the bases of both Lie algebras g and gk . The notation [a] corresponds to the list of structure constants dening the law of Lie algebra gn and expressed as in (3). From this point, we work over the Lie algebra gk , denoted by L1 by the package. We rst implement a subroutine called listposi , which computes the position of an element within a list. The implementation of this routine is the same as the one given in [7]. Then, we also implement the routine expr , which expresses all the vectors in the basis of g as a linear combination of basis Bk of gk . In order to do so, we use the central series of both g and gk and, by applying Proposition 1, the procedure returns as output two lists: the rst contains all the expressions according to the second step of the method, that is, Equation (6); and in the second, the conditions over the coecients. In comparison to the implementation developed in [7], we have reduced the number of operations and, as a consequence of this, we will obtain less computing time and memory used in the computational and complexity study shown in Section 5 . L1 > expr:=proc() L1 > L:=[];M:=[]; L1 > for i from nops(LS_g) - 1 to 1 by -1 do L1 > for j from 1 to nops(LS_g[i]) do L1 > if member(LS_g[i][j],LS_g[i+1])=false then member(LS_g[i][j],LS_g[1],'p'); L1 > N:=[seq(listposi(LS_gn[i][k],LS_gn[1]),k=1..nops(LS_gn[i]))]; L1 > L:=[op(L),LS_g[i][j]=sum(b[p,N[k]]*LS_gn[i][k],k=1..nops(LS_gn[i]))]; L1 > M:=[op(M),sum(b[p,N[k]]^2,k=1..nops(LS_gn[i]))<>0]; L1 > end if; end do; end do; L1 > return L,M; L1 > end proc: After expressing the basis of g with respect to basis Bk of gk , we impose the law of both Lie algebras. In this way, we have the routine Listeq , which returns two lists: the rst one containing the elements to be equal to zero; and the second one with the conditions to assure the linear independence of the basis. The implementation of this routine was given in [7]. Finally, we consider the routine sys in order to solve the system of equations from the previous expressions. This routine must receive as input the outputs of the previous routine Listeq . It returns as output the set of solutions which determine the coecients of the representation of g by using Lie algebra gk . The implementation of sys was given in [7] 3.3 Practical example In this subsection, we show an example of the implementation of our method with the 3 -dimensional nilpotent Lie algebra given by [e1, e3] = e2 . We must run all the routines. Here, we only reproduce the most important outputs and those sentences to be modied 8
for this specic example. In order to dene the nilpotent Lie algebra g and compute its lower central series, we execute the following sentences > L:= _DG([["LieAlgebra", g, [3]], [[[1, 3, 2], 1]]]); > DGsetup(L); > LS_g:=Series("Lower"); From the routine DimRepresentation , we obtain > DimRepresentation(); 3 Consequently, we must use k= 3 in Step 1 of the algorithm. We look for a representation of g as a Lie subalgebra of g3 . We start loading Lie algebra g3 and computing its lower central series: > DGsetup(law_gn(3),[x],[a]); > LS_gn:=Series("Lower"); By executing the routine expr , we obtain the following output L1 > expr(); [e1=b[1,1]*x1+b[1,2]*x2+b[1,3]*x3,e2=b[2,2]*x2,e3=b[3,1]*x1+b[3,2]*x2+b[3,3]*x3], [b[1,1]^2+b[1,2]^2+b[1,3]^2<>0,b[2,2]^2<>0,b[3,1]^2+b[3,2]^2+b[3,3]^2<>0] The output of Listeq is L1 > Listeq(); [b[2,2]*x2-(b[1,1]*b[3,3]-b[1,3]*b[3,1])*x2], [b[1,1]^2+b[1,2]^2+b[1,3]^2<>0,b[2,2]^2<>0,b[3,1]^2+b[3,2]^2+b[3,3]^2<>0] Finally, the routine sys is executed L1 > sys(Listeq()[1],Listeq()[2]); {b[1,1]=b[1,1],b[1,2]=b[1,2],b[1,3]=b[1,3],b[2,2]=b[1,1]*b[3,3]- b[1,3]*b[3,1],b[3,1]=b[3,1],b[3,2]=b[3,2],b[3,3]=b[3,3]} 4 Application to Nilpotent Lie algebras of dimension less than 8 In this section, we show an application of our algorithm computing minimal faithful matrix representation of nilpotent Lie algebras. These algebras have been classied until dimension 7 . We will consider the minimal matrix representation of indecomposable nilpotent Lie algebras of dimensions 6 and 7 . This is due to the results obtained in [5] for reductive Lie algebras and the minimal representatives obtained and for nilpotent Lie algebras of dimension less than 6 in [2]. 9
g Expression of the law µ Representation g7,2.12 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e2, e4] = e7,[e3, e5] = e7 .6 e1=X1,2+X2,3 , e2=X2,3+X3,5 , e3=X2,4+X4,5 , e4=X1,3+X2,5 , e5=X1,4 , e6=X1,5 , e7=−X1,5 g7,2.13 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6, [e2, e3] = e6,[e2, e6] = e7,[e4, e5] = −e7 .7 e1=P5 i=1 Xi,i+1 , e2=X6,7 , e3=X5,7 , e4=X4,7 , e5=X3,7 , e6=X2,7 , e7=X1,7 g7,2.14 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6 , [e1, e6] = e7,[e2, e3] = e5 , [e3, e4] = −e6,[e3, e5] = −e7 . 6 e1=X1,2+X2,3 , e2=X2,3+X3,5 , e3=X2,4+X4,5 , e4=X1,3+X2,5 , e5=X1,4 , e6=X1,5 , e7=−X1,5 g7,2.15 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e6,[e3, e4] = −e7 .7 e1=P5 i=1 Xi,i+1 , e2=X6,7 , e3=X5,7 , e4=X4,7 , e5=X3,7 , e6=X2,7 , e7=X1,7 g7,2.16 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e6, [e1, e6] = e7,[e2, e3] = e7 .6 e1=X1,2+X2,3+X3,4+X4,5 , e2=X5,6 , e3=−X1,5 , e4=X4,6 , e5=X3,6 , e6=X2,6 , e7=X1,6 g7,2.17 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7, [e2, e3] = e5,[e2, e4] = e6,[e3, e4] = −e7 .6 e1=X1,2+X2,3+X3,5 , e2=X2,5−X4,6+X5,6 , e3=X1,4 , e4=X1,5+X3,6 , e5=X2,6 , e6=−X1,6 , e7=X1,6 g7,2.18 [e1, e2] = e4,[e1, e4] = e5,[e1, e5] = e7, [e2, e3] = e7,[e2, e4] = e6 .6 e1=X1,2+X2,3+X3,5 , e2=X2,5−X4,6+X5,6 , e3=X1,4 , e4=X1,5+X3,6 , e5=X2,6 , e6=−X1,6 , e7=X1,6 g7,2.19 [e1, e2] = e4,[e1, e3] = e6,[e1, e4] = e5, [e1, e5] = e7,[e2, e4] = e6 .6 e1=X1,2+X2,3+X3,5+X4,6 , e2=X2,5+X5,6 , e3=X1,4 , e4=X1,5+X3,6 , e5=X2,6 , e6=−X1,6 , e7=X1,6 g7,2.20 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6, [e1, e6] = e7,[e3, e5] = e7 .6 e1=X1,2+X2,3+X5,6 , e2=X2,6 , e3=X1,4+X3,5+ 3X4,5 , e4=X1,6 , e5=X2,5−X3,6−3X4,6 , e6=X1,5−2X2,6 , e7=−3X1,6 g7,2.21 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e1, e6] = e7,[e2, e3] = e7 .6 e1=X1,2+X3,5+X4,6+X5,6 , e2=X2,3−3X4,6 , e3=X1,4 , e4=X1,3−X2,5 , e5=−X1,6 , e6=−2X1,5+X2,6 , e7= 3X1,6 g7,2.22 [e1, e3] = e4,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e5,[e3, e4] = e7 .6 e1=X1,2+X2,3+X5,6 , e2=−X4,6 , e3= 3X1,3+X1,4+X3,5 , e4=X2,5−X3,6 , e5=X1,6 , e6=X1,5−2X2,6 , e7=−3X1,6 g7,2.23 [e1, e4] = e6,[e1, e6] = e7,[e2, e3] = e5, [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X2,3 , e2=−X4,5+X5,6 , e3=X1,4 , e4=−X3,6−X4,6 , e5=X1,5 , e6=−X2,6 , e7=−X1,6 g7,2.24 [e1, e2] = e4,[e1, e4] = e6,[e1, e5] = −e7, [e1, e6] = e7,[e2, e3] = e5,[e3, e4] = e7 .6 e1=X1,2+X2,3−X3,5 , e2=−X4,5+X5,6 , e3=X1,4 , e4=−X3,6 , e5=X1,5 , e6=−X2,6 , e7=−X1,6 g7,2.25 [e1, e2] = e5,[e1, e5] = e6,[e1, e6] = e7, [e2, e3] = e6,[e3, e4] = −e7,[e3, e5] = −e7 .6 e1=X1,2+X4,6 , e2=X1,3+ 2X3,4 , e3=X1,4+X2,4+X5,6 , e4= 2X1,5 , e5=−2X3,6 , e6=X1,4−X1,6−X2,6 , e7=−2X1,6 g7,2.26 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6, [e2, e5] = e7,[e3, e4] = e7,[e3, e5] = e6 .6 e1=X1,2+X3,5+X4,6 , e2=X1,3 , e3= 2X1,2+X1,4+X2,4+X5,6 , e4=−X1,5 , e5=X1,4−X1,6−X2,6+ X3,6, e6=−2X1,6 , e7=X1,6 16
g Expression of the law µ Representation g7,2.27 [e1, e2] = e5,[e1, e3] = e7,[e1, e5] = e6, [e2, e4] = e7,[e2, e5] = e7 .6 e1=X1,2+X3,5+X4,6+X5,6 , e2=X1,3+X3,4 , e3=X1,4 , e4=−X3,6 , e5=−X1,5−X3,6 , e6=X1,6 , e7=−X1,6 g7,2.28 [e1, e2] = e5,[e1, e3] = e6,[e1, e6] = e7, [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X4,6 , e2=X1,3+ 2X3,4 , e3=X1,4+X2,4+X5,6 , e4= 2X1,5 , e5=−2X3,6 , e6=X1,4−X1,6−X2,6 , e7=−2X1,6 g7,2.29 [e1, e2] = e5,[e1, e5] = e6,[e2, e3] = e6, [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X3,5+X5,6 , e2=X1,3−X4,6+1 2X5,6 , e3=X1,4 , e4=X4,6 , e5=−X1,5+1 2X3,6 , e6=X1,6 , e7=X1,6 g7,2.30 [e1, e2] = e5,[e1, e5] = e6,[e1, e6] = e7, [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X2,3−X3,5 , e2=X2,5+X5,6 , e3=X1,4 , e4=−X4,6 , e5=X1,5−X3,6 , e6=−X2,6 , e7=−X1,6 g7,2.31 [e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e5,[e2, e5] = e7 .6 e1=X1,2+X2,3−X3,5 , e2=−X4,5+X5,6 , e3=X1,4 , e4=−X3,6 , e5=X1,5 , e6=−X2,6 , e7=−X1,6 g7,2.32 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e1, e6] = e7,[e3, e5] = e7 .6 e1=X1,2+X3,5+X5,6 , e2=X2,3 , e3=X1,4−3X4,5 , e4=X1,3−X2,5 , e5= 3X4,6 , e6=−2X1,5+X2,6 , e7= 3X1,6 g7,2.33 [e1, e2] = e4,[e1, e4] = e6,[e1, e6] = e7, [e2, e3] = e5,[e3, e5] = e7 .6 e1=X1,2−X2,3+X3,5 , e2=X5,6 , e3=X1,4+X4,5 , e4=X3,6 , e5=−X4,6 , e6=−X2,6 , e7=−X1,6 g7,2.34 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e7, [e2, e4] = e6,[e3, e5] = e7 .6 e1=X1,2+X3,5−1 2X4,6+X5,6 , e2=X1,3+X5,6 , e3= 6X1,2−X1,3+X1,4+X2,4+X5,6 , e4=−X1,5+X3,6 , e5=X1,4+X1,5+1 2X1,6+1 2X2,6+X3,6 , e6= 2X1,6 , e7=X1,6 g7,2.35 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6, [e2, e4] = e6,[e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+ 2X2,3+X3,5 , e2=X2,3+X3,5+X4,5 , e3=−X1,3+X1,4−1 2X3,5 , e4=X1,3+X2,5 , e5=−1 2X1,4+X1,5−X2,5 , e6=−X1,5 , e7=1 2X1,5 g7,2.36 [e1, e3] = e5,[e1, e4] = e7,[e2, e3] = e6, [e2, e4] = −e5,[e3, e6] = −e7 .6 e1=X1,2+X3,5 , e2=X4,5 , e3=X1,4+X2,6 , e4=X1,3+X3,5 , e5=X1,6 , e6=−X1,5 , e7=−X1,5 g7,2.37 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e5, [e1, e6] = e7,[e2, e4] = e6, [e2, e5] = e7,[e3, e4] = e7 . 6 e1=X1,2+X3,5 , e2=1 2X2,3+X5,6 , e3=X1,3+X1,4 , e4=1 2X1,3−1 2X1,5+ X3,6 , e5=−X1,5 , e6=X2,6 , e7=X1,6 g7,2.38 [e1, e2] = e3,[e1, e3] = e6,[e1, e5] = e7, [e2, e3] = e7,[e2, e4] = e6 .6 e1=X1,2+X2,3 , e2=X2,3+X2,4+X3,4 , e3=X1,3+X1,4+X2,4 , e4=−X1,3 , e5=−X2,4 , e6=X1,4 , e7=−X1,4 g7,2.39 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e6, [e1, e5] = e7,[e2, e3] = e6 .6 e1=X1,2+X2,3 , e2=X3,5−X4,5 , e3=X1,4+X3,5 , e4=X2,5 , e5=X2,5 , e6=X1,5 , e7=X1,5 g7,2.40 [e1, e2] = e3,[e1, e3] = e5,[e1, e4] = e7, [e2, e3] = e6,[e2, e4] = e5 .6 e1=X1,2+X2,3 , e2=X2,3+X2,4+X3,4 , e3=X1,3+X1,4+X3,4 , e4=−X1,3+X2,5 , e5=X1,4 , e6=−X1,4 , e7=X1,5 17
g Expression of the law µ Representation g7,2.41 [e1, e2] = e3,[e1, e3] = e5,[e1, e4] = e6, [e1, e6] = e7,[e2, e3] = e6,[e2, e5] = e7 .6 e1=X1,2+X2,3 , e2=X2,4+X3,4−X4,5 , e3=X1,4+X2,4 , e4=X2,5+X3,5 , e5=X2,6 , e6=X1,5+X2,5 , e7=X1,5 g7,2.42 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e6, [e2, e3] = e6,[e2, e4] = e7 .6 e1=X1,2+X2,5 , e2=X2,3+X3,6−X4,6 , e3=X1,4+X5,6 , e4=X1,3 , e5=X2,6 , e6=X1,6 , e7=−X1,6 g7,2.43 [e1, e2] = e4,[e1, e3] = e5,[e1, e4] = e7, [e1, e5] = e6,[e3, e5] = e7 .6 e1=X1,2+X2,5−X5,6 , e2=X1,3−X2,5−X3,5+X3,6−X4,6+X5,6 , e3=X1,3+X1,4+X2,4−X3,5+X5,6 , e4=−X1,5−X3,6 , e5=X1,4+X2,6−X3,6 , e6=X1,6 , e7=−X1,6 g7,2.44 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e7, [e2, e3] = e6,[e2, e4] = e7 .6 e1=X1,2+X3,5−1 2X4,6 , e2=X1,3+X2,5+X4,6+X5,6 , e3=X1,4+X2,4+X5,6 , e4=X3,6 , e5=X1,4+1 2X1,6+1 2X2,6+X3,6 , e6=−X1,6 , e7=X1,6 g7,2.45 [e1, e2] = e5,[e1, e4] = e7,[e1, e5] = e6, [e2, e3] = e7,[e2, e4] = e6 .6 e1=X1,2+X2,5+X3,5+X5,6 , e2=−X4,6+X5,6 , e3=X1,4 , e4=−X1,5 , e5=X2,6+X3,6 , e6=X1,6 , e7=X1,6 g7,3.1(iλ) [e1, e2] = e4,[e1, e3] = e5 , [e1, e6] = e7,[e2, e3] = e6, [e2, e5] = λe7,[e3, e4] = e7(λ−1) . 6 e1=X1,2+X3,5 , e2=X5,6 , e3=X1,3+X1,4 , e4= (λ−1)X3,6 , e5=−λX1,5 , e6=X2,6 , e7=X1,6 g7,3.1(iii) [e1, e2] = e4,[e1, e3] = e5,[e1, e6] = e7, [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X3,5 , e2=X5,6 , e3=X1,3+X1,4 , e4=X3,6 , e5=−X1,5 , e6=X2,6 , e7=X1,6 g7,3.2[e1, e2] = e4,[e1, e3] = e5, [e1, e4] = e6,[e1, e6] = e7 .6 e1=X1,2+X2,3+X3,5 , e2=X5,6 , e3=X1,4+X2,4 , e4=X3,6 , e5=X1,4 , e6=X2,6 , e7=X1,6 g7,3.3[e1, e2] = e4,[e1, e4] = e6, [e1, e6] = e7,[e2, e3] = e5 .6 e1=X1,2+X2,3+X3,5 , e2=X4,6+X5,6 , e3=X1,4 , e4=X3,6 , e5=−X1,6 , e6=X2,6 , e7=X1,6 g7,3.4[e1, e2] = e4,[e1, e3] = e5, [e2, e4] = e6,[e3, e5] = e7 6 e1=X1,2+X3,5 , e2=X1,3+X5,6 , e3=−X1,3+X1,4+X5,6 , e4=−X1,5+X3,6 , e5=X1,5+X3,6 , e6= 2X1,6 , e7=−2X1,6 g7,3.5[e1, e2] = e4,[e1, e3] = e5,[e2, e4] = e6, [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X4,5 , e2=X3,4+X5,6 , e3=X1,4+X2,6 , e4=−X3,5+X4,6 , e5=−X1,5+X1,6 , e6= 2X3,6 , e7=X1,6 g7,3.6[e1, e2] = e4,[e1, e3] = e5, [e1, e5] = e7,[e2, e3] = e6 .6 e1=X1,2+X2,4 , e2=X2,5+X3,4 , e3=X4,5 , e4=X1,5 , e5=X2,5 , e6=X3,5 , e7=X1,5 g7,3.7[e1, e2] = e5,[e1, e5] = e6 , [e2, e4] = e6,[e3, e4] = −e7 .6 e1=X1,2+X2,3 , e2=X3,4 , e3=X5,6 , e4=−X1,3 , e5=X2,4 , e6=X1,4 , e7=X1,6 g7,3.8[e1, e2] = e5,[e1, e3] = e6 , [e1, e5] = e7,[e2, e4] = e7 .6 e1=X1,2+X3,6 , e2=X2,3+ 2X5,6 , e3=X2,4 , e4=X1,5 , e5=X1,3−X2,6 , e6=X1,4 , e7=−2X1,6 g7,3.9[e1, e2] = e5,[e1, e5] = −e7 , [e2, e3] = e6,[e2, e4] = e7 .6 e1=X1,2+X5,6 , e2=X3,5+X4,6 , e3=X5,6 , e4=X3,4 , e5=−X3,6 , e6=X3,6 , e7=−X3,6 g7,3.10 [e1, e2] = e5,[e1, e3] = e6,[e2, e4] = e6, [e2, e6] = e7,[e3, e5] = e7 .6 e1=X1,2+X2,4−X3,5+1 3X5,6 , e2=X1,3+X2,5+X4,6 , e3=X2,4+X5,6 , e4=X3,4 , e5= 2X1,5+2 3X2,6 , e6=X1,4−X3,6 , e7=−2X1,6 18
g Expression of the law µ Representation g7,3.11 [e1, e2] = e5,[e1, e3] = e6 , [e1, e5] = e7,[e2, e4] = e6 .6 e1=X1,2+X2,4−X3,5 , e2=X4,6 , e3=X5,6 , e4=X3,4 , e5=X2,6 , e6=−X3,6 , e7=X1,6 g7,3.12 [e1, e2] = e5,[e1, e3] = e6 , [e2, e4] = e6,[e3, e4] = e7 .6 e1=X1,2+X2,3+X4,5 , e2=X2,3+X4,5+X5,6 , e3=X4,5+X5,6 , e4=−X4,5 , e5=X1,3+X4,6 , e6=X4,6 , e7=X4,6 g7,3.13 [e1, e2] = e5,[e1, e5] = e6 , [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X2,3+X3,4+1 2X3,6 , e2=X2,3+X3,6 , e3=X5,6 , e4=X1,5 , e5=X1,3−X2,4+1 2X2,6 , e6=−2X1,4 , e7=−X1,6 g7,3.14 [e1, e2] = e5,[e1, e3] = e6 , [e1, e6] = e7,[e2, e4] = e7 .6 e1=X1,2+X2,4+X2,5+X5,6 , e2=X1,5+X3,6−X4,6+X5,6 , e3=X1,4−X5,6 , e4=X1,3 , e5=−X1,6 , e6=−X2,6 , e7=−X1,6 g7,3.15 [e1, e2] = e5,[e1, e3] = e6 , [e2, e5] = e7,[e3, e4] = e7 .6 e1=X1,2+X4,5 , e2=X2,3+X3,6 , e3=X5,6 , e4=X1,5 , e5=X1,3 , e6=X4,6 , e7=−X1,6 g7,3.16 [e1, e2] = e5,[e1, e5] = e7 , [e3, e4] = e6,[e3, e6] = e7 .6 e1=X1,2−X2,3+X2,4 , e2=X4,6 , e3=X1,3+X3,5+X4,5 , e4=X5,6 , e5=X2,6 , e6=X3,6+X4,6 , e7=X1,6 g7,3.17 [e1, e2] = e5,[e1, e5] = e6, [e1, e6] = e7,[e3, e4] = e7 .6 e1=X1,2+X2,3+X3,4 , e2=X4,6 , e3=−X5,6 , e4=X1,5 , e5=X3,6 , e6=X2,6 , e7=X1,6 g7,3.18 [e1, e2] = e6,[e1, e6] = e7, [e2, e5] = e7,[e3, e4] = e7 .5 e1=X1,2+X3,4+X4,5 , e2=X1,3 , e3=X1,2 , e4=X1,4+X2,5 , e5=X3,5 , e6=−X1,4 , e7=X1,5 g7,3.19 [e1, e2] = e6,[e1, e3] = e7 , [e3, e4] = e6,[e4, e5] = e7 .5 e1=X1,2 , e2=−X2,4 , e3=X2,4+X3,4 , e4=X1,3 , e5=X3,4 , e6=−X1,4 , e7=X1,4 g7,3.20 [e1, e2] = e4,[e1, e3] = e5 , [e1, e4] = e6,[e1, e5] = e7 .5 e1=X1,2+X3,4+X4,5 , e2=X1,3 , e3=X1,2+X1,3 , e4=−X1,4 , e5=−X1,4 , e6=X1,5 , e7=X1,5 g7,3.21 [e1, e2] = e4,[e1, e3] = e5 , [e1, e4] = e6,[e3, e5] = e7 .5 e1=X1,2+X3,4+X4,5 , e2=X1,3 , e3=X1,2+X1,3+1 2X2,4 , e4=−X1,4 , e5=−1 2X1,4−1 2X2,5 , e6=X1,5 , e7=−1 2X1,5 g7,3.22 [e1, e2] = e4,[e1, e3] = e5,[e1, e5] = e7, [e2, e5] = e6,[e3, e4] = e6 .5 e1=X1,2+X3,4+X4,5 , e2=X4,5 , e3=X1,3 , e4=X3,5 , e5=−X1,4 , e6=X1,5 , e7=X1,5 g7,3.23 [e1, e2] = e3,[e1, e3] = e5 , [e1, e4] = e7,[e2, e3] = e6 .5 e1=X2,3+X4,5 , e2=X3,4+X4,5 , e3=X2,4−X3,5 , e4=X3,5 , e5=−2X2,5 , e6=−X2,5 , e7=X2,5 g7,3.24 [e1, e2] = e5,[e2, e3] = e6 , [e2, e4] = e7,[e3, e4] = e5 .5 e1=X1,2 , e2=X2,4+X3,4 , e3=X1,3−X3,4 , e4=X1,3 , e5=X1,4 , e6=−X1,4 , e7=−X1,4 g7,4.1[e1, e2] = e5,[e1, e3] = e6,[e3, e4] = e7 . 5 e1=X1,2+X3,4 , e2=X2,4 , e3=X1,3 , e4=X3,5 , e5=X1,4 , e6=−X1,4 , e7=X1,5 g7,4.2[e1, e2] = e5,[e1, e3] = e6,[e1, e4] = e7 . 5 e1=X1,2+X3,4 , e2=X2,4 , e3=X1,3 , e4=X1,3 , e5=X1,4 , e6=−X1,4 , e7=−X1,4 g7,4.3[e1, e2] = e6,[e3, e5] = e6,[e4, e5] = e7 . 5 e1=X1,2 , e2=X2,4 , e3=X1,3 , e4=X1,3 , e5=X3,4 , e6=X1,4 , e7=X1,4 g7,4.4[e1, e4] = e7,[e2, e5] = e7,[e3, e6] = e7 . 5 e1=X1,2+X4,5 , e2=X4,5 , e3=X3,5 , e4=−X2,5 , e5=X1,4+X2,5 , e6=X1,3 , e7=−X1,5 19
5 Computational data and complexity Finally, we present a computational analysis of the algorithm which was introduced in the previous section and implemented in an Intel Core 2 Duo T 5600 with a 1.83 GHz processor and 2.00 GB of RAM. For this study, we have considered the Lie algebras fn generated by {ei}n i=1 with non-zero brackets [e1, eh] = eh−1,3≤h≤n. This family constitutes a special subclass of nilpotent Lie algebras, which allows us to check empirically the computational data given for both the computing time and the used memory. Therefore, Table 1 reproduces computational data about both the computing time and the memory used to obtain the minimal matrix representation of fn with respect to the dimension n of the algebra. Table 1 was obtained from computing matrix representations of the algebras fn up to dimension n= 10 inclusive. Note that the computing time is between two and three times greater when the dimension n is increased by one unit. Table 3: Computing time and used memory. Input Computing time Used memory n= 3 0.16 s 3.18 MB n= 4 0.37 s 4.38 MB n= 5 0.92 s 5.49 MB n= 6 2.53 s 6.37 MB n= 7 7.09 s 7.63 MB n= 8 20.19 s 9.87 MB n= 9 55.32 s 18.25 MB n= 10 115.79 s 32.18 MB Next we show a brief statistics about the relation between the computing time and the memory used by our algorithm for the Lie algebras fn . In this sense, Figure 1 shows the behavior of the computing time (C.T.) and used memory (U.M.) with respect to the dimension n of fn . Note that the computing time increases more quickly than the used memory. Additionally, whereas the increase of the computing time ts a positive exponential model, the used memory does not follow such a model. Finally, we have studied the quotients between used memory and computing time, obtaining the frequency diagram shown in Figure 2 . In this case, the behavior also ts an exponential model, but being negative this time. In order to conclude, we have determined the complexity order of the main routines of the algorithm, considering the number of operations carried out in the worst case. We have used the big O notation to express the complexity, which can be consulted in [17]. By Ni(n) we have denoted the order of the operations when running Step i . This function depends on the dimension n of the Lie algebra. Table 4 shows the number of 20
Figure 1: Graph for the C.T. with respect to dimension. Figure 2: Graph for the quotients U.M./C.T. with respect to dimension. computations and the complexity order of each step. In fact, we have determined that the algorithm is of polynomial complexity order, where the last two steps are the most computationally expensive. In order to compute the number of operations for each step, we have considered the worst case. In comparison to the table shown in [7], notice that we have obtained an improvement since our method has six routines instead of eight and we were able to reduce the number of operations. 6 Conclusions In this paper, we have shown an algorithmic procedure to obtain a minimal faithful matrix representation of any nilpotent Lie algebra by using the symbolic computation package MAPLE 12. We have also developed a computational study of the previous algorithm and we have obtained a list with explicit representatives of minimal faithful matrix representations for nilpotent Lie algebras of dimension less than 8 . In our opinion, the methods and results shown in this paper may be useful and helpful to compute minimal faithful matrix representations for a given Lie algebra. Moreover, the knowledge of this representation may involve a new tool to solve the problem of classifying Lie algebras. 21
Table 4: Complexity and number of operations. Routine Complexity Operations law_gn O(n3)N1(n) = n−1 X i=1 n X j=i+1 O(1) + n−2 X i=1 n−1 X j=i+1 n X k=j+1 O(1) DimRepresentation O(n3)N2(n) = n X i=1 n X i=1 n X j=1 O(1) listposi O(n)N3(n) = n X i=1 O(1) expr O(n2)N4(n) = n−1 X i=1 n X j=1 O(1) Listeq O(n4)N5(n) = n X i=1 n X j=1 (N4(n)) sys O(n)N6(n) = O(n) Acknowledgment This work has been partially supported by MTM2013-40455-P, MTM2016-75024-P and FEDER. References [1] J.C. Benjumea, F.J. Echarte, J. Núñez, A.F. Tenorio, A method to obtain the Lie group associated with a nilpotent Lie algebra, Comput. Math. Appl. 51 (2006) 1493 1506. [2] J.C. Benjumea, J. Núñez, A.F. Tenorio, Minimal linear representations of the lowdimensional nilpotent Lie algebras, Math. Scand. 102 (2008) 1726. [3] D. Burde, On a renement of Ado's Theorem. Arch. Math. (Basel) 70 (1998) 118127. [4] D. Burde. Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Central European J. of Math. 4 :3, (2006), 323-357. [5] D. Burde, W. Moens, Minimal faithful representations of reductive Lie algebras, Arch. Math. 89 (2007), 513523. [6] M. Ceballos, J. Núñez, A.F. Tenorio, Representing Filiform Lie Algebras Minimally and Faithfully by Strictly Upper-Triangular Matrices, Journal of Algebra and its Applications. 12 :4 (2013), 15 pp. DOI: 10.1142/S0219498812501964 22
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