About a family of naturally graded no p-filiform lie algebras

E extracta mathematicae Vol. 20, N´um. 1, 1 – 12 (2005) About a family of Naturally Graded no p-filiform Lie algebras † L.M. Camacho, J.R. G´ omez, A.J. Gonz´ alez Departamento de Matem´atica Aplicada I, Universidad de Sevilla Avda. Reina Mercedes s/n, 41012 Sevilla, Spain e-mail: lc[email protected], jr[email protected], [email protected] (Presented by Santos Gonz´alez) AMS Subject Class. (2000): 17B45, 17B70 Received December 20, 2004 1. Introduction The knowledge of naturally graded Lie algebras of a particular Lie algebras class gives a valuable information about the structure of the rest of algebras of that class. In 1970, Vergne [9] obtained the classification in finite arbitrary dimension, n, for the case of filiform (nilindex n−1). In [8, 7] Goze and Khakimdjanov gave the geometric description of the characteristically nilpotent filiform Lie algebras using the naturally graded filiform Lie algebras. In [6] G´omez and Jim´enez-Merch´an, obtained the classification in finite arbitrary dimension for the case 2-filiform (nilindex n−2). There are two subcases for the nilindex n−3: 3-filiform Lie algebras and the Lie algebras with characteristic sequence (n−3,2,1). In [4, 5], Cabezas, G´omez and Pastor gave the classification of naturally graded p-filiform Lie algebras. Consistently, for nilindex n−3, only rest to study the case of characteristic sequence (n−3,2,1). In this work we offer the classification in arbitrary finite dimension of the family of naturally graded Lie algebras gwith the above characteristic sequence such that the dimension of the derived ideal is minimum, that is, with dim[g,g] = n−3. The two first acceptable dimensions are 5 and 6, but the general situation occurs only for n≥7. y This paper has been partially supported by the PAICYT, of Junta de Andaluc´ıa (Spain), and by the Ministerio de Ciencia y Tecnolog´ıa (Spain), ref. BFM 2000-1047 1 2l.m. camacho, j.r. g´ omez, a.j. gonz´ alez 2. Preliminaries The descending central sequence of a Lie algebra gis defined by (Ci(g)), i∈N∪ {0}, where C0(g) = gand Ci(g) = [g,Ci−1(g)]. A Lie algebra gis called nilpotent if there exists k∈Nsuch that Ck(g) = {0}. The smallest integer verifying this equation is called the nilindex of g. A Lie algebra g, with dim(g) = n, is called filiform (or 1-filiform ) if it verifies dim(Ci(g)) = n−i−1 for 1 ≤i≤n−1. These algebras have maximal nilindex n−1. The Lie algebras with a nilindex n−2 are called quasifiliform (or 2-filiform ) and those whose nilindex is 1 are called abelian. Let gbe a nilpotent Lie algebra of dimension n. For all X∈g−[g,g], c(X)=(c1(X), c2(X), . . . , 1) is the sequence, in decreasing order, of the dimensions of the characteristic subspaces of the nilpotent operator ad(X), where the adjoint operator of an element X∈g, ad(X), is defined by ad(X) : g→g Y7→ [X, Y ]. The finite sequence c(g) = sup{c(X) : X∈g−[g,g]}is called the characteristic sequence or Goze invariant of the nilpotent Lie algebra g. The filiform, quasifiliform and abelian Lie algebras of dimension nhave as their Goze invariant (n−1,1), (n−2,1,1) and (1,1, . . . , 1), respectively. The Lie algebras with characteristic sequence (n−p, 1, . . . , 1) are known as p-filiform Lie algebras [3]. We know the classification of p-filiform for the integer values of pbetween n−5 and n−2 ([2, 1]). Remark that, for nilindex n−3, there are two families with Goze invariant (n−3,1,1,1) and (n−3,2,1) respectively. Note that a complex Lie algebra gis naturally filtered by the descending central sequence. This result leads to associate any Lie algebra gwith a graded Lie algebra, gr gwith equal nilindex: gr g=M i∈ Z gi,gi=Ci−1(g)/Ci(g). By nilpotency, the above graduation is finite, that is gr g=g1⊕g2⊕· · ·⊕gk with [gi,gj]⊂gi+j, for i+j≤k. A Lie algebra gis said to be naturally graded if gr gis isomorphic to g, what will be denoted henceforth by gr g=g. Let {X0, X1, . . . , Xn−3, Y1, Y2}be an adapted basis of g. We study the case where the dimension of the derived ideal is minimum, consistently dim[g,g] = n−3. Thus, Y1is not in [g,g] and, consequently, Y1∈g1. In general, if we denote as rto the position of the vector Y1into the subspaces of the natural about a family of lie algebras 3 graduation, we observe that the value of ris r= 1. We remark that the position of Y2is previously determined because we have that [X0, Y1] = Y2 and that implies Y2∈gr+1 with 1 ≤r≤n−4. Then, in this case Y2∈g2. From now, Jacobi identity for the vectors X, Y, Z will be denoted as Jac(X, Y, Z) and the laws of the algebras, g, of dimension nsuch that dim[g,g] is minimum will be denoted as µn. 3. Structure theorem In this section, we will obtain a first approximation to the structure of naturally graded Lie algebras with Goze invariant (n−3,2,1). Let gbe a naturally graded Lie algebra of Goze’s invariant (n−3,2,1) and let {X0, X1, . . . , Xn−3, Y1, Y2}be an adapted basis of g, that is: [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Xn−3] = 0 , [X0, Y1] = Y2, [X0, Y2] = 0 , where X0∈g−[g,g]. That implies C1(g)⊃ hX2, X3, . . . , Xn−3, Y2i, Ci(g)⊃ hXi+1, Xi+2, . . . , Xn−3i(2 ≤i≤n−4) . Lemma 3.1. Let gbe a Lie algebra of dimension nand Goze’s invariant (n−3,2,1) and let {X0, X1, . . . , Xn−3, Y1, Y2}be an adapted basis of g. Then, X1/∈ C1(g), Xn−3∈ Z(g), Y1/∈ Cn−4(g), Y2/∈ Cn−3(g). Proof. Obviously, Xn−3∈ Z(g), Y1/∈ Cn−4(g) and Y2/∈ Cn−3(g) because, otherwise, gcould not be of characteristic sequence (n−3,2,1). It is easy to prove that X1/∈[g,g] supposing that X1∈[Y1, Y2], or X1∈[Xi, Yj], 1≤i≤n−4, 1 ≤j≤2, or X1∈[Xi, Xj], 1 ≤i < j ≤n−3−i, and obtaining contradiction. Remark 3.2. We identify each vector with its class, and we call µ(n, r) the family of laws of Lie algebras with Goze invariant (n−3,2,1) where nis the dimension and ris the position of Y1in the subsets of the natural gradation. We remark that the position of Y2is previously determined because we have that [X0, Y1] = Y2and that implies Y2∈gr+1 with 1 ≤r≤n−4. 4l.m. camacho, j.r. g´ omez, a.j. gonz´ alez Remark 3.3. It is easy to see that g1⊃ hX0, X1iand gi⊃ hXii, 2 ≤i≤ n−3. Now, we obtain the general structure of laws of naturally graded Lie algebras of characteristic sequence (n−3,2,1) in arbitrary dimension. At first, we prove that if Y1∈gr, then ris odd. Lemma 3.4. If r is even, the case µ(n, r)is not admissible in any dimension. Proof. Let gbe a naturally graded Lie algebra of Goze invariant (n− 3,2,1), let {X0, X1, . . . , Xn−3, Y1, Y2}be an adapted basis of g, and let Y1∈gr be with reven. It is easy to prove that Y1/∈[g,g] so Y1∈g1and this is impossible because ris even. Theorem 3.5. (Structure theorem) Any complex naturally graded Lie algebra gof dimension n≥5, with Goze invariant (n−3,2,1) is isomorphic to one whose law can be expressed in an adapted basis {X0, X1, . . . , Xn−3, Y1, Y2}by: •If r= 1      [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [Xi, Xj] = aijXi+j(1 ≤i < j ≤n−3−i). •If 3≤r≤n−5 2,rodd                      [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [Xi, Xj] = aijXi+j(i+j /∈ {r, r + 1},1≤i < j ≤n−3−i), [Xi, Xr−i] = ai,r−iXr+ (−1)i−1Y1(1 ≤i≤r−1 2), [Xi, Xr+1−i] = ai,r+1−iXr+1 + (−1)i−1(r+1−2i) 2Y2(1 ≤i≤r−1 2), [Xi, Y1] = εXr+i(1 ≤i≤n−3−r), with ε∈ {0,1}. about a family of lie algebras 5 •If n−4 2≤r≤n−4,rodd                                    [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [Xi, Xj] = aijXi+j(i+j /∈ {r, r + 1},1≤i < j ≤n−3−i), [Xi, Xr−i] = ai,r−iXr+ (−1)i−1Y1(1 ≤i≤r−1 2), [Xi, Xr+1−i] = ai,r+1−iXr+1 +(−1)i−1(r+1−2i) 2Y2(1 ≤i≤r−1 2), [Xi, Y1] = (c1−(i−1)c2)Xr+i(1 ≤i≤n−3−r≤n−2 2), [Xi, Y2] = c2Xr+1+i(1 ≤i≤n−4−r≤n−4 2), [Y1, Y2] = hXn−3(h= 0 if r6=n−4 2), with c1, c2∈C. Proof. If gis in the condition of theorem, then a first general expression of gis given by:                                    [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [Xi, Xj] = aijXi+j(i+j /∈ {r, r + 1},1≤i < j ≤n−3−i), [Xi, Xr−i] = ai,r−iXr+bi1Y1(1 ≤i≤r−1 2), [Xi, Xr+1−i] = ai,r+1−iXr+1 +bi2Y2(1 ≤i≤r−1 2), [X1, Y1] = c11Xr+1 +dY2, [Xi, Y1] = ci1Xr+i(2 ≤i≤n−3−r), [Xi, Y2] = ci2Xr+1+i(1 ≤i≤n−4−r), [Y1, Y2] = hX2r+1 (si r≤n−4 2). Some elementary changes of basis jointly with Jacobi identity implies that: •If 1 ≤r≤n−5 2the coefficients can be expressed by ci,1=c1(1 ≤i≤n−3−r) and ci,2= 0 (1 ≤i≤n−4−r). •If n−4 2≤r≤n−4 the coefficients can be expressed by ci,1=c1−(i−1)c2(1 ≤i≤n−3−r) and ci,2=c2(1 ≤i≤n−4−r). By using Jacobi identity it is posible to obtain that bi,2= (−1)(i−1) r+ 1 −2i 2b1,1≤i≤r−1 2. 6l.m. camacho, j.r. g´ omez, a.j. gonz´ alez Furthermore, b16= 0 (in other case Y1/∈ C1(g) and then Y1/∈gr=< Xr, Y1> with r≥3). Next, an easy change of basis allows to suppose b1= 1. Then, •If 3 ≤r≤n−5 2. As b16= 0, if c16= 0 an easy change of basis allows to suppose c1= 1, and consistently c1∈ {0,1}. •If r= 1, the case must be studied separately. 4. Dimensions n= 5 and n= 6. Even if our main aim is to study the case of dimension nfinite arbitrary, the low dimensional cases are special and we will study them previously. The lowest cases are for dimensions n= 5 and n= 6 and they have a special treatment. Theorem 4.1. Any complex naturally graded Lie algebra of dimension 5 with Goze invariant (2,2,1) is isomorphic to one whose law can be expressed in an adapted basis {X0, X1, X2, Y1, Y2}by: µ5:([X0, X1] = X2, [X0, Y1] = Y2. Proof. The proof is trivial. Theorem 4.2. Any complex naturally graded Lie algebra of dimension 6 with Goze invariant (3,2,1) is isomorphic to one whose law can be expressed in an adapted basis {X0, X1, X2, X3, Y1, Y2}by: µ1 6:([X0, Xi] = Xi+1 (1 ≤i≤2) , [X0, Y1] = Y2,µ2 6:     [X0, Xi] = Xi+1 (1 ≤i≤2) , [X0, Y1] = Y2, [X1, X2] = X3. Proof. In dimension six the graduation is hX0, X1, Y1i⊕hX2, Y2i ⊕ hX3i, and by Theorem 3.5 the laws of these algebras are the following:          [X0, X1] = X2, [X0, X2] = X3, [X0, Y1] = Y2, [X1, X2] = a12X3. about a family of lie algebras 7 By using a generic change of basis we prove that nullity of coefficient a12 is an invariant. •If a12 6= 0, it is easy to obtain the algebra of law µ2 6. •If a12 = 0, we obtain the algebra of law µ1 6. 5. Dimension n≥7. Now, we present the classification of the naturally graded Lie algebras with Goze invariant (n−3,2,1), dimension n≥7 and dim[g,g] minimum, that is, equal to n−3. The first expression of this family is given by the following lemma: Lemma 5.1. Let gbe a naturally graded Lie algebra with Goze invariant (n−3,2,1),dim(g) = n≥7and dim[g,g] = n−3. Then, there exists a characteristic vector X0and an adapted basis {X0, X1, . . . , Xn−3, Y1, Y2}, which lead us to express the laws of gby: µa n:     [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [X1, Xi] = aXi+1 (2 ≤i≤n−4) , if nis odd, or µa,b n:                [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [X1, Xi] = aXi+1 (2 ≤i≤n−5) , [X1, Xn−4] = (a+b)Xn−3, [Xi, Xn−3−i] = (−1)i+1bXn−3(2 ≤i≤n−4 2), if nis even. Proof. By using Teorema 3.5 it follows that, in this case (r= 1), there exists a characteristic vector X0and an adapted basis, {X0, X1, . . . , Xn−3, Y1, Y2}, such that the laws of the algebra are given by µn:     [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [X1, Xi] = aijXi+j(2 ≤i < j ≤n−3−i). 8l.m. camacho, j.r. g´ omez, a.j. gonz´ alez Now, we use an inductive procedure on n. Dimension n= 7 :In dimension seven the graduation is hX0, X1, Y1i⊕hX2, Y2i⊕hX3i ⊕ hX4i, and by using the Jacobi identity in the family µ7we obtain µa 7. Dimension n= 8 :Analogously, by using the Jacobi identity it is easy to obtain that µ8is µa,b 8. The inductive procedure is realized in function of the parity of the dimension. That is the reason why we study the cases of dimension neven and n odd separately. Dimension n > 7,nodd : If we suppose that the result is true for n=k even, we will prove it for n=k+ 1 odd. If kis even, we suppose that it is possible to express µkby µa,b k:                [X0, Xi] = Xi+1 (1 ≤i≤k−4) , [X0, Y1] = Y2, [X1, Xi] = aXi+1 (2 ≤i≤k−5) , [X1, Xn−4] = (a+b)Xn−3, [Xi, Xn−3−i] = (−1)i+1bXn−3(2 ≤i≤k−4 2). Now, for n=k+ 1, we add the brackets [X0, Xk−3] = α0Xk−2, [Xi, Xk−2−i] = αiXk−2(1 ≤i≤k−4 2), [Xk−3, Y1] = β1Xk−2, [Xk−4, Y2] = β2Xk−2. By using Jacobi identity we prove the result. Dimension n > 8,neven : We suppose that the result is true for n=k odd and we will prove it for n=k+ 1 even. If kis odd, we suppose that it is possible to express µkby µa k:     [X0, Xi] = Xi+1 (1 ≤i≤k−4) , [X0, Y1] = Y2, [X1, Xi] = aXi+1 (2 ≤i≤k−4) . For n=k+ 1 it is necessary to add the same brackets as in the odd case and analogously we obtain the result. about a family of lie algebras 9 6. Classification theorem Finally, we give the theorem of classification for naturally graded Lie algebras with Goze invariant (n−3,2,1), r= 1 and n≥7. Theorem 6.1. Any complex naturally graded Lie algebra of dimension n, n ≥7, with Goze invariant (n−3,2,1) and laws µ(n)is isomorphic to one whose law can be expressed in suitable adapted basis by µ1 (n−3,2,1) (n≥5) :([X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2; µ2 (n−3,2,1) (neven, n ≥6) :     [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [Xi, Xn−3−i] = (−1)i+1Xn−3(1 ≤i≤n−4 2) ; µ3 (n−3,2,1) (n≥7) :     [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [X1, Xi] = Xi+1 (2 ≤i≤n−4) ; µ4 (n−3,2,1) (neven, n ≥8) :         [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [X1, Xi] = Xi+1 (2 ≤i≤n−5) , [Xi, Xn−3−i] = (−1)iXn−3(2 ≤i≤n−4 2) ; µ5 (n−3,2,1) (neven, n ≥8) :                [X0, Xi] = Xi+1 (1 ≤i≤n−4) , [X0, Y1] = Y2, [X1, Xi] = Xi+1 (2 ≤i≤n−5) , [X1, Xn−4] = 2Xn−3, [Xi, Xn−3−i] = (−1)i+1Xn−3(2 ≤i≤n−4 2). Proof. By using the above lemma we will obtain the result. In function of the dimension of the algebra it is necessary to consider two different cases. Let gbe a naturally graded Lie algebra of dimension nodd, n≥7, with Goze invariant (n−3,2,1) and laws µn. Then, the natural graduation is given by hX0, X1, Y1i⊕hX2, Y2i⊕hX3i ⊕ · · · ⊕ hXn−3i.