Breadth and characteristic sequence of nilpotent Lie algebras
aa r X i v : . [ m a t h . R A ] M a y BREADTH AND CHARACTERISTIC SEQUENCE OF NILPOTENT LIEALGEBRAS
ELISABETH REMM
Abstract.
The notion of breadth of a nilpotent Lie algebra was introduced and used toapproach problems of classification up to isomorphism in [6]. In the present paper, we studythis invariant in terms of characteristic sequence, another invariant introduced by Goze andAncochea in [2]. This permits to complete the determination of Lie algebras of breadth 2studied in [6] and to begin the work for Lie algebras with breadth greater than 2.
Introduction
The breadth of a Lie algebra was introduced in [6]. As it is well explained in this paper,initially, the notion of breadth was defined to investigate the p -groups: the breadth of a p -group is the size of the largest conjugacy class in this p -group. Recently characterizationsof finite p-groups of breadth 1 and 2 have been done. In [6], the authors translate this notionto finite dimensional nilpotent Lie algebras: the breadth of a nilpotent Lie algebra is themaximum of the ranks of the adjoint operators and in their paper, they describe nilpotentLie algebras of breadth 1 and, for dimensions 5 and 6, of breadth 2.We know that the classification, up to isomorphism, of finite dimensional nilpotent Liealgebras is a very difficult problem. It is solved up to the dimension 7 in the complex andreal cases and for some special families of nilpotent Lie algebras in bigger dimensions. Inrecent papers [8, 9], since the classification problem in large dimensions seems unrealistic,we choose to describe classes of nilpotent Lie algebras, in any dimension, associated withthe characteristic sequence. For these families, we have reduced the notion of Chevalley-Eilenberg cohomology and given, sometimes, topological properties of these families. Inthis paper, we consider these results to complete in any dimension the characterization ofnilpotent Lie algebras of breadth equal to 2, completing, in some way, the work [6]. Wedevelop also the case where the breadth is 3.1. breadth and characteristic sequence Breadth of a nilpotent Lie algebra.
Let g be a finite dimensional Lie algebra overfield K of characteristic 0. Definition 1. [6]
The breadth of g is the invariant, up to isomorphism, b ( g ) = max { rank(ad X ) , X ∈ g } . More generally, if I is a not trivial ideal of g , then the breadth of g on I is b I ( g ) = max { rank(ad X | I ) , X ∈ g } . Date : Chevat 16, 5775.
We have b g ( g ) = b ( g ) and clearly b I ( g ) ≤ b ( g ). For example, g is abelian if and onlyif b ( g ) = 0 and g is filiform (see [5]) if and only if b ( g ) = dim g − . Recall that a finitedimensional nilpotent Lie algebra is called filiform, notion introduced by Michele Vergne, if g satisfies dim C i ( g ) = i for i = 0 , · · · , n − n = dim g and C i ( g ) are the terms of the ascending sequence of g that is (cid:26) C ( g ) = { } , C i ( g ) = { X ∈ g , [ X, g ] ∈ C i − ( g ) } . The following proposition is proved in [6]
Proposition 2.
For any finite dimensional Lie algebra g , we have b ( g ) ≤ dim( g /Z ( g )) − . In the case of a filiform Lie algebra, the upper bound of this inequality is satisfied. In factif g is filiform, then dim Z ( g ) = 1 and b ( g ) = n − g /Z ( g )) − . Assume now that g is finite dimensional and nilpotent. We have seen that b ( g ) = 0 if andonly if g is abelian. Now if b ( g ) = 1, then, from [6] we have dim C ( g ) = 1 where C ( g ) is thederived subalgebra of g , that is generated by the Lie brackets [ X, Y ] for any
X, Y ∈ g . Then,if g is indecomposable, that is without direct factor, then g is an Heisenberg Lie algebra.The non trivial study begins with b ( g ) = 2. This is a large part of the work of [6]. Wewill complete their result using and comparing with an another invariant of nilpotent Liealgebras, the characteristic sequence, presented in the following section.1.2. Characteristic sequence.
Let g be a finite dimensional nilpotent Lie algebra over K .Let C ( g ) and Z ( g ) be respectively the derived subalgebra and center of g . The characteristicsequence of g , sometimes called the Goze’s invariant and introduced in [2], is an usefulinvariant in the classification theory of finite-dimensional nilpotent Lie algebras. It is defineas follows: for any X ∈ g , let c ( X ) be the ordered sequence, for the lexicographic order,of the dimensions of the Jordan blocks of the nilpotent operator adX . The characteristicsequence of g is the invariant, up to isomorphism, c ( g ) = max { c ( X ) , X ∈ g − C ( g ) } . Then c ( g ) is a sequence of type ( c , c , · · · , c k ,
1) with c ≥ c ≥ · · · ≥ c k ≥ c + c + · · · + c k + 1 = n = dim g . For example,(1) c ( g ) = (1 , , · · · ,
1) if and only if g is abelian,(2) c ( g ) = (2 , , · · · ,
1) if and only if g is a direct product of an Heisenberg Lie algebraby an abelian ideal,(3) If g is 2-step nilpotent then there exists p and q such that c ( g ) = (2 , , · · · , , , · · · , p + q = n , that is p is the occurrence of 2 in the characteristic sequence and q the occurrence of 1.(4) g is filiform if and only if c ( g ) = ( n − , X ∈ g − C ( g ) is called a characteristic vector if c ( X ) = c ( g ). Of course, such avector is not unique. READTH AND CHARACTERISTIC SEQUENCE OF NILPOTENT LIE ALGEBRAS 3
Breadth and characteristic sequence.
Let g be a finite dimensional nilpotent Liealgebra, b ( g ) its breadth and c ( g ) its characteristic sequence. Lemma 3.
Let X be a characteristic vector with c ( g ) = c ( X ) = ( c , · · · , c k , . Then (1) rank(ad X ) = k X i =1 c i − k In fact, if we consider the Jordan block J i of dimension c i of the operator ad X , since 0 isthe only eigenvalue, we have rank( J i ) = c i − . We deduce
Theorem 4.
Let g be a finite dimensional Lie algebra whose characteristic sequence is c ( g ) = ( c , · · · , c k , . Then, its breadth satisfies (2) b ( g ) = k X i =1 c i − k. Consequences.
1. If g is n -dimensional 2-step nilpotent, then c ( g ) = (2 , · · · , , , · · · , p times, then b ( g ) = p.
2. If g is n -dimensional 3-step nilpotent, then c ( g ) = (3 , · · · , , , · · · , , , · · · , p times, 2 appears p times in this sequence, then b ( g ) = 2 p + p .
3. More generally, if g is n -dimensional k -step nilpotent, then its characteristic sequenceis c ( g ) = ( k, · · · , k, · · · , , · · · , , , · · · , , , · · · , k appears p k -times, k − p k − -times, up to 2 which appears p -times and 1 which appears p -times. Then b ( g ) = ( k − p k + ( k − p k − + · · · + 2 p + p = k X i =1 ( i − p i . Proposition 5. A n -dimensional nilpotent Lie algebra satisfies b ( g ) = 2 if and only if oneof these conditions is satisfied: (1) g is -step nilpotent with c ( g ) = (2 , , , · · · , . (2) g is -step nilpotent with c ( g ) = (3 , , · · · , . In fact, if g is k -step nilpotent with k ≥
4, then b ( g ) = ( k − p k +( k − p k − + · · · +2 p + p with p k >
0. Then b ( g ) ≥ . Now if g is 3-step nilpotent, then b ( g ) = 2 p + p with p > b ( g ) = 2 implies p = 1 and p = 0. If g is 2-step nilpotent, then b ( g ) = p this gives p = 2. ELISABETH REMM Nilpotent Lie algebras with breadth equal to
2A lot of results concerning nilpotent Lie algebras with breadth equal to 2 are presentedin [6], principally for the small dimensions. In this section, we propose to complete thisstudy using Proposition 5. But, before begining this study, we recall some properties of animportant tool used in [8] and based on deformation theory of Lie algebras. Let g = ( K n , µ )be a n -dimensional Lie algebra over K where µ denotes the Lie bracket of g and K n theunderlying linear space to g . A formal deformation g of g is given by a pair g = ( K n , µ )where µ is a formal series µ = µ + X t i ϕ i with skew-symmetric bilinear maps ϕ i on K n and µ satisfying the Jacobi identity. To simplifynotations, we denote for any bilinear map ϕ on K n with values on K n , ϕ • ϕ the trilinearmap ϕ • ϕ ( X, Y, Z ) = ϕ ( ϕ ( X, Y ) , Z ) + ϕ ( ϕ ( Y, Z ) , X ) + ϕ ( ϕ ( Z, X ) , Y )and by ϕ ◦ ϕ the comp -operation (the composition on the first element), that is ϕ ◦ ϕ ( X, Y, Z ) = ϕ ( ϕ ( X, Y ) , Z ) . Then the Jacobi identity for ϕ is equivalent to ϕ • ϕ = 0 . If µ is a formal deformation of µ ,then µ • µ = 0 implies (cid:26) δ C,µ ϕ = 0 ,ϕ • ϕ = δ C,µ ϕ , where δ C,µ is the coboundary operator associated to the Chevalley-Eilenberg complex of g : δ C,µ ( ϕ ) = µ • ϕ + ϕ • µ . We shall denote by H ∗ C ( g , g ) or sometimes H ∗ C ( µ , µ ) the corresponding cohomology spaces.A special class of formal deformation is the class of linear deformations. We call linear de-formation of g = ( K n , µ ) a formal deformation µ = µ + P t i ϕ i with ϕ k = 0 for any k ≥ µ = µ + tϕ . In this case, the Jacobi identity for µ is equivalent to(3) (cid:26) δ C,µ ϕ = 0 ,ϕ • ϕ = 0 . Lie algebras with characteristic sequence (3 , , · · · , . Let g = ( K n , µ ) be a n -dimensional nilpotent Lie algebra with characteristic sequence c ( g ) = (3 , , · · · , g , ,n − = ( K n , µ ) whoseLie brackets are[ X , X ] = X , [ X , X ] = X , [ X , X i ] = 0 for i = 4 , · · · , n, [ X i , X j ] = 0 , ≤ i < j ≤ n. So µ = µ + tϕ where ϕ is a bilinear map satisfying the conditions (3) and conditions whichimply that µ is a Lie bracket of a 3-step nilpotent Lie algebra. All these conditions aredescribed in [8] and imply that g is in one of the following cases:(1) g is isomorphic to g , ,n − ,(2) g is a direct product of an abelian ideal and of the 5-dimensional Lie algebra and itsLie bracket is isomorphic to µ + ϕ with ϕ ( X , X ) = aX , READTH AND CHARACTERISTIC SEQUENCE OF NILPOTENT LIE ALGEBRAS 5 (3) g is a direct product of an abelian ideal and of the 5-dimensional Lie algebra and itsLie bracket is isomorphic to µ + ϕ with ϕ ( X , X ) = bX (4) If g is indecomposable and of dimension greater than 5, then g is isomorphic to theLie algebra [ X , X ] = X , [ X , X ] = X , [ X , X k ] = a ,k X , k ≥ , [ X l , X k ] = a l,k X , ≤ l < k ≤ n. This last family of Lie algebras, parametrized by the constants structure a ,k and a l,k can bereduced to obtain a classification up to isomorphism. Proposition 6. [4]
Any n -dimensional nilpotent Lie algebra, n ≥ with breadth equal to and characteristic sequence (3 , , · · · , is isomorphic to one of the following Lie algebraswhose Lie brackets are (1) [ X , X ] = X , [ X , X ] = X that is g , ,n − . (2) [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , here n ≥ . (3) [ X , X ] = X , [ X , X ] = X , [ X i +1 , X i +2 ] = X , i = 1 , · · · , p − n = 2 p , (4) [ X , X ] = X , [ X , X ] = X , [ X , X n ] = X , [ X i +1 , X i +2 ] = X , i = 1 , · · · , p − n = 2 p + 1 , Lie algebras whose characteristic sequence is (2 , , , · · · , . In [7], we have stud-ied nilpotent Lie algebras with characteristic sequence (2 , , , , ,
1) in order to constructsymplectic form on these Lie algebras. We will use notations and results which are in thispaper. Let F n, be the set of n -dimensional 2-step nilpotent Lie algebras and F n, ,n − the sub-set corresponding to Lie algebras of characteristic sequence (2 , , , · · · ,
1) (the subscriptscorrespond to the numbers of 2 and 1 in the characteristic sequence). Let g ,n − be the n -dimensional Lie algebra given by the brackets[ X , X i ] = X i +1 , i = 1 , . It is obvious that g ,n − ∈ F n, ,n − . From [7], any n -dimensional 2-step nilpotent Lie algebrawith characteristic sequence (2 , , , · · · ,
1) is isomorphic to a linear deformation of g ,n − .This means that its Lie bracket is isomorphic to µ = µ + tϕ , where µ is the Lie bracket of g ,n − , and ϕ is a skew-bilinear map such that ( ϕ ∈ Z CH ( g ,n − , g ,n − ) ,ϕ ◦ ϕ = 0 . Recall that Z CH ( g ,n − , g ,n − ) is constituted of bilinear maps ϕ on K n with values in thisspace such that ϕ ◦ µ + µ ◦ ϕ = 0 . In fact, in [8], we have define a sub-complex of the Chevalley-Eilenberg complex whose mainproperty is that for any deformation µ = µ + P t i ϕ i of a 2-step nilpotent Lie algebra g = ( µ , K n ) which is also 2-step nilpotent, then ϕ ∈ Z CH ( µ , µ ).We deduce that the family F n, ,n − of n -dimensional 2-step nilpotentLie algebras with char-acteristic sequence (2 , , , · · · ,
1) is the union of two algebraic components, the first one,
ELISABETH REMM C ( F n, ,n − ), corresponding to the cocycles(4) ϕ ( X i , X j ) = X k =1 a k +1 i,j X k +1 , ≤ i < j ≤ n, i, j = 3 , C ( F n, ,n − ), to the cocyles(5) (cid:26) ϕ ( X i , X j ) = P k =1 a k +1 i,j X k +1 , ( i, j ) = (2 , , ≤ i < j ≤ n − , i, j / ∈ { , } ϕ ( X , X ) = X n where the undefined products ϕ ( X, Y ) are nul or obtained by skew-symmetry. Each of thesecomponents is a regular algebraic variety. These components can be characterized by thenumber of generators.
Proposition 7.
Any n -dimensional nilpotent Lie algebra with breadth equal to and char-acteristic sequence (2 , , , · · · , is isomorphic to a Lie algebra belonging to one of thecomponents C ( F n, ,n − ) or C ( F n, ,n − ) . Nilpotent Lie algebras with breadth equal to n -dimensional Lie algebra of breadth b ( g ) = 3 is equal to one of the following:(1) c ( g ) = (4 , , · · · , , ( n ≥ c ( g ) = (3 , , , · · · , n ≥ c ( g ) = (2 , , , , · · · , n ≥ Nilpotent Lie algebras of characteristic sequence (4 , , · · · , . Let us considerthe n -dimensional nilpotent Lie algebra g , , ,n − = ( K n , µ ) with characteristic sequence(4 , , · · · ,
1) whose Lie brackets in the basis { X , · · · , X n } are µ ( X , X i ) = X i +1 , i = 2 , , n -dimensional Lie algebra g whose characteristicsequence is also (4 , , · · · ,
1) is isomorphic to a Lie algebra whose Lie bracket µ is a lineardeformation of µ , that is µ = µ + tϕ where ϕ is a 2-cocycle of the Chevalley-Eilenbergcomplex H ∗ C ( µ , µ ) satisfying(6) δ C,µ ( ϕ ) = 0 ,ϕ • ϕ = 0 ,δ R,µ ( ϕ ) = 0 ,µ ◦ ϕ + µ ◦ ϕ ◦ µ ◦ ϕ + µ ◦ ϕ ◦ µ + ϕ ◦ µ ◦ ϕ ++ ϕ ◦ µ ◦ ϕ ◦ µ + ϕ ◦ µ = 0 ,µ ◦ ϕ + ϕ ◦ µ ◦ ϕ + ϕ ◦ µ ◦ ϕ + ϕ ◦ µ = 0 ,ϕ = 0 , where δ R,µ ( ϕ ) is the 5-linear map δ R,µ ( ϕ ) = µ ◦ ϕ + µ ◦ ϕ ◦ µ + µ ◦ ϕ ◦ µ + ϕ ◦ µ , READTH AND CHARACTERISTIC SEQUENCE OF NILPOTENT LIE ALGEBRAS 7 and ϕ k = ϕ ◦ · · · ◦ ϕ. Moreover, ϕ satisfies ϕ ( X , Y ) = 0 for any Y and X is not in thederived subalgebra of g . This implies that ( µ ◦ ϕ )( X i , X j , X k ) = 0 if 1 < i < j < k . Then(6) gives(7) ϕ ( X , X i ) = 0 , ∀ i,ϕ ( X , X ) = aX + P nk =6 d k X k ,ϕ ( X , X k ) = a k X + b k X ≤ k ≤ n,ϕ ( X , X k ) = a k X ≤ k ≤ n,ϕ ( X i , X j ) = c i,j X ≤ i < j ≤ n. Proposition 8.
Every n -dimensional nilpotent Lie algebra ( n ≥ ) of breadth and charac-teristic sequence (4 , , · · · , is isomorphic to a Lie algebra belonging to one of the families [ X , X i ] = X i +1 , i = 2 , , , [ X , X i ] = 0 , ∀ i, [ X , X ] = aX , [ X , X k ] = a k X + b k X , ≤ k ≤ n, [ X , X k ] = a k X , ≤ k ≤ n, [ X i , X j ] = c i,j X , ≤ i < j ≤ n. [ X , X i ] = X i +1 , i = 2 , , , [ X , X i ] = 0 , ∀ i [ X , X ] = aX + P nk =6 d k X k , [ X , X k ] = b k X , ≤ k ≤ n, [ X i , X j ] = c i,j X , ≤ i < j ≤ n, with j − X i =6 c i,j d i − n X i = j +1 c j,i d i = 0 for 6 ≤ j ≤ n. The first family is a plane of dimension ( n − n − . The second family defines an algebraicvariety of codimension n − ( n − n − . Proposition 8 gives all the nilpotent Lie algebras of breadth 3 and characteristic se-quence (4 , , · · · , . The nilpotent Lie algebras of breadth 3 and characteristic sequences(3 , , , · · · ,
1) and (2 , , , , · · · ,
1) are determinated in [8].
Remark.
We have just seen that the set F n, , , ,n − of n -dimensional Lie algebras, n ≥ , , · · · ,
1) is an algebraic variety which is the union oftwo irreducible components, the first one is the orbit by the linear group of a ( n − n − plane then it is reduced (the affine shema which corresponds to this component is reduced).Then, if there exists a rigid Lie algebra g in F n, , , ,n − , that is any deformation of g in F n, , , ,n − is isomorphic to g , then dim H CR ( g , g ) = 0, where H CR ( g , g ) = Z CR ( g , g ) B CR ( g , g ) andthe two cocycles of Z CR ( g , g ) are bilinear maps satisfying (6) and B CR ( g , g ) the subspace ELISABETH REMM { δf, f ∈ gl ( n, K ) } ∩ Z CR ( g , g ). Now, if we consider the Lie algebra [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X n − ] = X , [ X , X n ] = X , [ X , X n ] = X , [ X , X ] = · · · = [ X r , X n ] = X , r = 2 p if n = 2 p + 1 or r = 2 p − n = 2 p. we have dim Z CR ( g , g ) = ( n − n − = dim B CR ( g , g ) . We deduce that g is rigid in F n, , , ,n − and the first component is the Zariski closure of the orbit, for the natural actionof the linear group, of g . A similar computation for the second component shows that theLie algebra g given by [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X n , [ X , X n ] = X , [ X k , X k +1 ] = X ≤ k ≤ n − , belongs to the second component and it is rigid in this component. So any n -dimensionalLie algebras of characteristic sequence (4 , , · · · ,
1) is a contraction (or degeneration) of oneof these two Lie algebras.3.2.
Lie algebras of characteristic sequence (2 , , , , · · · , . Let n ≥ g ,n − be the n -dimensional Lie algebra given by the brackets[ X , X i ] = X i +1 , ≤ i ≤ , and other non defined brackets are equal to zero. Any 2-step nilpotent n -dimensional Liealgebra with characteristic sequence (2 , , , , · · · ,
1) is isomorphic to a linear deformationof g ,n − . Its Lie bracket is isomorphic to µ = µ + tϕ , where µ is the Lie bracket of g ,n − ,and ϕ is a skew-bilinear form such that ( ϕ ∈ Z CH ( g ,n − , g ,n − ) ,ϕ ◦ ϕ = 0 . From the construction of the linear deformation, we can assume that ϕ ( X , X ) = 0 for any X . Now the first relation is equivalent to ϕ ( X i , X j ) = X k =1 a k +12 i, j X k +1 + n X k =8 a k i, j X k , ≤ i < j ≤ ,ϕ ( X s , X l ) = X k =1 b k +1 s,l X k +1 , s, l ≥ . In this case, the identity ϕ ◦ ϕ = 0 is satisfied and with such cocycles we describe, upto isomorphism, all the deformations of g ,n − which are 2-step nilpotent. But, we haveto restrict these deformations only to Lie algebras with characteristic sequence equal to(2 , , , , · · · , READTH AND CHARACTERISTIC SEQUENCE OF NILPOTENT LIE ALGEBRAS 9 (1) n ≥ ϕ ( X , X ) = X k =1 a k +12 , X k +1 + X ,ϕ ( X , X ) = X k =1 a k +12 , X k +1 + X ,ϕ ( X , X ) = X k =1 a k +14 , X k +1 ,ϕ ( X s , X l ) = 0 s, l ≥ . (2) n ≥ ϕ ( X , X ) = X k =1 a k +12 , X k +1 + X ,ϕ ( X i , X j ) = X k =1 a k +12 i, j X k +1 , ≤ i < j ≤ ,ϕ ( X s , X l ) = a s,l X , s, l ≥ . (3) n ≥ ϕ ( X i , X j ) = X k =1 a k +12 i, j X k +1 , ≤ i < j ≤ ,ϕ ( X s , X l ) = P k =1 a s,l X k +1 , s, l ≥ . Proposition 9.
Any n -dimensional nilpotent Lie algebra whose breadth is and character-istic sequence (2 , , , , · · · , is isomorphic to a Lie algebra whose Lie bracket is µ + ϕ i where ϕ i , i = 1 , , are defined above. For the 7-dimensional case, since the classification of 7-dimensional nilpotent Lie alge-bras is known, from this list we find the following 7-dimensional nilpotent Lie algebras ofcharacteristic sequence (2 , , ,
1) (the notations are those of [ ? ]) n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X . n : [ X , X i ] = X i +1 . i = 2 , , n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . Lie algebras of characteristic sequence (3 , , , · · · , . Consider now the case ofcharacteristic sequence (3 , , , · · · , g is greaterthan 6. In dimension 6 any Lie algebra with characteristic sequence (3 , ,
1) is isomorphic to n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X [ X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X − X [ X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X [ X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , , , ,
1) isisomorphic to n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , X , X ] = X n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = − X , [ X , X ] = X . n : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X . n : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . n : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . n : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . READTH AND CHARACTERISTIC SEQUENCE OF NILPOTENT LIE ALGEBRAS 11 n : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = − X . n ( α ) : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = αX . n : (cid:26) [ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = − X . Let us consider now the general case. We know that any n -dimensional nilpotent Liealgebra with characteristic sequence (3 , , , · · · ,
1) is isomorphic to a linear deformation of g , ,n − whose Lie bracket µ is given by µ ( X , X i ) = X i +1 , i = 2 , , , other non defined product are equal to zero. Since the classification up an isomorphism seemsalso in this case very utopic (see the previous example of dimension 7), we shall determinea reduced family containing only the algebras with this characteristic sequence. For this itis sufficient to compute the 2-cocycle of the Chevalley cohomology of g , ,n − satisfying(8) δ C,µ ( ϕ ) = 0 ,ϕ • ϕ = 0 ,δ R,µ ( ϕ ) = 0 ,µ ◦ ϕ + ϕ ◦ µ ◦ ϕ + ϕ ◦ µ = 0 ,ϕ = 0 , with δ R,µ ( ϕ ) = µ ◦ ϕ + µ ◦ ϕ ◦ µ + ϕ ◦ µ = 0 . But any linear deformation of µ associatedwith a such cocycle belong to F n, that is the family of 3-step nilpotent Lie algebras andnot necessarily in the sub-family F n, , ,n − of 3-step nilpotent Lie algebras with characteristicsequence (3 , , , · · · , Lemma 10.
We have F n, , ,n − = { µ ∈ F n, , with breadth( µ ) = 3 } . From the construction of the linear deformation between any Lie algebra of F n, , ,n − and µ , we can assume that ϕ ( X , X ) = 0 for any X . Now, since µ = µ + ϕ ∈ F n, , ,n − , we havenecessarily ϕ ( X i , X j ) ∈ K { X , X , X } . The previous identities and lemma imply that ϕ satisfies:(9) ϕ ( X , X ) = c X + e X + f X ,ϕ ( X , X ) = c X + e X + f X ,ϕ ( X , X ) = c X + e X + f X ,ϕ ( X , X ) = ( − c + b ) X + ( d − e X ) + f X ,ϕ ( X , X ) = c X + f X ,ϕ ( X , X i ) = c i X + e i X , i ≥ ,ϕ ( X , X i ) = b i X + c i X + e i X , i ≥ ,ϕ ( X , X i ) = b i X References [1] Ancochea-Berm´udez J.M., G´omez-Martin J.R., Valeiras G., Goze M. Sur les composantes irr´eductiblesde la vari´et´e des lois d’alg`ebres de Lie nilpotentes. J. Pure Appl. Algebra 106 (1996), no. 1, 11–22.[2] Ancochea-Berm´udez J. M., Goze M. Sur la classification des alg`ebres de Lie nilpotentes de dimension7.
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302 (1986), 611–613.[3] Cabezas J. M., G´omez J. R. (n-4)-filiform Lie algebras. Comm. Algebra 27 (1999), no. 10, 4803-4819.[4] Cabezas J. M., G´omez J. R., Jimenez-Merch´an A. Family of p -filiform Lie algebras. Algebra and operatortheory (Tashkent, 1997), 93-102, Kluwer Acad. Publ., Dordrecht, 1998.[5] Goze M., Khakimdjanov Y. Nilpotent Lie algebras.
Mathematics and its Applications, 361. KluwerAcademic Publishers Group, Dordrecht, 1996. xvi+336 pp. ISBN: 0-7923-3932-0[6] Khuhirun B., Misra K.C., Stitzinger E. On nilpotent Lie algebras of small breadth.
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