Faithful Lie algebra modules and quotients of the universal enveloping algebra
aa r X i v : . [ m a t h . R T ] J un FAITHFUL LIE ALGEBRA MODULES AND QUOTIENTS OFTHE UNIVERSAL ENVELOPING ALGEBRA
DIETRICH BURDE AND WOLFGANG ALEXANDER MOENS
Abstract.
We describe a new method to determine faithful representations ofsmall dimension for a finite dimensional nilpotent Lie algebra. We give variousapplications of this method. In particular we find a new upper bound on the mi-ninmal dimension of a faithful module for the Lie algebras being counter examplesto a well known conjecture of J. Milnor. Introduction
Let g be a finite-dimensional complex Lie algebra. Denote by µ ( g ) the minimaldimension of a faithful g -module. This is an invariant of g , which is finite by Ado’stheorem. Indeed, Ado’s theorem asserts that there exists a faithful linear represen-tation of finite dimension for g . There are many reasons why it is interesting tostudy µ ( g ), and to find good upper bounds for it. One important motivation comesfrom questions on fundamental groups of complete affine manifolds and left-invariantaffine structures on Lie groups. A famous problem of Milnor in this area is relatedto the question whether or not µ ( g ) ≤ dim( g ) + 1 holds for all solvable Lie algebras.For the history of this problem, and the counter examples to it see [9], [2] and thereferences given therein.It is also interesting to find new proofs and refinements for Ado’s theorem. Wewant to mention the work of Neretin [10], who gave a proof of Ado’s theorem, whichappears to be more natural than the classical ones. This gives also a new insightinto upper bounds for arbitrary Lie algebras.From a computational view, it is also very interesting to construct faithful repre-sentations of small degree for a given nilpotent Lie algebra g . In [6] we have givenvarious methods for such constructions. In this paper we present another methodusing quotients of the universal enveloping algebra, which has many applicationsand gives even better results than the previous constructions. We obtain new up-per bounds on the invariant µ ( g ) for complex filiform nilpotent Lie algebras g . Inparticular, we find new upper bounds on µ ( g ) for the counter examples to Milnor’sconjecture in dimension 10. Date : October 14, 2018.2000
Mathematics Subject Classification.
Primary 17B10, 17B25.The authors were supported by the FWF, Projekt P21683. The second author was also sup-ported by a Junior Research Fellowship of the ESI, Vienna.
The paper is organized as follows. After some basic properties we give estimates on µ ( g ) in terms of dim( g ) according to the structure of the solvable radical of g . Inthe third section we describe the new construction of faithful modules by quotientsof the universal enveloping algebra. We decompose the Lie algebra g as a semidi-rect product g = d ⋉ n , for some ideal n and a subalgebra d ⊆ Der( n ), and thenconstructing faithful d ⋉ n -submodules of U ( n ). This is illustrated with two easyexamples.In the fourth section we give some applications of this construction. First we provea bound on µ ( g ) for an arbitrary Lie algebra g in terms of the dim( g / n ) and dim( r ),where n denotes the nilradical of g , and r the solvable radical. Then we apply theconstruction to show that µ ( g ) ≤ dim( g ) for all 2-step nilpotent Lie algebras. Fi-nally we apply the method to obtain new estimates on µ ( g ) for filiform Lie algebras g , in particular for dim( g ) = 10. As for the counter examples to Milnor’s conjecturein dimension 10, we give an example in 4.13. It is quite difficult to see that this Liealgebra satisfies µ ( f ) ≥
12, so that it does not admit an affine structure, see [2]. Onthe other hand, it was known that µ ( f ) ≤
22. Our new method gives µ ( f ) ≤ Definitions and basic properties
All Lie algebras are assumed to be complex and finite-dimensional, if not statedotherwise. Denote by c the nilpotency class of a nilpotent Lie algebra. Definition 2.1.
Let g be a Lie algebra. We denote by µ ( g ) the minimal dimensionof a faithful g -module, and by e µ ( g ) the minimal dimension of a faithful nilpotent g -module.Note that e µ ( g ) is only well-defined, if g is nilpotent. On the other hand, everynilpotent Lie algebra admits a faithful nilpotent g -module of finite dimension [1].Recall the following lemma from [5]. Lemma 2.2.
Let h be a subalgebra of g . Then µ ( h ) ≤ µ ( g ) . Furthermore, if a and b are two Lie algebras, then µ ( a ⊕ b ) ≤ µ ( a ) + µ ( b ) . Definition 2.3.
Denote by b m the subalgebra of gl m ( C ) consisiting of all upper-triangular matrices, by n m = [ b m , b m ] the subalgebra of all strictly upper-triangularmatrices, and by t m the subalgebra of diagonal matrices.The following result is in principle well known. However, it appears in differentformulations, e.g., compare with Theorem 2 . Proposition 2.4.
Let n be a nilpotent Lie algebra and ρ : n → gl ( V ) be a linearrepresentation of n of degree m . Then there exists a basis of V such that ρ can bewritten as the sum of representations ρ = δ + ν , such that (1) δ ( n ) ⊆ t m and ν ( n ) ⊆ n m . (2) δ ([ n , n ]) = 0 , and δ and ν commute. AITHFUL MODULES 3 (3) [ ρ ( x ) , ρ ( y )] = [ ν ( x ) , ν ( y )] for all x, y ∈ n .Proof. By the weight space decomposition for modules of nilpotent Lie algebras wecan write V = s M i =1 V λ i ( n ) , where λ ∈ Hom( n , C ) are the different weights of ρ , and V λ i ( n ) are the weightspaces. In an appropriate basis of V the operators ρ ( x ) are given by block matriceswith blocks λ i ( x ) ∗ . . .0 λ i ( x ) . Then let δ ( x ) the diagonal part given by ⊕ i λ i ( x ) id | V λi , and put ν = ρ − δ . Nowit is easy to see that δ and ν are representations. In fact, the λ i are characters, sothat δ ([ n , n ]) = 0. Also, δ commutes with ν , since it is a multiple of the identity oneach block. This shows (1) and (2), which in turn imply (3). (cid:3) The next proposition gives an lower bound on µ ( n ) in terms of the nilpotencyclass of n . As a special case we recover the well known estimate n ≤ µ ( f ) for afiliform Lie algebra f of dimension n . Proposition 2.5.
Let n be a nilpotent Lie algebra of class c and dimension n ≥ .Then we have c + 1 ≤ µ ( n ) .Proof. If n is abelian, then µ ( n ) ≥ ⌈ √ n − ⌉ ≥ c + 1 by proposition 2 . n is not abelian. Consider a faithful representation ρ : n ֒ → gl ( V )of degree m . Let ρ = δ + ν be a decomposition according to proposition 2.4. Then[ ρ ( x ) , ρ ( y )] = [ ν ( x ) , ν ( y )] for all x, y ∈ n . Hence the non-trivial nilpotent Lie algebras ρ ( n ) and ν ( n ) have the same nilpotency class c . Since the nilpotency class of n m is m −
1, and ν ( n ) ⊆ b , it follows c ≤ m −
1. If we take ρ to be of minimal degree, weobtain c + 1 ≤ µ ( n ). (cid:3) Corollary 2.6.
Let f be a filiform nilpotent Lie algebra of dimension n . Then n ≤ µ ( f ) . There has been some interest lately in determining e µ ( n ) for nilpotent Lie algebras n . We find that e µ ( n ) coincides with µ ( n ) for a broad class of nilpotent Lie algebras. Lemma 2.7.
Let n be a nilpotent Lie algebra satisfying Z ( n ) ⊆ [ n , n ] . Consider alinear representation ρ of n with above decomposition ρ = δ + ν . Then ρ is faithfulif and only if ν is.Proof. A representation of a nilpotent Lie algebra n is faithful if and only if thecenter Z ( n ) acts faithfully. Since ρ ( x ) = ν ( x ) for all x, y ∈ [ n , n ], and Z ( n ) ⊆ [ n , n ], ρ and ν coincide on Z ( n ). Hence the center acts faithfully by ρ if and only if it actsfaithfully by ν . (cid:3) D. BURDE AND W. MOENS
Corollary 2.8.
Let n be a nilpotent Lie algebra satisfying Z ( n ) ⊆ [ n , n ] . Then µ ( n ) = e µ ( n ) .Remark . The condition Z ( n ) ⊆ [ n , n ] on nilpotent Lie algebras n is not toorestrictive. In fact, n always splits as C ℓ ⊕ m with Z ( m ) ⊆ [ m , m ]. In particular,if the center is 1-dimensional, or if n is indecomposable, the condition is satisfied.This includes n being filiform nilpotent.We are also interested in estimating µ ( g ) in terms of dim( g ). We present resultswhich depend on the structure of the solvable radical of g . A first result is thefollowing. Lemma 2.10.
For any Lie algebra g we have p dim( g ) ≤ µ ( g ) .Proof. Suppose that g can be embedded into some gl m ( C ), thendim( g ) ≤ dim( gl m ( C )) = m . In particular this holds for m = µ ( g ). (cid:3) Lemma 2.11.
Let g be represented as b ⋉ δ a for a Lie algebra b and an abelian Liealgebra a , such that the homomorphism δ : b → gl ( a ) is faithful. Then we have µ ( g ) ≤ dim( a ) + 1 . Proof.
Let dim( a ) = r and aff ( a ) = gl r ( C ) ⋉ id C r ⊆ gl r +1 ( C ) be the Lie algebra ofaffine transformations of a = C r . Define ϕ : b ⋉ δ a → aff ( a ) , ( b, a ) ( δ ( b ) , a ) . Then it is obvious that ϕ is faithful if and only if δ is faithful. Moreover the degreeof the representation is r + 1. (cid:3) Denote by rad( g ) the solvable radical of g . Proposition 2.12.
Let g be a Lie algebra such that rad( g ) is abelian. Then we have µ ( g ) ≤ dim( g ) , and the only Lie algebras which satisfy equality are the abelian Lie algebras of di-mension n ≤ and the Lie algebras e ⊕ · · · ⊕ e .Proof. The claim is clear for simple and abelian Lie algebras, see [5]. Since the µ -invariant is subadditive, it also follows for reductive Lie algebras. Now suppose that g is not reductive. Then we can even show that µ ( g ) ≤ dim( g ) −
2. Let a = rad( g ),and s ⋉ δ a be a Levi decomposition, where the homomorphism δ : s → gl ( a ) is givenby δ ( x ) = ad( x ) | a . Since s is semisimple we can choose an ideal s ′ in s such that s = ker( δ ) ⊕ s ′ and g = ker( δ ) ⊕ ( s ′ ⋉ δ ′ a ), where δ ′ = δ | s ′ . Note that δ ′ : s ′ → gl ( a )is faithful. Now s ′ is non-trivial, since otherwise g = ker( δ ) ⊕ a would be reductive. AITHFUL MODULES 5
This implies dim( s ′ ) ≥ δ )) = dim( s ) − dim( s ′ ) ≤ dim( s ) −
3. Sinceker( δ ) is semisimple, and by lemma 2.11 we obtain µ ( g ) ≤ µ (ker( δ )) + µ ( s ′ ⋉ δ ′ a ) ≤ dim(ker( δ )) + dim( a ) + 1 ≤ dim( s ) − a ) + 1= dim( g ) − . Finally we assume that µ ( g ) = dim( g ). By the above inequality, g needs to bereductive. If g is simple, then only g = e satisfies the condition, see [5]. For asemisimple Lie algebra s = s ⊕· · ·⊕ s ℓ we have µ ( s ) = P i µ ( s i ) and µ ( s i ) ≤ dim( s i ).This implies that the only semisimple Lie algebras s satisfying µ ( s ) = dim( s ) aredirect sums of e . Also, the only abelian Lie algebras satisfying the condition are theones of dimension n ≤
4. On the other hand, any reductive Lie algebra g satisfying µ ( g ) = dim( g ) must be either semisimple or abelian: if g = s ⊕ C ℓ +1 with ℓ ≥ s , then µ ( s ⊕ C ) = µ ( s ), see [5], and µ ( g ) ≤ µ ( s ⊕ C ) + µ ( C ℓ ) ≤ µ ( s ) + ℓ ≤ dim( s ) + ℓ ≤ dim( g ) − . This is a contradiction, and we are done. (cid:3)
Our next result is that µ ( g ) ≤ dim( g ) + 1 for any Lie algebra with rad( g ) abelianor 2-step nilpotent. We need the following two lemmas. Lemma 2.13.
Let g be a nilpotent Lie algebra and D a derivation of g that inducesan isomorphism on the center. Then µ ( g ) ≤ dim( g ) + 1 .Proof. The center Z ( g ) is a nonzero characteristic ideal of g , such that D ( Z ( g )) ⊆ Z ( g ). Denote by d the 1-dimensional Lie algebra generated by D , and form the splitextension d ⋉ g . By assumption this is a Lie algebra of dimension dim( g ) + 1 withtrivial center. Hence its adjoint representation ad : d ⋉ g → gl ( d ⋉ g ) is faithful.Together with the embedding g ֒ → d ⋉ g we obtain a faithful representation of g ofdegree dim( g ) + 1. (cid:3) Lemma 2.14.
Let g be a Lie algebra with Levi-decomposition g = s ⋉ r , such that s ≤ Der( r ) . Suppose D is a derivation of the radical r . Then the map π : s ⋉ r → s ⋉ r given by ( X, t ) (0 , D ( t )) is a derivation of g if and only if [ D, s ] = 0 .Proof. Consider any pair a = ( X, t ) and b = ( Y, s ) of elements in g . We need toshow that π ([ a, b ]) = [ π ( a ) , b ] + [ a, π ( b )]. The commutator of a and b is given by[( X, t ) , ( Y, s )] = ([
X, Y ] , X ( s ) − Y ( t ) + [ t, s ]) so that π ([( X, t ) , ( Y, s )]) = (0 , D ([ t, s ]) + ( D ◦ X )( s ) − ( D ◦ Y )( t ))= (0 , [ D ( t ) , s ] + [ s, D ( t )] + ( D ◦ X )( s ) − ( D ◦ Y )( t )) . D. BURDE AND W. MOENS
We have π (( X, t )) = (0 , D ( t )) and π (( Y, s )) = (0 , D ( s )), hence[ π (( X, t )) , ( Y, s )] + [(
X, t ) , π (( Y, s ))] = [(0 , D ( t )) , ( Y, s )] + [(
X, t ) , D ( s )]= (0 , [ D ( t ) , s ] + [ t, D ( s )] + ( X ◦ D )( s ) − ( Y ◦ D )( t )) . We see that π is a derivation of g if and only if these two expressions coincide forall X, Y ∈ s and all s, t ∈ n . This is the case iff [ D, X ]( s ) = 0 for all X ∈ s and all s ∈ n . This finishes the proof. (cid:3) Proposition 2.15.
Let g be a Lie algebra such that rad( g ) is nilpotent of class atmost two. Then we have µ ( g ) ≤ dim( g ) + 1 .Proof. Let s ⋉ n be a Levi-decomposition for g . If rad( g ) is abelian, the claimfollows from proposition 2.12. Now assume that n is nilpotent of class two. Asin the proof of proposition 2.12 we may assume that s acts faithfully on n andthat s ⊆ Der( n ). Now n = [ n , n ] is an s -submodule of n , since s acts on n byderivations, and n is invariant under these derivations, becuase it is a characteristicideal. Since s is semisimple, there exists an s -invariant complement n to [ n , n ].The s -module decomposition n + n of n defines a linear transformation D of n asfollows: D | n = id n and D | n = 2 id n . This is in fact a derivation of n . Note that D commutes with s in Der( n ). The derivation D then extends to a derivation π of g = s ⋉ n by lemma 2.14. Since D is an isomorphism, π | Z ( g ) is also an isomorphism.By lemma 2.13, we may then conclude that µ ( g ) ≤ dim( g ) + 1. (cid:3) Quotients of the universal enveloping algebra
Order and length functions.
Let n be a nilpotent Lie algebra of dimension n and class c . Consider a strictly descending filtration of n of the following form n = n [1] ⊃ n [2] ⊃ · · · ⊃ n [ C +1] = 0 , where the n [ i ] are subalgebras satisfying [ n [ i ] , n [ j ] ] ⊆ n [ i + j ] for all 1 ≤ i, j ≤ C + 1.We say that the filtration is of length C , and we call it an adapted filtration . Forexample, such a filtration is given by the descending central series n i for n of length c . To any such filtration associate a order function o : n → N ∪ {∞} , x max t ∈ N { x ∈ n [ t ] } . If we let n [ t ] = 0 for all t ≥ C + 1, then it makes sense to define o (0) = ∞ . It is easyto see that the order function o satisfies the following two properties o ( x + y ) ≥ min { o ( x ) , o ( y ) } ,o ([ x, y ]) ≥ o ( x ) + o ( y )for all x, y ∈ n .For a given subalgebra m of n satisfying m ⊃ n [2] we obtain an induced filtration m ⊃ n [2] ⊃ · · · ⊃ n [ C +1] = 0 , AITHFUL MODULES 7 and an associated order function. We extend the order function to the universalenveloping algebra U ( n ) of n as follows. Choose a basis x , . . . , x n of n such thatthe first n elements span a complement of n [2] in n , the next n elements spana complement of n [3] in n [2] , and so on. We identify the basis elements x i of n with the images X i in U ( n ) by the natural embedding. The Poincar´e-Birkhoff-Witttheorem states that the monomials X α = X α · · · X α n n form a basis for U ( n ). Nowwe set o ( X α ) = P nj =1 α j o ( X j ). For a linear combination W = P α c α X α we define o ( W ) = min α { o ( X α ) | c α = 0 } .Furthermore we define a length function λ : U ( n ) → N ∪ {∞} by λ (0) = ∞ , λ (1) = 0 and λ ( X α ) = λ ( X α · · · X α n n ) = P ni =1 α i . Here 1 denotesthe unit element of U ( n ). For a linear combination W = P α c α X α we set λ ( W ) =min α { λ ( X α ) | c α = 0 } .The following result is well known for functions o and λ with respect to the standardfiltration of n . It easily generalizes to all adapted filtrations we have defined. Lemma 3.1.
For all
X, Y ∈ U ( n ) we have the following inequalities: (1) o ( X + Y ) ≥ min { o ( X ) , o ( Y ) } . (2) o ( XY ) ≥ o ( X ) + o ( Y ) . (3) λ ( X + Y ) ≥ min { λ ( X ) , λ ( Y ) } . (4) λ ( X ) ≤ o ( X ) . Note that the elements of length 1 are just the nonzero elements of n . Let V t = { X ∈ U ( n ) | o ( X ) ≥ t } . This is a n -submodule of U ( n ), where the action is given by left-multiplication.Furthermore we have n ∩ V t = { } for all t ≥ C + 1.3.2. Actions on U ( n ) . The Lie algebra n acts naturally on U ( n ) by left multipli-cations. We denote this action by xY , for x ∈ n and Y ∈ U ( n ). We will show thatsemidirect products d ⋉ n for subalgebras d ≤ Der( n ) also act naturally on U ( n ).First of all, d acts on n by derivations. Thus we already have an action of d on theelements of length one in U ( n ). For D ∈ Der( n ) let D (1) = 0 and define recursively D ( XY ) = D ( X ) Y + XD ( Y ) for all X, Y ∈ U ( n ). Then the action of d ⋉ n on U ( n )is given by ( D, x ) .Y = D ( Y ) + xY for all ( D, x ) ∈ d ⋉ n , and all Y ∈ U ( n ). This is well-defined, and we have thefollowing useful lemma concerning faithful quotients. Lemma 3.2.
Suppose that W is a d ⋉ n -submodule of U ( n ) such that W ∩ n = 0 .Then the quotient module U ( n ) /W is faithful. D. BURDE AND W. MOENS
Consider a nilpotent Lie algebra n together with the standard filtration given bythe lower central series. We have the following result. Proposition 3.3.
Let n be a nilpotent Lie algebra of dimension n and nilpotencyclass c . Let d be a subalgebra of Der( n ) acting completely reducibly on n . Then V c +1 is a d ⋉ n -submodule of U ( n ) such that the quotient module U ( n ) /V c +1 is faithful ofdimension at most √ n n .Proof. Choose a basis for n associated to the standard filtration of n as in section 3 . C [ i ] to n [ i ] is also invariantunder the action of d , i.e., D ( C [ i ] ) ⊆ C [ i ] for all D ∈ d . This is possible since the n [ i ] are characteristic ideals, hence invariant under d , so that they are submodules,which have a complementary submodule by the complete reducibility. Associate aPBW-basis for U ( n ) as before. Consider a basis element x j ∈ C [ i ] . Then o ( x j ) = i and o ( D ( x j )) ≥ i , since D ( x j ) is again in C [ i ] , so has order i or ∞ . Hence it followsthat o ( D ( W )) ≥ o ( W ) for all W ∈ U ( n ). This means that V c +1 is a d -submodule of U ( n ). Since we already know that V c +1 is a n -submodule, it is a d ⋉ n -submoduleof U ( n ). The quotient is faithful by lemma 3.2. Its dimension is bounded by √ n n ,which was shown in [3], where it was considered just as an n -module. (cid:3) The construction of faithful quotients.
Let n be a nilpotent Lie algebra,together with some adapted filtration n [ t ] of length C , and a subalgebra d ≤ Der( n ). Definition 3.4.
An ideal J of n is called compatible , with respect to n [ t ] and d , if itsatisfies(1) D ( J ) ⊆ J for all D ∈ d ,(2) J is abelian.(3) n [ t ] ⊆ J ⊆ n [ t +1] for some t ≥ hh J ii the linear subspace of U ( n ) generated by all Xy for X ∈ U ( n )and y ∈ J . By assumption J satisfies n = n [1] ⊃ · · · ⊃ n [ t +1] ⊇ J ⊇ n [ t ] ⊃ · · · ⊃ n [ C +1] = 0 . For the rest of this section choose a basis x , . . . , x n of n such that the first n elements span a complement of n [2] in n , the next n elements span a complementof n [3] in n [2] , and so on, including a basis of a complement of J in n [ t +1] , and acomplement of n [ t ] in J . A basis for J is then of the form x m , . . . , x n for some m ≥
1. By the PWB-theorem we obtain standard monomials X α in U ( n ) accordingto this basis. Lemma 3.5.
Let J be a compatible ideal in n . Then hh J ii is the linear span of thestandard monomials X α · · · X α n n with ( α m , . . . , α n ) = (0 , . . . , . For any W ∈ U ( n ) and any y ∈ J we have λ ( W y ) ≥ λ ( W ) + 1 .Proof. First note that the monomials X α · · · X α n n with ( α m , . . . , α n ) = (0 , . . . , hh J ii . They even span hh J ii : assume that T = X i · · · X i ℓ is a standard AITHFUL MODULES 9 monomial of length ℓ , and x k be a basis vector of J , i.e., m ≤ k . If i ℓ ≤ k then T x k isone of our fixed standard monomials of length ℓ +1, and obviously contained in hh J ii .Otherwise there exists a minimal i r such that i r − ≤ k < i r . Then, by definitionof our basis for n , all X i r , · · · , X i ℓ are in J . Since J is abelian, X i r · · · X i ℓ x k = x k X i r · · · X i ℓ . Then we obtain T x k = X i · · · X i r − x k X i r · · · X i ℓ . This is a standardmonomial as above, contained in hh J ii , and of length ℓ + 1. For an arbitrary element W = P c α X α in U ( n ) we have, using (3) of lemma 3.1, λ ( W x k ) = λ (cid:0)X α c α X α x k (cid:1) ≥ min α { λ ( X α x k ) }≥ min α { λ ( X α ) + 1 } = λ ( W ) + 1 . Since the standard monomials T = X α span U ( n ) as a vector space, the claim followsby a similar computation. (cid:3) We define a subset L = h W ∈ U ( n ) | λ ( W ) ≥ i of U ( n ). Note that it is a vector space since a linear combination of elements of itis an element again of length at least two. We have n ∩ L = 0, since the nonzeroelements of n have length 1. Lemma 3.6.
Let J be a compatible ideal in n , and d be a subalgebra of Der( n ) .Then W J = hh J ii ∩ L is a d ⋉ n -submodule of U ( n ) , such that the quotient U ( n ) /W J is faithful.Proof. By the above remark, W J is a vector space. Let x ∈ n , W ∈ U ( n ) and x k ∈ J such that W x k ∈ W J . We want to show that x ( W x k ) = ( xW ) x k againis in W J . By definition it is in hh J ii . For the length we obtain, using lemma 3.5, λ (( xW ) x k ) ≥ λ ( xW ) ≥
2. Hence W J is invariant under the action of n . Nowwe will show that W J is invariant under d , so that it is a d ⋉ n -submodule of U ( n ).Let D ∈ d be a derivation. Then D ( W x k ) = D ( W ) x k + W D ( x k ). Both terms onthe RHS are in hh J ii by definition, and since D ( x k ) ∈ J . It remains to show thattheir length is at least 2. Since by assumption W x k ∈ W J , we have λ ( W ) ≥
1. Thisimplies λ ( D ( W )) ≥
1, and λ ( D ( W ) x k ) ≥ λ ( D ( W )) + 1 ≥
2. For the second termwe obtain λ ( W D ( x k )) ≥ λ ( W ) + 1 ≥
2. Since the sum of two elements of length atleast 2 has lenght at least 2, we obtain D ( W x k ) ∈ W J . Finally, we show that thequotient U ( n ) /W J is faithful. By lemma 3.2 is suffices to show that n ∩ W J = 0.This follows from n ∩ W J ⊆ n ∩ L = 0. (cid:3) We remark that the above quotient module will not yet be finite-dimensionalin general. We will achieve this by enlarging the submodule via V C , where again V t = { X ∈ U ( n ) | o ( X ) ≥ t } , and C denotes the length of the filtration attached toa compatible ideal J . Proposition 3.7.
Let J be a compatible ideal in n . Suppose that o ( D ( x )) ≥ o ( x ) + 1 for all x ∈ n and all D ∈ d .Then Z J = h W J , V C ∩ L i is a d ⋉ n -submodule of U ( n ) , such that the quotient U ( n ) /Z J is faithful and finite-dimensional.Proof. We first show that h V C ∩ L i is a d ⋉ n -submodule of U ( n ). The assumptionalso implies that o ( D ( W )) ≥ o ( W ) + 1 for all W ∈ U ( n ). Then for every ( D, x ) in d ⋉ n and every W ∈ V C ∩ L we have o (( D, x ) .W ) = o ( D ( W ) + xW ) ≥ min { o ( D ( W )) , o ( xW ) }≥ o ( W ) + 1 ≥ C + 1 . Hence h V C ∩ L i is mapped into V C +1 under the action of d ⋉ n . But we have V C +1 ⊆ V C ∩ L , because V C +1 ⊆ V C and V C +1 ⊆ L . For the latter inclusion wenote that all elements of V C +1 must have length at least 2, since all elements oflength at most 1 are contained in n , and n ∩ V C +1 = 0. Hence all ( D, x ) .W arecontained in h V C ∩ L i . This implies that Z J is a d ⋉ n -submodule, using lemma3.6. Since V C +1 ⊆ Z J we have dim( U ( n ) /Z J ) ≤ dim( U ( n ) /V C +1 ). Since the latterdimension is finite, we obtain that U ( n ) /Z J is finite-dimensional. Finally we showthat the quotient module is faithful. Since n ∩ Z J ⊆ n ∩ L = 0 it follows fromlemma 3.2. (cid:3) Algorithmic construction.
We want to apply proposition 3.7 to constructfaithful modules of small dimension for a given nilpotent Lie algebras g . The input is the Lie algebra g with a given basis, together with a decomposition g = d ⋉ n , forsome ideal n , a subalgebra d ⊆ Der( n ), and choices of an admissible filtration n [ t ] ,a compatible ideal J , and so on, such that the assumptions of the proposition aresatisfied. The output will be a faithful g -module of finite dimension. How small thisdimension is, will depend on clever choices of n , d , J , g [ t ] , and so on. The algorithmicconstruction can be derived from the proof of proposition 3.7. Let us illustrate thisexplicitly for the standard filiform Lie algebra g of dimension 4, with two differentchoices. We choose a basis x , . . . x of g such that [ x , x i ] = x i +1 for i = 2 , Example 3.8.
Write g = d ⋉ n with n = h x , x , x i and d = h ad( x ) | n i . Choosethe filtration n = n [1] ⊃ n [2] ⊃ n [3] ⊃ n [4] = 0 of length C = 3 by n [2] = h x , x i and n [3] = h x i . Choose J = n [2] as the compatible ideal. Then all conditions of theproposition are satisfied, and we obtain a faithful g -module of dimension . AITHFUL MODULES 11
First note that we really have a filtration, J is indeed a compatibe ideal, and theassumption for the derivations in d is satisfied. Now the basis elements of order atmost 3 in U ( n ) are given as follows: 1 has order 0; X has order 1; X , X haveorder 2, and X , X X , X have order 3. Also, 1 has length 0, and X , X , X havelength 1. Then we obtain U ( n ) = h , X , X , X , X , X X , X i + V , hh J ii = h X , X X , X i + V ′ ,W J = h X X i + V ′′ Z J = h X X , X i + V . where V ′ , V ′′ are subspaces of V . Hence we obtain that U ( n ) /Z J = h , X , X , X , X i where the bar denotes the cosets. This is a faithful g -module of dimension 5. Wecan compute it explicitly, giving the action of the generators x , x of g . x · X , x · X = X , x · X = 0 , x · X = 0 , x · X = 0 ,x · , x · X = [ X , X ] = − X , x · X = 0 , x · X = X , x · X = 0 . Here we have x · X = [ X , X ]= [ X , X ] X + X [ X , X ]= − X X − X X = − [ X , X ] − X X = X − X X , so that x · X = X . Note that this g -module has a submodule, generated by X with a faithful quotient of dimension 4. Since µ ( g ) = 4, the result is optimal.In the second example we will directly obtain a faithful 4-dimensional g -module. Itwill not be isomorphic to the above quotient module. Example 3.9.
Write g = d ⋉ n with n = h x , x , x i and d = h ad( x ) | n i . Choosethe filtration n = n [1] ⊃ n [2] ⊃ n [3] ⊃ n [4] = 0 of length C = 3 by n [2] = h x , x i and n [3] = h x i . Choose J = n [1] as the compatible ideal. Then all conditions of theproposition are satisfied, and we obtain a faithful g -module of dimension . Note that J is an abelian ideal of codimension 1 in g . With D = ad( x ) | n we have D ( x ) = x and D ( x ) = x . The elements of order at most 3 in U ( n ) are given asfollows: 1 has order 0; X has order 1; X , X have order 2, and X , X X , X have order 3. Then we obtain U ( n ) = h , X , X , X , X , X X , X i + V , hh J ii = h X , X , X , X , X X , X i + V ′ ,W J = h X , X , X X i + V ′′ Z J = h X , X , X X i + V . where V ′ , V ′′ are subspaces of V . Hence we obtain that U ( n ) /Z J = h , X , X , X i . This is a faithful g -module of dimension 4. It is given by x · , x · X = X , x · X = X , x · X = 0 ,x · X , x · X = 0 , x · X = 0 , x · X = 0 . Applications
A general bound.
It is interesting to ask for good estimates on µ ( g ) forarbitrary Lie algebras. So far, general bounds have only been given for nilpotentLie algebras. For example, if g is nilpotent of dimension r and of class c , then µ ( g ) ≤ (cid:0) r + cc (cid:1) , see [8]. Independently of c we have µ ( g ) ≤ √ r r , see [3]. Therehave been some attempts to find similar estimates for solvable Lie algebras. We willpresent here such a bound for arbitrary Lie algebras g . Denote by n the nilradicalof g , and by r its solvable radical. We may assume that r is non-trivial, becauseotherwise the adjoint representation is faithful. Hence let dim( r ) = r ≥
1. We willshow that µ ( g ) ≤ µ ( g / n ) + √ r · r .We start with the following result of Neretin [10], which we have slightly reformulatedfor our purposes. Proposition 4.1.
Let g be a complex Lie algebra with solvable radical r and Levidecomposition g = s ⋉ r . Let p be a reductive subalgebra of g and m a nilpotent idealsatisfying the following properties: (a) p ∩ m = 0 , (b) [ g , r ] ⊆ m and s ⊆ p , (c) p acts completely reducibly on m .Then there exists a nilpotent Lie algebra h of dimension dim( g ) − dim( p ) such that g embedds into a Lie algebra ( p ⊕ C ℓ ) ⋉ h , with ℓ = dim( g / ( p ⋉ m )) , and the actionof p ⊕ C ℓ on h is completely reducible. We note the following corollary.
Corollary 4.2.
Let g be a complex Lie algebra with solvable radical r and nilradical n . Then there exists a nilpotent Lie algebra h of dimension dim( r ) such that g AITHFUL MODULES 13 embedds into a Lie algebra ( g / n ) ⋉ h , and the action of g / n on h is completelyreducible.Proof. In the notation of the above proposition write g = s ⋉ r and choose p = s ,and m = n . Then the conditions ( a ) − ( c ) are satisfied. Indeed, s ∩ n ⊆ s ∩ r =0. Furthermore [ g , r ] is a nilpotent ideal, hence is contained in n . Finally, s actscompletely reducibly on n , because s is semisimple. The result follows. (cid:3) We obtain the following bound on µ ( g ): Proposition 4.3.
Let g be a complex Lie algebra with nilradical n and solvableradical r . Assume that dim( r ) = r ≥ . Then we have µ ( g ) ≤ µ ( g / n ) + 3 √ r · r Proof.
We can embedd g into a Lie algebra ( g / n ) ⋉ h as in the corollary, where h isa nilpotent Lie algebra of dimension dim( r ), and q = g / n is reductive. This means g ⊆ q ⋉ h , and hence µ ( g ) ≤ µ ( q ⋉ h ) by lemma 2.2. Now we want to apply proposition3.3 to q ⋉ h . For that we need that q is a subalgebra of Der( h ), or equivalently, that q acts faithfully on h . However, we may always decompose the reductive Lie algebra q as q = q ⊕ q , where q commutes with h , and q acts faithfully and completelyreducibly on h . Again by lemma 2.2, we obtain µ ( q ⋉ h ) ≤ µ ( q )+ µ ( q ⋉ h ). We have µ ( q ) ≤ µ ( q ) because of q ⊆ q . Furthermore we have µ ( q ) ≤ dim( q ) by proposition2.12. Now proposition 3.3 can be applied to q ⋉ h , and we obtain µ ( g ) ≤ µ ( q ⋉ h ) ≤ µ ( q ) + µ ( q ⋉ h ) ≤ dim( q ) + 3 √ r · r (cid:3) Two-step nilpotent Lie algebras.
It is well known that we have µ ( g ) ≤ dim( g ) + 1 for all two-step nilpotent Lie algebras g , see [3]. It is not so easy toimprove this bound in general. Of course, for certain classes of two-step nilpotentLie algebras better bounds can be produced. We show the following result. Proposition 4.4.
It holds µ ( g ) ≤ dim( g ) for all two-step nilpotent Lie algebras g .Proof. We can write g = g ⊕ g with Z ( g ) ⊆ [ g , g ] and g abelian. Assumethat we already know that µ ( g ) ≤ dim( g ). Then, by lemma 2.2, it follows µ ( g ) ≤ µ ( g ) + µ ( g ) ≤ dim( g ) − dim( g ) + µ ( g ) ≤ dim( g ). Hence we may assume that g satisfies Z ( g ) ⊆ [ g , g ]. Let dim( g ) = n and choose an ideal n ⊆ g of codimension 1containing the commutator of g . Let x , . . . , x n be a basis of g , such that x , . . . , x n span n . Then g = h x i ⊕ n as a vector space. Let d = h ad( x ) | n i , and we maywrite g = d ⋉ n . Let n [1] ⊃ n [2] ⊃ C = 2 given by n [1] = n and n [2] = Z ( g ) = [ g , g ]. Recall here that n ⊃ [ g , g ]. Choose J = Z ( g ) as a compatible ideal. It satifies the conditions of definition 3.4, since it is invariantunder all derivations of d , and it is abelian. Note that we have D ( n [1] ) ⊆ [ g , g ] = n [2] for all D ∈ d , so that o ( D ( x )) ≥ o ( x )+1 for all x ∈ n . Now we can apply proposition3.7 with these choices. We obtain a faithful module U ( n ) /Z J = U ( n ) / L , which hasdimension n , since it is spanned by the classes of 1 , x , . . . , x n . (cid:3) Filiform nilpotent Lie algebras.
We wish to apply proposition 3.7 to filiformnilpotent Lie algebras f of dimension n in order to improve the known upper boundsfor µ ( f ). Let f = f and f i = [ f , f i − ]. Let β ( f ) be the maximal dimension of anabelian ideal of f . It is well known that n/ ≤ β ( f ) ≤ n −
1. Denote by p k ( j ) thenumber of partitions of j in which each term does not exceed k . Let p k (0) = 1 forall k ≥ p ( j ) = 0 for all j ≥ Proposition 4.5.
Let f be a filiform nilpotent Lie algebra of dimension n having anabelian ideal J of dimension ≤ β ≤ n − . Then we have µ ( f ) ≤ f ( n, β ) , where f ( n, β ) = β + n − X j =0 p n − − β ( j ) . Proof.
Let x , . . . , x n be an adapted basis of f in the sense of [11]. Then choose n = h x , . . . , x n i and d = h ad( x ) | n i , so that f = d ⋉ n . Define a filtration n [1] ⊃ n [2] · · · ⊃ n [ C ] ⊃ C = n − n [1] = n and n [ i ] = f i for i ≥
2. Wemay write J = h x m , . . . , x n i with m ≥ n − m + 1 = β . It is easy to seethat J is a compatible ideal in the sense of definition 3.4. Furthermore we have o ( D ( x )) ≥ o ( x ) + 1 for all x ∈ n and all D ∈ d . Now we can apply proposition3.7. We obtain a faithful module U ( n ) /Z J . We will show that its dimension is β + P n − j =0 p n − − β ( j ). It is generated by the classes { X m , . . . , X n } ∪ { X α = X α · · · X α m − m − | o ( X α ) ≤ n − } . There are β monomials in the first set. The cardinality of the second set is given by { ( α , . . . , α m − ) ∈ Z m − ≥ | · α + 2 · α + · · · + ( m − · α m − ≤ n − } = n − X j =0 { ( α , . . . , α m − ) | · α + 2 · α + · · · + ( m − · α m − = j } = n − X j =0 p m − ( j ) . Since m − n − − β we obtain the required dimension. (cid:3) Note that for β = 1 we obtain the bound from [3]: µ ( f ) ≤ f ( n,
1) = 1 + n − X j =0 p ( j ) < e √ π ( n − / . AITHFUL MODULES 15
Here p ( j ) denotes the unrestricted partition function, and p (0) = 1. The followingresult shows that our bound from the above proposition yields an improvement. Proposition 4.6.
Let n ≥ . Then f ( n, β ) is monotonic in β , i.e., it holds f ( n, n − ≤ f ( n, n − ≤ · · · ≤ f ( n,
2) = f ( n, , with equality for β = 1 and β = 2 . The proof is easy, and we leave it to the reader. We can also determine f ( n, β )explicitly for large β : Proposition 4.7.
Let n ≥ . Then it holds f ( n, n −
1) = n,f ( n, n −
2) = 2 n − ,f ( n, n −
3) = n + 3 n −
12 + 2 ⌊ n/ ⌋ . If β = n −
1, then β = β ( f ), and f is the standard graded filiform Lie algebra.Then the bound µ ( f ) ≤ f ( n, n −
1) = n is optimal, since we already know that µ ( f ) = n in this case. See also example 3.9 for the case n = 4. Remark . It is also easy to show that f ( n, β ) ≤ β + (2 n − β − n − β − ( n − β − n ≥ ≤ β ≤ n − µ ( f ) which only depends on n . For this we takethe smallest possible β = β ( f ) in terms of n , which is given by β = ⌈ n/ ⌉ . Then n − − β = ⌊ n/ ⌋ −
1, and we obtain the following result:
Corollary 4.9.
Let f be a filiform nilpotent Lie algebra of dimension n ≥ . Then µ ( f ) ≤ n − n − X j =0 p ⌊ n ⌋ − ( j ) . Filiform Lie algebras of dimension 10.
We may represent all complexfiliform Lie algebras of dimension 10 with respect to an adapted basis ( x , . . . , x )as a family of Lie algebras f = f ( α , . . . , α ), with 13 parameters satisfying thefollowing polynomial equations: α (2 α + α ) − α = 0 ,α (2 α − α − α ) = 0 ,α (2 α + α ) − α (2 α + α ) = 3 α ( α + α ) − α α . We call the parameters admissible , if they define a Lie algebra, i.e., if they satisfythese equations. Note that we obtain other equations as consequences, such as α ( α − α ) = 0 . The explicit Lie brackets are given as follows:[ x , x i ] = x i +1 , ≤ i ≤ x , x ] = α x + α x + α x + α x + α x + α x [ x , x ] = α x + α x + α x + α x + α x [ x , x ] = ( α − α ) x + ( α − α ) x + ( α − α ) x + ( α − α ) x [ x , x ] = ( α − α ) x + ( α − α ) x + ( α − α ) x [ x , x ] = ( α − α + α ) x + ( α − α + α ) x [ x , x ] = ( α − α + 3 α ) x [ x , x ] = − α x [ x , x ] = α x + α x + α x + α x [ x , x ] = α x + α x + α x [ x , x ] = ( α − α ) x + ( α − α ) x [ x , x ] = ( α − α ) x [ x , x ] = α x [ x , x ] = α x + α x [ x , x ] = α x [ x , x ] = − α x [ x , x ] = α x We want to determine as good as possible upper bounds on µ ( f ), for all Liealgebras f = f ( α , . . . , α ). The results will depend on the parameters, and we haveto introduce a case distinction. For each case we choose a particular constructionwhich yields a faithful f -module V of some dimension 10 ≤ dim( V ) ≤
18. Thisimproves the known bound 10 ≤ µ ( f ) ≤
22 from [2] for such Lie algebras. We canalso construct a faithful f -module V = V ( α , . . . , α ), which does not depend on acase distinction for the parameters. In other words, such a module gives an upper AITHFUL MODULES 17 bound on µ ( f ) for all admissible parameters at the same time. We call such a modulea general f -module. We will give such a module explicitly. Proposition 4.10.
There is a general faithful f -module V = V ( α , . . . , α ) ofdimension .Proof. The faithful f -module V is obtained by proposition 4.5 as follows. Take J = h x , . . . , x i as compatible ideal. This means β = 5 and the construction yieldsa module with a basis consisting of f (10 ,
5) = 58 monomials. The computation of f (10 ,
5) uses ( p (0) , · · · , p (8)) = (1 , , , , , , , , x i for X i .order monomials0 1 , x , x , x , x , x x , x , x , x x , x , x x , x , x , x x , x x , x x , x x x , x x , x , x , x , x x , x x x , x , x x , x x , x x , x x , x , x , x x , x x , x x , x x x , x x x , x x , x x , x x , x x , x x , x , x , x x , x , x x x , x x , x x , x x x , x , x x x , x x x ,x x , x x , x x , x x , x x , x , x . Denote this basis by v , . . . , v , ordered lexicographically. Note that v = x generates the center of f . The module is determined by the action of the generators x and x of the Lie algebra f = f ( α , . . . , α ). It is given by x .v = 0 ,x .v = v ,x .v = v ,x .v = 2 v − α v − α v − α v − α v − α v − α v ,x .v = v ,x .v = v + v ,x .v = 3 v − α v + α ( α − α ) v + (2 α α − α α − α α ) v + (2 α α − α α + α α + α − α α − α α ) v + (2 α α − α α + 3 α α + α α − α α + 2 α α − α α − α α − α α ) v , x .v = v ,x .v = v + v ,x .v = 2 v − α v − α v − α v − α v ,x .v = v + 2 v − α v + α α v + ( α α − α α + α α ) v + ( α α − α α − α α + α α + α α + α α ) v ,x .v = 4 v − α v + α (4 α α − α − α α ) v + (4 α α − α α − α α − α α α + 3 α α α + α α α + 2 α α α − α α α + 11 α α α − α α α + α α α + 6 α α α + α α − α α α − α α α − α α ) v ,x .v = v ,x .v = v + v ,x .v = v ,x .v = 2 v + v − α v + α α v + ( α α + α α − α α ) v ,x .v = 2 v + v ,x .v = v + 3 v − α v + (2 α α − α α − α α α − α α α + α α α + α α + 2 α α α ) v ,x .v = 5 v − α v + α α (4 α α − α − α α ) v ,x .v = v ,x .v = 2 v − α v − α v ,x .v = v ,x .v = v + v + v ,x .v = 3 v + ( α − α α + α α ) v ,x .v = 2 v − α v + α α v ,x .v = 3 v + v − α v + α α ( α − α ) v ,x .v = 2 v + 2 v − α v − α α α v ,x .v = v + 4 v − α v ,x .v = 6 v − α v ,x .v = v ,x .v = v ,x .v = v + 2 v ,x .v = 2 v + v − α α v ,x .v = v + v ,x .v = v + 2 v + v , AITHFUL MODULES 19 x .v = 3 v + v ,x .v = 3 v ,x .v = 4 v + v ,x .v = 2 v + 3 v ,x .v = v + 5 v ,x .v = 7 v ,x .v = v ,x .v = 0 ,x .v = − α v ,x .v = · · · = x .v = 0 .x .v = v , x .v = v , x .v = v , x .v = v , x .v = v ,x .v = v , x .v = v , x .v = v , x .v = v , x .v = v ,x .v = v , x .v = v , x .v = 0 , x .v = v , x .v = v ,x .v = v , x .v = v , x .v = v , x .v = v , x .v = 0 ,x .v = v , x .v = v , x .v = v , x .v = v , x .v = v ,x .v = v , x .v = v , x .v = v , x .v = v , x .v = 0 ,x .v = v , x .v = v , x .v = v , x .v = v , x .v = v ,x .v = v , x .v = v , x .v = v , x .v = v , x .v = v ,x .v = v , x .v = · · · = x .v = 0 . (cid:3) Corollary 4.11.
There is a general faithful f -module V = V ( α , . . . , α ) of di-mension .Proof. We apply the algorithm
Quotient from [6] to the module V . This works asfollows. The space of invariants is given by V f = h α v + v , v , v , . . . , v i , with dim( V f ) = 16 for all parameters α , . . . , α . We choose a complement U of Z ( f ) = h v i in V f by taking the above basis for V f except for v . Then U is asubmodule such that the quotient V = V /U is a faithful module of dimension 43.For the quotient, we may write the following relations v = 0 ,v = − α v ,v = · · · = v = 0 . In other words, we may view v , . . . , v , v as a basis of V . Now we repeat thisprocedure. We have V f = h α v + v , v , v + α α v , v , . . . , v , v i , with dim( V f ) = 12 for all parameters α , . . . , α . We choose U from V f by omitting v , and obtain a faithful quotient V = V /U of dimension 32. We can take thefollowing quotient relations v = − α v ,v = 0 ,v = − α α v ,v = · · · = v = 0 . In the next step we obtain dim( V f ) = 10 for all parameters α , . . . , α . Choosing acomplement U as above we obtain a faithful module V = V /U of dimension 23,where the relations are given by v = − α v − α v − α v ,v = − α v , ... = ... v = 0 . The dimension of the space of invariants V f however does depend on the parameters.It can be of dimension 5,6 or 7, depending on certain case distinctions. Withoutcase distinction we can still choose some subspace U of invariants not containing v ,which need not be a maximal with this property. This way we arrive at a faithfulquotient V of dimension 20. If we continue with case distinctions we obtain manydifferent faithful quotients V of dimensions 10 ≤ dim( V ) ≤
18. The quotientalgorithm stops if the space of invariants is 1-dimensional, spanned by v . Thenthere is no faithful quotient of lower dimension. (cid:3) Remark . Note that the choice of the complements U in the quotient algorithmis not unique. For our choice we obtained faithful modules of dimensions 58, 43, 32and 23. In general, the dimensions might depend on U . However, taking quotientsby invariants is no restriction. In fact, the following result is easy to show: let n be anilpotent Lie algebra, and V be a nilpotent n -module. Then every faithful quotientof V can be obtained by taking successive quotients by invariants. Example 4.13.
Consider the Lie algebra f = f ( α , . . . , α ) with ( α , . . . , α ) = (1 , , , , , , − , , , , , − , . We have µ ( f ) ≥ , and f admits no affine structure, see [2] . The above algorithmyields a faithful quotient of V of dimension . Hence we have µ ( f ) ≤ , and thisis up to now the best known estimate. AITHFUL MODULES 21
Note that the above Lie algebra has minimal β -invariant, namely β ( f ) = 5. TheBetti numbers are given by ( b , . . . , b ) = (1 , , , , , , , , , , µ ( f ) for all filiform Liealgebras f = f ( α , . . . , α ) of dimension 10. Therefore we need to consider differentchoices of admissible parameters, which give well-defined classes of filiform Lie al-gebras. The cases are as follows: Case : α + α = 0. Case : α + α = 0. Case a : α = 0, α = α = 0. Case a : α = α . Case a : α = − α . Case a a : α + α = 0. Case a b : α + α = 0. Case b : α = 0. Case b : α = α . Case b : α = α . Case b a : α = α . Case b b : α = − α . Case b b : α + α = 0. Case b b : α + α = 0. Lemma 4.14.
All above conditions are isomorphism invariants. In particular, al-gebras of different cases are non-isomorphic.Proof.
Using the β -invariant we have α = 0 ⇔ β ( f / f ) = 4 ,α = 0 ⇔ β ( f / f ) = 5 ,α = 0 ⇔ β ( f / f ) = 6 ,α = 0 ⇔ β ( f / f ) = 6 ,α = α ⇔ β ( f / f ⋉ f / f ) = 4 . The Lie algebras of case 1 satisfy 2 α + α = 0, which is equivalent to α = α = 0.The above table shows that these conditions are isomorphism invariants. Hence theLie algebras of case 1 and case 2 are well-defined. The same applies to case 2 a and case 2 b , because α = 0 and α = 0 are isomorphism invariants. Recall that α = 0 implies α = α . The claim is also clear for the cases 2 a
1, 2 a
2. Note that α = α = 0 is also equivalent to the conditions β ( f / f ) = 5 and [ f , f ] = f . As wewill see in proposition 4.18, the Lie algebras of case 2 b α = − α = 0 the condition 3 α + α = 0 is equivalent to the fact, that the Lie algebra f / f admits an invertible derivation. Hence this conditionis also an isomorphism invariant. (cid:3) For each case we have a result on µ ( f ). Let us start with the first case. Proposition 4.15.
Let f = f ( α , . . . , α ) be a filiform nilpotent Lie algebra ofdimension satisfying α + α = 0 . Then µ ( f ) = 10 .Proof. The parameters are admissible iff α = α = 0 and α ( α + α ) = 0. Toconstruct a module for f we need to find two operators L ( x ) and L ( x ), which define L ( x i ) := [ L ( x ) , L ( x i − )] for i ≥
3, so that the conditions L ([ x i , x j )] = [ L ( x i ) , L ( x j )]are satisfied for all i, j ≥
1. This module is faithful if and only if L ( x ) is nonzero.It is easy to see that we can always find such operators, by taking L ( x ) = ad( x )and L ( x ) some 10 ×
10 lower-triangular matrix. However, the construction dependson different cases, such as α = 0, or α = 0 with α = 0 , α = 0, with α =0 , α = 0, or with α = 0. For more details see [2]. (cid:3) For case 2 a we have the following results, see [2]: Proposition 4.16.
Let f = f ( α , . . . , α ) be a filiform nilpotent Lie algebra ofdimension satisfying α + α = 0 , α = 0 and α = α . Then µ ( f ) ≤ . Proposition 4.17.
Let f = f ( α , . . . , α ) be a filiform nilpotent Lie algebra ofdimension satisfying α + α = 0 , α = 0 and α = − α . Then µ ( f ) ≤ ifand only if α + α = 0 . Otherwise we have µ ( f ) ≤ . In this case the module V from proposition 4.10 always has a faithful quotient ofdimension 18. This can be seen by applying the quotient algorithm as in corollary4.11. For 3 α + α = 0 this is the best bound known so far. The example given in4.13 belongs to this class.For case 2 b we have the following results, see [2] and [4]: Proposition 4.18.
Let f = f ( α , . . . , α ) be a filiform nilpotent Lie algebra ofdimension satisfying α + α = 0 . Then f admits a central extension → Z ( h ) → h → f → by some filiform nilpotent Lie algebra h if and only if α = 0 and α = α , in which case we have µ ( f ) = 10 . Proposition 4.19.
Let f = f ( α , . . . , α ) be a filiform nilpotent Lie algebra ofdimension satisfying α + α = 0 , α = 0 and α = α . Then µ ( f ) ≤ . Proposition 4.20.
Let f = f ( α , . . . , α ) be a filiform nilpotent Lie algebra ofdimension satisfying α + α = 0 , α = 0 and α = − α . Then µ ( f ) ≤ ifand only if α + α = 0 . Otherwise we have µ ( f ) ≤ . Here we use proposition 4.10 for the subcase 3 α + α = 0. Then the module V has a faithful quotient of dimension 14. In fact, for some cases, it even has a faithfulquotient of dimension 12, 13 or 14. AITHFUL MODULES 23
References [1] G. Birkhoff:
Representability of Lie algebras and Lie groups by matrices . Annals of Mathe-matics (1937), 526–532.[2] D. Burde: Affine structures on nilmanifolds . Int. J. of Math. (1996), no. 5, 599–616.[3] D. Burde: A refinement of Ado’s Theorem . Archiv Math. (1998), 118–127.[4] D. Burde: Affine cohomology classes for filiform Lie algebras . Contemporary Mathematics (2000), 159–170.[5] D. Burde, W. Moens:
Minimal faithful representations of reductive Lie algebras . Archiv derMathematik (2007), no. 6, 513–523.[6] D. Burde, B. Eick, W. de Graaf: Computing faithful representations for nilpotent Lie algebras .J. Algebra (2009), no. 3, 602–612.[7] L. Cagliero, N. Rojas: Faithful representations of minimal dimension of current HeisenbergLie algebras . Internat. J. Math. (2009), no. 11, 1347–1362.[8] W. A. de Graaf: Constructing faithful matrix representations of Lie algebras . Proceedingsof the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC’97,(1997), 54–59. ACM Press New York.[9] J. Milnor,
On fundamental groups of complete affinely flat manifolds , Adv. Math. (1977),178-187.[10] Y. Neretin: A construction of finite-dimensional faithful representation of Lie algebra . Rend.Circ. Mat. Palermo (2) Suppl. No. (2003), 159–161.[11] M. Vergne: Cohomologie des alg`ebres de Lie nilpotentes. Application `a l’´etude de la vari´et´edes alg`ebres de Lie nilpotentes . Bull. Soc. Math. France (1970), 81-116. Fakult¨at f¨ur Mathematik, Universit¨at Wien, Nordbergstrasse 15, 1090 Wien,Austria
E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Wien, Nordbergstrasse 15, 1090 Wien,Austria
E-mail address ::