Almost inner derivations of 2-step nilpotent Lie algebras of genus 2
aa r X i v : . [ m a t h . R A ] A p r ALMOST INNER DERIVATIONS OF 2-STEP NILPOTENT LIE ALGEBRASOF GENUS 2
DIETRICH BURDE, KAREL DEKIMPE, AND BERT VERBEKE
Abstract.
We study almost inner derivations of 2-step nilpotent Lie algebras of genus 2, i.e.,having a 2-dimensional commutator ideal, using matrix pencils. In particular we determine allalmost inner derivations of such algebras in terms of minimal indices and elementary divisorsover an arbitrary algebraically closed field of characteristic not 2 and over the real numbers. Introduction
One of the fundamental problems in spectral geometry is, to what extent the eigenvaluesdetermine the geometry of a given manifold. A classical question here, going back to HermannWeyl, asks whether or not isospectral manifolds need to be isometric. A first example witha negative answer was given in 1964 by John Milnor. He constructed a pair of isospectralbut non-isometric flat tori of dimension 16 using arithmetic lattices. Several other exampleswere given later and in 1984 Gordon and Wilson [6] even constructed continuous families ofisospectral non-isometric manifolds. These manifolds were compact Riemannian manifolds ofthe form G/ Γ with a simply connected exponential solvable group G and a discrete cocompactsubgroup Γ. The isospectral non-isometric property was derived from the existence of almostinner but non-inner automorphisms of the Lie group G . Such automorphisms of G can bestudied on the Lie algebra level. They correspond to almost inner but non-inner derivations ofthe Lie algebra g of G .Denote by AID( g ) the subalgebra of the derivation algebra Der( g ) consisting of almost innerderivations, for a given finite-dimensional Lie algebra over a field K . The motivation fromspectral geometry and other applications leads us to study the algebras AID( g ) in general.Several classes of Lie algebras do not possess an almost inner but non-inner derivation. Clearlyany semisimple Lie algebra g over a field of characteristic zero satisfies AID( g ) = Inn( g ). Thesame is true for free-nilpotent Lie algebras or almost abelian Lie algebras, see [2], and for manyother classes of Lie algebras. Also all complex Lie algebras of dimension n ≤ { x , . . . , x } and brackets [ x , x i ] = x i +1 for 2 ≤ i ≤ x , x ] = x . As it turns out,we can construct many almost inner but non-inner derivations for certain nilpotent Lie algebrasand in particular for certain 2 -step nilpotent Lie algebras.In this article we compute AID( g ) for all 2-step nilpotent Lie algebras with a 2-dimensionalcommutator ideal over R and over an algebraically closed field K of characteristic not 2. SuchLie algebras can be described by skew-symmetric matrix pencils and their invariant polynomials Date : April 23, 2020.2000
Mathematics Subject Classification.
Primary 17B30, 17D25.
Key words and phrases.
Almost inner derivation, matrix pencil, nilpotent Lie algebra. and elementary divisors due to Weierstrass and Kronecker. Using these canonical invariantshelps a lot in computing the almost inner derivations of such Lie algebras and we obtain anexplicit formula.In section 3 we review all necessary notions concerning matrix pencils, minimal indices andelementary divisors. We give several examples of 2-step nilpotent Lie algebras of genus 2 withtheir associated invariants. In section 4 we introduce the subalgebra of central derivations C ( g )in Der( g ), which contains AID( g ). We give necessary and sufficient criteria for a derivation D ∈ C ( g ) to be almost inner. Finally, in section 5, we compute AID( g ) separately for each typeof elementary divisor or minimal index and combine this for a general formula for AID( g ) inTheorems 5 .
10 and 5 .
11. 2.
Preliminaries
Let g denote a Lie algebra over an arbitrary field K if not said otherwise. We will alwaysassume that g is finite-dimensional over K . The lower central series of g is given by g = g ⊇ g ⊇ g ⊇ g ⊇ · · · where the ideals g i are recursively defined by g i = [ g , g i − ] for all i ≥
1. ALie algebra g is called c -step nilpotent if g c = 0. The genus of a Lie algebra g is the numberdim( g ) − | S | , where S is a minimal system of generators. If g is nilpotent, then the genus isgiven by dim( g ) − dim ( g / [ g , g ]) = dim([ g , g ]).We recall the definition of an almost inner inner derivation [6, 1]. Definition 2.1.
A derivation D ∈ Der( g ) of a Lie algebra g is said to be almost inner , if D ( x ) ∈ [ g , x ] for all x ∈ g . The space of all almost inner derivations of g is denoted by AID( g ).Let us denote by Inn( g ) the ideal of inner derivations in Der( g ). Certainly every innerderivation of g is almost inner. The converse need not be true. So we obtain a chain ofsubalgebras Inn( g ) ⊂ AID( g ) ⊂ Der( g ).Let g be a 2-step nilpotent Lie algebra. Then g = [ g , [ g , g ]] = 0 so that [ g , g ] ⊆ Z ( g ). If g has genus 1, then g is a central extension of a 2 n + 1-dimensional Heisenberg Lie algebra andwe have AID( g ) = Inn( g ) by Lemma 3 . . Pencils, elementary divisors and minimal indices
Two-step nilpotent Lie algebras have not been classified in general so far. For certain sub-classes however there is a complete description. We are interested in two-step nilpotent Lie alge-bras g of genus 2, i.e., satisfying dim([ g , g ]) = 2. Such algebras can be described in terms of ma-trix pencils. This has been studied for several purposes in the literature, see [5, 9, 4, 11, 7, 3, 8].We mainly follow the notation of [5]. Let K [ λ, µ ] be the polynomial ring in two variables. Definition 3.1.
Let
A, B ∈ M n ( K ). A polynomial matrix µA + λB ∈ M n ( K [ λ, µ ]) is calleda matrix pencil or just a pencil . Two such pencils µA + λB and µC + λD are called strictlyequivalent if there are matrices S, T ∈ GL n ( K ) satisfying S ( µA + λB ) T = µC + λD. The pencil is called skew if both A and B are skew-symmetric. Two skew-symmetric pencils µA + λB and µC + λD are called strictly congruent if there is a matrix S ∈ GL n ( K ) such that LMOST INNER DERIVATIONS 3 S t ( µA + λB ) S = µC + λD . A pencil is called regular or non-singular if its determinant is notthe zero polynomial in K [ λ, µ ].It is known that skew-symmetric pencils over an algebraically closed field of characteristicdifferent from two are strictly equivalent if and only if they are strictly congruent [5]. The sameis true over the field of real numbers [7].Let g be a two-step nilpotent Lie algebra of genus 2 with a basis ( x , . . . , x n , y , y ), where( y , y ) is a basis of [ g , g ]. Denote by A = ( a ij ) and B = ( b ij ) the skew-symmetric matrices ofstructure constants determined by [ x i , x j ] = a ij y + b ij y for all 1 ≤ i, j ≤ n . Definition 3.2.
Let g be a two-step nilpotent Lie algebra of genus 2 over an arbitrary field K .Then µA + λB ∈ M n ( K [ λ, µ ]) is called the pencil associated to g with respect to a given basisas above.The following proposition is a special case of [9, Proposition 4.1] Proposition 3.3.
Let g and h be two-step nilpotent Lie algebras of genus over an arbitraryfield K . If the pencils associated to g and h with respect to some bases of g and h are strictlycongruent, then g and h are isomorphic. From this proposition it follows that it is important for our study to be able to classify skewpencils up to strict equivalence. For regular pencils this was solved by Weierstrass in terms of elementary divisors . For a pencil µA + λB of rank r let G m ( µ, λ ) be the greatest common divisorof all its minor determinants of order m . Then G m ( µ, λ ) | G m +1 ( µ, λ ) for all 1 ≤ m ≤ r − i ( µ, λ ) = G ( µ, λ ) and i m ( µ, λ ) = G m ( µ, λ ) G m − ( µ, λ )for 1 < m ≤ r . Definition 3.4.
The homogeneous polynomials { i m ( µ, λ ) } m are called the invariant factors of the pencil µA + λB . Each polynomial i m ( µ, λ ) can be written as a product of powers ofprime polynomials because K [ λ, µ ] is a unique factorization domain. These prime power factors(which are only determined up to scalar multiple) are called the elementary divisors e a ( µ, λ ),for a = 1 , , . . . , t of the pencil µA + λB . An elementary divisor is said to have multiplicity ν ifit appears exactly ν times in the factorizations of the invariant factors i m ( µ, λ ) for 1 ≤ m ≤ r .Suppose that K is algebraically closed. In this case, the elementary divisors are all linear.Since the elementary divisors are only determined up to scalar multiple, each elementary divisoris either of type ( bµ + λ ) e or of type µ f . The first one is called of finite type . The second oneis called of infinite type , which means that the divisor belongs to K [ µ ]. Elementary divisorsof infinite type exist if and only if det( B ) = 0. The elementary divisors e a ( µ, λ ) of finite typecorrespond to the elementary divisors of the pencil A + λB ∈ K [ λ ] as follows. Setting µ = 1 in e a ( µ, λ ) we clearly obtain the elementary divisors e a ( λ ) of A + λB . These can be computed bythe Smith normal form because K [ λ ] is a PID. The diagonal elements of the Smith normal formare just the invariant polynomials. Conversely, from each elementary divisor e a ( λ ) of A + λB ofdegree e we obtain the corresponding elementary divisor e a ( µ, λ ) by e a ( µ, λ ) = µ e e a ( λµ ). In case D. BURDE, K. DEKIMPE, AND B. VERBEKE K = R , apart from these elementary divisors of degree 1, there are also elementary divisors ofdegree 2 which are of the form ( λ − µ ( a + bi ))( λ − µ ( a − bi )) = λ − aλµ + ( a + b ) µ , with a ∈ R , b ∈ R × . Example 3.5.
Let g be the -dimensional real Lie algebra with basis { x , . . . , x , y , y } andLie brackets defined by [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = − y . Then the associated pencil is given by µA + λB = µ λ − µ λ − λ − µ − λ µ . The pencil is regular because det( µA + λB ) = ( µ + λ ) is not the zero polynomial. The Smithnormal form of A + λB is given by diag(1 , , λ + 1 , λ + 1) . Hence there is one elementarydivisor e ( µ, λ ) = µ (1 + λ µ ) = µ + λ of finite type of multiplicity and no elementary divisorof infinite type.When we consider the complexification g ⊗ C of this Lie algebra, then the corresponding pencilhas two elementary divisors: λ − iµ and λ + iµ both of multiplicity 2. For singular pencils we still need another invariant. Let µA + λB be a singular pencil ofsize n . Then ( A + λB ) x = 0 has a nonzero solution in K [ λ ] n . Let x ( λ ) be such a nonzerosolution of minimal degree ε . Of all solutions which are K [ λ ]-independent of x ( λ ) let us selecta solution x ( λ ) of minimal degree ε . It is obvious that ε ≤ ε . By continuing this process weobtain a set x ( λ ) , . . . , x k ( λ ) of solutions, which is a maximal set of elements in K [ λ ] n satisfying( A + λB ) x i ( λ ) = 0 for i = 1 , . . . , k and being K [ λ ]-independent. We have k ≤ n . Note thatthis set is not uniquely determined, but that different sets have the same minimal degrees ε ≤ ε ≤ . . . ≤ ε k . Hence the following notion is well-defined. Definition 3.6.
Let µA + λB be a singular pencil. The associated numbers ε , . . . , ε k are calledthe minimal indices of the pencil µA + λB . Example 3.7.
Let g be the -dimensional real Lie algebra with basis { x , . . . , x , y , y } andLie brackets defined by [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y . Then the associated pencil is given by µA + λB = µ λ
00 0 0 µ λ − µ − λ − µ − λ . Since det( µA + λB ) = 0 the pencil is singular. The equation ( A + λB ) x = 0 has a non-zerosolution x ( λ ) = (0 , , λ , − λ, t , and the set is maximal. Hence there is one minimal index ε = 2 . The Smith normal form of A + λB is given by diag(1 , , , , . LMOST INNER DERIVATIONS 5
The following well-known result classifies skew pencils up to congruence, see Corollary 6 . . Proposition 3.8.
Let K be an algebraically closed field of characteristic not or the field ofreal numbers. Two skew-symmetric pencils of the same dimension are strictly congruent if andonly if they have the same elementary divisors and the same minimal indices. For a skew pencil µA + λB over an algebraically closed field K the elementary divisors occurin pairs and we can arrange them as µ e , µ e , · · · , µ e s , µ e s , ( λ − µα ) f , ( λ − µα ) f , · · · , ( λ − µα t ) f t , ( λ − µα t ) f t where α , α , . . . , α t ∈ K .When K = R , the elementary divisors still occur in pairs, and apart from the above set ofelementary divisors (where of course α , . . . , α t ∈ R ), we can also have pairs of the form ξ ( a , b ) m , ξ ( a , b ) m , · · · , ξ ( a p , b p ) m p , ξ ( a p , b p ) m p , where a , . . . , a p ∈ R , b , . . . , b p ∈ R × and ξ ( a, b ) = ( λ − µ ( a + ib ))( λ − µ ( a − bi )) with a, b ∈ R .Since elementary divisors occur in pairs, we introduce a notation to indicate such pairs. Let K be an algebraically closed field or K = R . For given α ∈ K ∪ {∞} or α ∈ C ∪ {∞} in case K = R and e ∈ N we denote by ( α, e ) the following pairs of elementary divisors( α, e ) := µ e , µ e if α = ∞ ( λ − µα ) e , ( λ − µα ) e if α ∈ K ξ ( a, b ) e , ξ ( a, b ) e if K = R and α = a + bi ∈ C \ R . Hence for a skew pencil µA + λB over an algebraically closed field K we can associate in aunique way a set of elementary divisors as follows:( ∞ , e ) , . . . , ( ∞ , e s ) , ( α , f ) , . . . , ( α t , f t )with α , . . . , α t ∈ K and for a skew pencil µA + λB over R we find a set of elementary divisorsof the form ( ∞ , e ) , . . . , ( ∞ , e s ) , ( α , f ) , . . . , ( α t , f t ) , ( β , m ) , . . . , ( β p , m p ) , where α i ∈ R and β i ∈ C \ R .For a giving pair of elementary divisors ( α, e ) as above or a minimal index ε , there exists acanonical skew pencil having exactly that one pair of elementary divisors ( α, e ) (and no minimalindices or other elementary divisors) or having no elementary divisors and exactly one minimalindex ε .These skew pencils are given by the following cases: Case 1:
For ( α, e ) = ( ∞ , e ) the skew pencil is given by F ( ∞ , e ) := (cid:18) µ ∆ e + λ Λ e − µ ∆ e − λ Λ e (cid:19) ∈ M e ( K [ µ, λ ]) , D. BURDE, K. DEKIMPE, AND B. VERBEKE where ∆ e = . . . , Λ e =
00 1 . . . . . . ∈ M e ( K ) . Case 2:
For ( α, f ) = ( λ − µα ) f , ( λ − µα ) f the skew pencil is given by F ( α, f ) := (cid:18) λ − µα )∆ f + µ Λ f − ( λ − µα )∆ f − µ Λ f (cid:19) ∈ M f ( K [ µ, λ ]) . Case 3:
Only in case K = R and for ( α, m ) = ( a + bi, m ) = ξ ( a, b ) m , ξ ( a, b ) m (with a + bi ∈ C \ R )the real skew pencil is given by C ( a, b, m ) := (cid:18) T m − T m (cid:19) ∈ M m ( R [ µ, λ ]) , where T m = R. . . R µ ∆ . . . R . . .R µ ∆ ∈ M m ( R [ µ, λ ])for m ≥ T = R = (cid:18) − µb λ − µaλ − µa µb (cid:19) ∈ M ( R [ µ, λ ]) . Case 4:
For each minimal index ε ≥ M ε = (cid:18) ε +1 L ε ( µ, λ ) −L ε ( µ, λ ) t ε (cid:19) ∈ M ε +1 ( K [ µ, λ ]) , where L ε ( µ, λ ) = λ . . . µ λ . . . . . . µ λ . . . µ ∈ M ε +1 ,ε ( K [ µ, λ ]) . For minimal index ε = 0, the skew pencil is just M = (0), the 1 × Proposition 3.9.
Let K be an algebraically closed field of characteristic not 2. Any skew pencilover K with elementary divisors ( ∞ , e ) , . . . , ( ∞ , e s ) , ( α , f ) , . . . , ( α t , f t ) and minimal indices ε , . . . , ε k is strictly congruent to the pencil consisting of a matrix with the blocks F ( ∞ , e ) , . . . , F ( ∞ , e s ) , F ( α , f ) , . . . , F ( α t , f t ) , M ε , . . . , M ε k LMOST INNER DERIVATIONS 7 on the diagonal. Any skew pencil over R having elementary divisors ( ∞ , e ) , . . . , ( ∞ , e s ) , ( α , f ) , . . . , ( α t , f t ) , ( a + b i, m ) , . . . , ( a p + b p i, m p ) , and minimal indices ε , . . . , ε k is strictly congruent to the pencil consisting of a matrix with theblocks F ( ∞ , e ) , . . . , F ( ∞ , e s ) , F ( α , f ) , . . . , F ( α t , f t ) ,C ( a , b , m ) , . . . , C ( a p , b p , m p ) , M ε , . . . , M ε k on the diagonal. Definition 3.10.
A 2-step nilpotent Lie algebra of genus 2 over an algebraically closed field K of characteristic not 2 or K = R is called canonical if its associated skew pencil is of a blockeddiagonal form as in Proposition 3.9 above.As an immediate consequence of Proposition 3.3 we obtain the following corollary: Corollary 3.11.
Let K be an algebraically closed field of characteristic not 2 or K = R .Any -step nilpotent Lie algebra of genus over K is isomorphic to a canonical one with thesame elementary divisors and minimal indices. It follows that the computation of AID( g ) for 2-step nilpotent Lie algebras of genus 2 over K can be reduced to canonical Lie algebras.4. Applications to almost inner derivations
Any almost inner derivation D of a 2-step nilpotent Lie algebra g maps the center Z ( g ) tozero and g to [ g , g ], see Proposition 2 . C ( g ) defined below containsAID( g ) and is a subalgebra of Der( g ). Definition 4.1.
Let g be a 2-step nilpotent Lie algebra over a field K . Then C ( g ) = { D ∈ End( g ) | D ( Z ( g )) = 0 , D ( g ) ⊆ [ g , g ] } is a subalgebra of Der( g ) with AID( g ) ⊆ C ( g ). It is called the algebra of central derivations .Since [ g , g ] ⊆ Z ( g ) for 2-step nilpotent Lie algebras, any D ∈ C ( g ) is a derivation of g .Let us now assume that g be a 2-step nilpotent Lie algebra of genus 2. We fix a basis { x , . . . , x n , y , y } , where y , y spans [ g , g ]. Let µA + λB be the associated pencil. Every element in v ∈ g canbe written as v = x + y in this basis, where x = a x + . . . , a n x n and y = b y + b y for a i , b i ∈ K . For every D ∈ C ( g ) we have D ( y ) = 0 and D ( x ) = d ( x ) y + d ( x ) y for some d , d ∈ Hom( U, K ), where U = h x , x , . . . , x n i . Recall that D ∈ AID( g ) if and only if thereexists a map ϕ D : g → g such that D ( v ) = [ v, ϕ D ( v )]for all v ∈ g . We may assume that ϕ D ( v ) = ϕ D ( x ) ∈ h x , . . . , x n i for v = x + y as above, i.e.,that v = x and ϕ D ( x ) = c ( x ) x + · · · + c n ( x ) x n D. BURDE, K. DEKIMPE, AND B. VERBEKE for some c i ( x ) ∈ K for i = 1 , . . . , n . Let c ( x ) = ( c ( x ) , . . . , c n ( x )) t ,a ( x ) = ( a , . . . , a n ) t ,L ( x ) = (cid:18) a ( x ) t Aa ( x ) t B (cid:19) ∈ M ,n ( K ) ,d ( x ) = (cid:18) d ( x ) d ( x ) (cid:19) . Lemma 4.2.
Let g be a -step nilpotent Lie algebra of genus over K with the notations asabove. Then a given map D ∈ C ( g ) is in AID( g ) if and only if L ( x ) c ( x ) = d ( x ) has a solutionin the unknowns c i ( x ) , i = 1 , . . . , n for all x = a x + · · · + a n x n in g .Proof. We have D ∈ AID( g ) if and only if for all x = a x + · · · + a n x n there exists a map ϕ D : g → g such that D ( x ) = [ x, ϕ D ( x )], i.e., if and only if we can find c i ( x ) for i = 1 , . . . , n such that D ( x ) = " n X i =1 a i x i , n X j =1 c j ( x ) x j = n X i =1 n X j =1 a i c j ( x )( a ij y + b ij y ) . This is equivalent to the system of linear equations L ( x ) c ( x ) = d ( x ) for all x = a x + · · · + a n x n . (cid:3) We obtain the following result.
Proposition 4.3.
Let g be a -step nilpotent Lie algebra of genus over K with associatedpencil µA + λB and the notations as above. Then a given map D ∈ C ( g ) is in AID( g ) if andonly if for all x = a x + · · · + a n x n in g and all λ, µ ∈ K we have the condition ( µA + λB ) a ( x ) = 0 = ⇒ µd ( x ) + λd ( x ) = 0 . (1) Proof.
Suppose that D ∈ AID( g ) and ( µA + λB ) a ( x ) = 0. Since A and B are skew-symmetric,we have 0 = ( µA + λB ) a ( x ) = − ( µa ( x ) t A + λa ( x ) t B ) t , so that µa ( x ) t A + λa ( x ) t B = 0. Now a ( x ) t A is the first row of L ( x ) and a ( x ) t B the secondrow of it. Since D ∈ AID( g ), both L ( x ) and the extended matrix ( L ( x ) | d ( x )) have the samerank by Lemma 4 .
2. Hence for any linear combination of rows of L ( x ) which equals zero, thesame linear combination of rows of ( L ( x ) | d ( x )) equals zero. Hence µd ( x ) + λd ( x ) = 0. Theconverse direction follows similarly. (cid:3) We will apply this result to Example 3 . Example 4.4.
Let g be the -dimensional real Lie algebra with basis { x , . . . , x , y , y } andLie brackets defined by [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y . Let D ∈ C ( g ) and x = a x + · · · + a x . Then the matrix of D is given by D = r r r r r s s s s s LMOST INNER DERIVATIONS 9 and d ( x ) = a r + · · · + a r , d ( x ) = a s + · · · + a s . Condition (1) then yields that D ∈ AID( g ) if and only if r = s = 0 and s = r , s = r . Thus we have dim(AID( g )) = 6 and dim Inn( g ) = 5 . In fact, the kernel of µA + λB for this Lie algebra is 1-dimensional for all ( µ, λ ) = (0 , a ( x ) = (0 , , λ , − µλ, µ ) t . Therefore condition (1) applied to this vector yields0 = µd ( x ) + λd ( x )= µ ( a r + · · · + a r ) + λ ( a s + · · · + a s )= µ r + µ λ ( s − r ) + µλ ( r − s ) + λ s . for all λ, µ ∈ K , which shows the claim. Corollary 4.5.
Let g be a -step nilpotent Lie algebra of genus over R , whose associatedpencil µA + λB satisfies det( µA + λB ) = 0 if and only if µ = λ = 0 . Then AID( g ) = C ( g ) and dim(AID( g )) = 2 dim(Inn( g )) = 2 n .Proof. By assumption the system ( µA + λB ) a ( x ) = 0 only has the trivial solution x = 0.Hence condition (1) is satisfied and it follows AID( g ) = C ( g ) from Proposition 4 .
3. Since theassumptions imply that [ g , g ] = Z ( g ) we have dim Inn( g ) = n , spanned by ad( x ) , . . . , ad( x n ),and dim C ( g ) = 2 n . It follows that dim(AID( g )) = 2 n . (cid:3) We can apply this corollary to Example 3 . n = 4. Example 4.6.
Let g be the Lie algebra over a field K with basis { x , . . . , x , y , y } and Liebrackets defined by [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = − y . Then for K = R , we have that det( µA + λB ) = ( µ + λ ) = 0 if and only if µ = λ = 0 , so that AID( g ) = C ( g ) and dim(AID( g )) = 2 · dim(Inn( g )) = 8 . However, we have AID( g ) = Inn( g ) for K being an algebraically closed field of characteristic not 2. This will follow from our mainresult Theorem . . Almost inner derivations of Lie algebras of genus 2
In this section we determine the algebra AID( g ) for canonical Lie algebras in the sense ofDefinition 3 .
10 over K , where K is either R or an algebraically closed field of characteristic not2. We will start with the case that the canonical pencil only consists of one block. Definition 5.1.
Let g be a Lie algebra with a given basis B . We say that a linear map D : g → g is B -almost inner , if D ( x ) ∈ [ x, g ] for all basis elements x ∈ B .A derivation, which is B -almost inner for some basis B need not be almost inner. Lemma 5.2.
Let g be a canonical Lie algebra over K with one pair of elementary divisors ( ∞ , e ) . Then we have dim(Inn( g )) = 2 e and dim(AID( g )) = 4 e − .Proof. By assumption the matrix pencil of g is F ( ∞ , e ), so that the Lie brackets of g in theusual basis B are given by [ x i , x e +1 − i ] = y , ≤ i ≤ e, [ x j , x e +2 − j ] = y , ≤ j ≤ e. We have dim(Inn( g )) = 2 e . We will compute AID( g ) by Proposition 4 .
3. A basis for C ( g ) isgiven by the maps D i,j : g → g for 1 ≤ i ≤ e and j = 1 , e X k =1 a k x k + ( b y + b y ) a i y j . We have dim C ( g ) = 4 e . It is easy to see that all D i,j are B -almost inner except for D , and D e +1 , . It is easy to see that the span of D , and D e +1 , has only trivial intersection withInn( g ). Then we have dim AID( g ) = 4 e − e − D ∈ C ( g ) be B -almost inner. We have det( µA + λB ) = µ e . For µ = 0 condition (1) issatisfied, so that we may assume that µ = 0. Then the kernel of µA + λB = λB = F ( ∞ , e ) isequal to the set of all vectors a ( x ) = ( k , , . . . , , k e +1 , , . . . , t with k , k e +1 ∈ K . For thesevectors we have d ( a ( x )) = 0, so that condition (1) is satisfied and the proof is finished. (cid:3) Lemma 5.3.
Let g be a canonical Lie algebra over K with one pair of elementary divisors ( α, f ) . Then we have dim(Inn( g )) = 2 f and dim(AID( g )) = 4 f − .Proof. The Lie brackets of g with respect to the usual basis { x , . . . , x f , y , y } and matrixpencil F ( α, f ) are given by [ x i , x f +1 − i ] = y − αy , ≤ i ≤ f, [ x j , x f +2 − j ] = y , ≤ j ≤ e. We may pass to the basis { x , . . . , x f , y − αy , y } so that g coincides with the Lie algebra ofLemma 5 .
2. This finishes the proof. (cid:3)
The next Lemma is only for the case K = R . Lemma 5.4.
Let g be a canonical Lie algebra over R with one pair of elementary divisors ( β, m ) , where β = a + bi and b = 0 . Then we have dim Inn( g ) = 4 m and dim AID( g ) = 8 m .Proof. The Lie brackets of g with respect to the usual basis { x , . . . , x m , y , y ′ } and matrixpencil µA + λB = C ( a, b, m ) are given by[ x i − , x m − i +1 ] = − by , ≤ i ≤ m, [ x i , x m − i +2 ] = by , ≤ i ≤ m, [ x j , x m +1 − j ] = y ′ − ay , ≤ j ≤ m, and in addition, for m ≥
2, [ x k , x m − k +3 ] = y , ≤ k ≤ m. We can pass to a basis B = { x , . . . , x n , y , y } by setting y := y ′ − ay . Then a basis for C ( g )is given by the maps D i,j : g → g for 1 ≤ i ≤ m and j = 1 , m X k =1 a k x k + ( b y + b y ) a i y j . We have dim( C ( g )) = 8 m and det( µA + λB ) = ( λ + µ b ) m , so that AID( g ) = C ( g ) over R by Corollary 4 . (cid:3) Let us again consider Example 3 . LMOST INNER DERIVATIONS 11
Example 5.5.
Let g be the -dimensional Lie algebra over R with basis { x , . . . , x , y , y } andLie brackets defined by [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = − y . This is a canonical Lie algebra with one pair of elementary divisors ( β, m ) = ( i, . Hence,Lemma 5.4 says that dim(Inn( g )) = 4 and dim(AID( g )) = 8 which coincides with what weobtained in Example 4.6. Lemma 5.6.
Let g be a canonical Lie algebra over K with minimal index ε ≥ . Then it holds dim(Inn( g )) = 2 ε + 1 and dim(AID( g )) = 3 ε .Proof. The Lie brackets of g with respect to the usual basis B = { x , . . . , x ε +1 , y , y } andmatrix pencil µA + λB = M ε are given by[ x i , x i + ε +1 ] = y , ≤ i ≤ ε, [ x j +1 , x j + ε +1 ] = y , ≤ j ≤ ε. It is easy to see that Z ( g ) = h y , y i , so we have dim(Inn( g )) = 2 ε + 1. For ε = 1 we haveAID( g ) = Inn( g ) by Theorem 4 . g is determined by a graph. For ε ≥ C ( g ) is given by the maps D i,j : g → g for 1 ≤ i ≤ ε + 1 and j = 1 , ε +1 X k =1 a k x k + ( b y + b y ) a i y j . Hence we have dim( C ( g )) = 4 ε + 2 and AID( g ) ⊆ C ( g ). Suppose now that D = ε +1 X i =1 α i D i, + ε +1 X i =1 β i D i, is an element of AID( g ). Then for any b ∈ K we have D ε +1 X i =1 b i x i ! = ε +1 X i =1 α i b i y + ε +1 X i =1 β i b i y . (2)Since D ∈ AID( g ) there exist c j ( b ) ∈ K for all ε + 2 ≤ j ≤ ε + 1 such that D ε +1 X i =1 b i x i ! = " ε +1 X i =1 b i x i , ε +1 X j = ε +2 c j ( b ) x j = ε +1 X i =2 b i c i + ε ( b ) y + ε X i =1 b i c i + ε +1 ( b ) y . (3)We also have b ε X i =1 b i c i + ε +1 ( b ) ! = ε +1 X i =2 b i c i + ε ( b ) . (4)Comparing coefficients of y and y in (2) and (3) and using (4) we find that b ε +1 X i =1 β i b i − ε +1 X i =1 α i b i = 0 , so that − α b + ε +1 X i =2 ( β i − − α i ) b i + β ε +1 b ε +2 = 0 . Since this holds for all b ∈ K we obtain α = 0 , β ε +1 = 0 , α i = β i − for 2 ≤ i ≤ ε + 1 . This means that AID( g ) is contained in the subspace V = { D = ε +1 X i =1 α i D i, + ε +1 X i =1 β i D i, ∈ C ( g ) | α = β ε +1 = 0 , α i = β i − for 2 ≤ i ≤ ε + 1 } . Note that dim( V ) = dim( C ( g )) − ( ε + 2) = 4 ε + 2 − ( ε + 2) = 3 ε . Hence dim(AID( g )) ≤ ε . Weclaim that there holds equality. More precisely we will show that each D j, for ε + 2 ≤ j ≤ ε is almost inner. Here we do not consider D ε +1 , because it already coincides with the innerderivation ad( x ε +1 ) and hence is almost inner. Let x = ε +1 X i =1 a i x i + ( b y + b y )be an element in g . If a j = 0, then D j, ( x ) = [ x,
0] = 0. Otherwise we have D j, ( x ) = (cid:20) x, − a j a ℓ x ℓ − ε (cid:21) = a j y for ℓ := max { j ≤ k ≤ ε +1 | a k = 0 } . This shows that D j, is almost inner for all ε +2 ≤ j ≤ ε .Consider the subspace W of AID( g ) generated by all D j, for ε + 2 ≤ j ≤ ε . We claim that W ∩ Inn( g ) = 0 . Then we are done. We know that dim(Inn( g )) = 2 ε + 1 and dim( W ) = ε −
1, so that Inn( g ) ⊕ W is a 3 ε -dimensional subspace of AID( g ). This implies dim(AID( g )) ≥ ε and hence thereholds equality. So assume that D = P εi = ε +2 α i D i, ∈ W ∩ Inn( g ) with D = ad( x ) for some x = P ε +1 i =1 k i x i . We will show that all k i = 0 and all α i = 0. Because of0 = ad( x )( x j ) = D ( x j ) = " ε +1 X i =1 k i x i , x j = − k ε + j +1 y − k ε + j y for 2 ≤ j ≤ ε we have k ε +2 = k ε +3 = · · · = k ε +1 = 0. Also we have α ε + j y = D ( x ε + j ) = [ k x + · · · + k ε +1 x ε +1 , x ε + j ] = k j − y + k j y for all 2 ≤ j ≤ ε + 1, where we take α ε +1 = 0. Hence all α i and all k i are zero and we haveshown that W ∩ Inn( g ) = 0. (cid:3) Remark . For a minimal index ε = 0, the corresponding Lie algebra g is just the abelian3-dimensional Lie algebra (with basis x , y , y ) over K and so in this case Inn( g ) = AID( g ) = 0. Example 5.8.
For ε = 2 the canonical Lie algebra g of Lemma . is isomorphic to the Liealgebra of Example . . We have dim(Inn( g )) = 2 ε + 1 = 5 and dim(AID( g )) = 3 ε = 6 , whichcoincides with the result of Example . . LMOST INNER DERIVATIONS 13
For the next lemma, let g be a 2-step nilpotent Lie algebra over an arbitrary field K withbasis { x , . . . , x n , y , . . . , y m , z , . . . , z p } , where the z i span [ g , g ]. Define Lie subalgebras by g x = h x , . . . , x n , z , . . . , z p i , g y = h y , . . . , y m , z , . . . , z p i . Lemma 5.9.
Let g be a -step nilpotent Lie algebra over a field K with the above basis suchthat [ x i , y j ] = 0 for all ≤ i ≤ n and ≤ j ≤ m . Then we have dim(AID( g )) = dim(AID( g x )) + dim(AID( g y )) . Proof.
Let D ∈ AID( g ) and write e = x + y + z , where x is a linear combination of the x i , y is a linear combination of the y i and z a linear combination of the z i . Then there are maps ϕ D x : g x → g x and ϕ D y : g y → g y such that D ( e ) = D ( x + y + z ) = [ x, ϕ D x ( x )] + (cid:2) y, ϕ D y ( y ) (cid:3) , This means that D x : g x → g x , x D | g x ( x ) ∈ AID( g x ) with determination map ϕ D x and D y : g y → g y , y D | g y ( y ) ∈ AID( g y ) determined by ϕ D y . Conversely, any almost innerderivation of g x or g y can be extended to an almost inner derivation of g . (cid:3) Finally we can state our main result of this paper by combining the previous lemmas. Forclarity, we formulate this result as two separate theorems depending on the type of field K weare considering. We only give a proof for the last theorem in case K = R . The proof for theother case is similar. Theorem 5.10.
Let g be a -step nilpotent Lie algebra of genus over an algebraically closedfield K of characteristic not 2 with minimal indices ε , . . . , ε k and elementary divisors ( ∞ , e ) , . . . , ( ∞ , e s ) , ( α , f ) , . . . , ( α l , f l ) with α j ∈ K for all ≤ j ≤ l . Then we have dim(AID( g )) = dim(Inn( g )) + k X j =1 ,ε j =0 ( ε j −
1) + 2 s X j =1 ( e j −
1) + 2 l X j =1 ( f l − . Theorem 5.11.
Let g be a -step nilpotent Lie algebra of genus over R with minimal indices ε , . . . , ε k and elementary divisors ( ∞ , e ) , . . . , ( ∞ , e s ) , ( α , f ) , . . . , ( α l , f l ) , ( β , m ) , . . . , ( β p , m p ) with α j ∈ R and β r = a r + b r i ∈ C \ R for all ≤ j ≤ l and ≤ r ≤ p . Then we have dim(AID( g )) = dim(Inn( g )) + k X j =1 ,ε j =0 ( ε j −
1) + 2 s X j =1 ( e j −
1) + 2 l X j =1 ( f l −
1) + 4 p X j =1 m j . Proof.
We may assume that g is canonical with N := k + s + l + p blocks. We will show theresult by induction over N . For N = 1 the claim follows from the previous lemmas. If we have acanonical Lie algebra g with N + 1 blocks, we take a basis { x , . . . , x n , y , . . . , y m , z , z } , where z , z span [ g , g ] and where { x , . . . , x n } corresponds to the first N blocks and { y , . . . , y m } tothe last block. Since we have [ x i , y j ] = 0 for all i, j , we can apply Lemma 5 . N + 1 blocks if it holds for N blocks. (cid:3) Acknowledgments
Dietrich Burde is supported by the Austrian Science Foundation FWF, grant I3248. KarelDekimpe and Bert Verbeke are supported by a long term structural funding, the Methusalemgrant of the Flemish Government and the Research Foundation Flanders (Project G.0F93.17N)
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