Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2014), 065, 16 pages Bicrossed Products, Matched Pair Deformationsand the Factorization Index for Lie Algebras
Ana-Loredana AGORE †‡ and Gigel MILITARU §† Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
E-mail: [email protected], [email protected]
URL: http://homepages.vub.ac.be/~aagore/ ‡ Department of Applied Mathematics, Bucharest University of Economic Studies,Piata Romana 6, RO-010374 Bucharest 1, Romania § Faculty of Mathematics and Computer Science, University of Bucharest,Str. Academiei 14, RO-010014 Bucharest 1, Romania
E-mail: [email protected], [email protected]
URL: http://fmi.unibuc.ro/ro/departamente/matematica/militaru_gigel/
Received January 20, 2014, in final form June 10, 2014; Published online June 16, 2014http://dx.doi.org/10.3842/SIGMA.2014.065
Abstract.
For a perfect Lie algebra h we classify all Lie algebras containing h as a subalgebraof codimension 1. The automorphism groups of such Lie algebras are fully determined assubgroups of the semidirect product h (cid:110) ( k ∗ × Aut
Lie ( h )). In the non-perfect case theclassification of these Lie algebras is a difficult task. Let l (2 n + 1 , k ) be the Lie algebrawith the bracket [ E i , G ] = E i , [ G, F i ] = F i , for all i = 1 , . . . , n . We explicitly describeall Lie algebras containing l (2 n + 1 , k ) as a subalgebra of codimension 1 by computing allpossible bicrossed products k (cid:46)(cid:47) l (2 n + 1 , k ). They are parameterized by a set of matricesM n ( k ) × k n +2 which are explicitly determined. Several matched pair deformations of l (2 n + 1 , k ) are described in order to compute the factorization index of some extensions ofthe type k ⊂ k (cid:46)(cid:47) l (2 n + 1 , k ). We provide an example of such extension having an infinitefactorization index. Key words: matched pairs of Lie algebras; bicrossed products; factorization index
The theory of Lie algebras is among the most developed fields in algebra due to his broad appli-cability in differential geometry, theoretical physics, quantum field theory, classical or quantummechanics and others. Besides the purely algebraic interest in this problem, the classificationof Lie algebras of a given dimension is a central theme of study in modern group analysis ofdifferential equations – for further explanations and an historical background see [21]. TheLevi–Malcev theorem reduces the classification of all finite-dimensional Lie algebras over a fieldof characteristic zero to the following three subsequent problems: (1) the classification of allsemi-simple Lie algebras (solved by Cartan); (2) the classification of all solvable Lie algebras(which is known up to dimension 6 [8]) and (3) the classification of all Lie algebras that aredirect sums of semi-simple Lie algebras and solvable Lie algebras.Surprisingly, among these three problems, the last one is the least studied and the mostdifficult. Only in 1990 Majid [16, Theorem 4.1] and independently Lu and Weinstein [15,Theorem 3.9] introduced the concept of a matched pair between two Lie algebras g and h .To any matched pair of Lie algebras we can associate a new Lie algebra g (cid:46)(cid:47) h called the a r X i v : . [ m a t h . R A ] J un A.L. Agore and G. Militaru bicrossed product (also called double Lie algebra in [15, Definition 3.3], double cross sum in [17,Proposition 8.3.2] or knit product in [19]). In light of this new concept, problem (3) can beequivalently restated as follows: for a given (semi-simple) Lie algebra g and a given (solvable)Lie algebra h , describe the set of all possible matched pairs ( g , h , (cid:47), (cid:46) ) and classify up to anisomorphism all associated bicrossed products g (cid:46)(cid:47) h . Leaving aside the semi-simple/solvablecase this is just the factorization problem for Lie algebras – we refer to [1] for more details andadditional references on the factorization problem at the level of groups, Hopf algebras, etc.The present paper continues our recent work [3, 4] related to the above question (3), in itsgeneral form, namely the factorization problem and its converse, called the classifying comple-ment problem , which consist of the following question: let g ⊂ L be a given Lie subalgebra of L .If a complement of g in L exists (that is a Lie subalgebra h such that L = g + h and g ∩ h = { } ),describe explicitly, classify all complements and compute the cardinal of the isomorphism classesof all complements (which will be called the factorization index [ L : g ] f of g in L ). Our startingpoint is [4, Proposition 4.4] which describes all Lie algebras L that contain a given Lie algebra h as a subalgebra of codimension 1 over an arbitrary field k : the set of all such Lie algebras L isparameterized by the space TwDer( h ) of twisted derivations of h . The pioneer work on this sub-ject was performed by K.H. Hofmann: [12, Theorem I] describes the structure of n -dimensionalreal Lie algebras containing a given subalgebra of dimension n −
1. Equivalently, this provesthat the set of all matched pairs of Lie algebras ( k , h , (cid:47), (cid:46) ) (by k we will denote the Abelian Liealgebra of dimension 1) and the space TwDer( h ) of all twisted derivations of h are in one-to-onecorrespondence; moreover, any Lie algebra L containing h as a subalgebra of codimension 1 isisomorphic to a bicrossed product k (cid:46)(cid:47) h = h ( λ, ∆) , for some ( λ, ∆) ∈ TwDer( h ). The clas-sification up to an isomorphism of all bicrossed products h ( λ, ∆) is given in the case when h is perfect. As an application of our approach, the group Aut Lie ( h ( λ, ∆) ) of all automorphismsof such Lie algebras is fully described in Corollary 3.3: it appears as a subgroup of a certainsemidirect product h (cid:110) ( k ∗ × Aut
Lie ( h )) of groups. At this point we mention that the clas-sification of automorphisms groups of all indecomposable real Lie algebras of dimension up tofive was obtained recently in [11] where the importance of this subject in mathematical physicsis highlighted. For the special case of sympathetic Lie algebras h , Corollary 3.5 proves that,up to an isomorphism, there exists only one Lie algebra that contains h as a Lie subalgebraof codimension one, namely the direct product k × h and Aut Lie ( k × h ) ∼ = k ∗ × Aut
Lie ( h ).Now, k is a subalgebra of k (cid:46)(cid:47) h = h ( λ, ∆) having h as a complement: for a 5-dimensionalperfect Lie algebra all complements of k in h ( λ, ∆) are described in Example 3.7 as matchedpair deformations of h . Section 4 treats the same problem for a given (2 n + 1)-dimensionalnon-perfect Lie algebra h := l (2 n + 1 , k ). Theorem 4.2 describes explicitly all Lie algebras con-taining l (2 n + 1 , k ) as a subalgebra of codimension 1. They are parameterized by a set T ( n ) ofmatrices ( A, B, C, D, λ , δ ) ∈ M n ( k ) × k × k n +1 : there are four such families of Lie algebras ifthe characteristic of k is (cid:54) = 2 and two families in characteristic 2. All complements of k in twosuch bicrossed products k (cid:46)(cid:47) l (2 n + 1 , k ) are described by computing all matched pair deforma-tions of the Lie algebra l (2 n + 1 , k ) in Propositions 4.4 and 4.8. In particular, in Example 4.6we construct an example where the factorization index of k in the 4-dimensional Lie algebra m (4 , k ) is infinite: that is k has an infinite family of non-isomorphic complements in m (4 , k ). Toconclude, there are three reasons for which we considered the Lie algebra l (2 n +1 , k ) in Section 4:on the one hand it provided us with an example of a finite-dimensional Lie algebra extension g ⊂ L such that g has infinitely many non-isomorphic complements as a Lie subalgebra in L .On the other hand, the Lie algebra l (2 n + 1 , k ) serves for constructing two counterexamples inRemark 4.10 which show that some properties of Lie algebras are not preserves by the matchedpair deformation. Finally, having [2, Corollary 3.2] as a source of inspiration we believe thatany (2 n + 1)-dimensional Lie algebra is isomorphic to an r -deformation of l (2 n + 1 , k ) associatedto a given matched pair: a more general open question is stated at the end of the paper.atched Pair Deformations and the Factorization Index 3 All vector spaces, Lie algebras, linear or bilinear maps are over an arbitrary field k . The AbelianLie algebra of dimension n will be denoted by k n . For two given Lie algebras g and h we denoteby Aut Lie ( g ) the group of automorphisms of g and by Hom Lie ( g , h ) the space of all Lie algebramaps between g and h . A Lie algebra L factorizes through g and h if g and h are Lie subalgebrasof L such that L = g + h and g ∩ h = { } . In this case h is called a complement of g in L ; if g isan ideal of L , then a complement h , if it exists, is unique being isomorphic to the quotient Liealgebra L / g . In general, if g is only a subalgebra of L , then we are very far from having uniquecomplements; for a given extension g ⊂ L of Lie algebras, the number of types of isomorphismsof all complements of g in L is called the factorization index of g in L and is denoted by [ L : g ] f –a theoretical formula for computing [ L : g ] f is given in [3, Theorem 4.5]. For basic concepts andunexplained notions on Lie algebras we refer to [9, 13].A matched pair of Lie algebras [15, 17] is a system ( g , h , (cid:47), (cid:46) ) consisting of two Lie algebras g and h and two bilinear maps (cid:46) : h × g → g , (cid:47) : h × g → h such that ( g , (cid:46) ) is a left h -module,( h , (cid:47) ) is a right g -module and the following compatibilities hold for all g, h ∈ g and x, y ∈ h x (cid:46) [ g, h ] = [ x (cid:46) g, h ] + [ g, x (cid:46) h ] + ( x (cid:47) g ) (cid:46) h − ( x (cid:47) h ) (cid:46) g, [ x, y ] (cid:47) g = [ x, y (cid:47) g ] + [ x (cid:47) g, y ] + x (cid:47) ( y (cid:46) g ) − y (cid:47) ( x (cid:46) g ) . Let ( g , h , (cid:47), (cid:46) ) be a matched pair of Lie algebras. Then g (cid:46)(cid:47) h := g × h , as a vector space, isa Lie algebra with the bracket defined by { ( g, x ) , ( h, y ) } := (cid:0) [ g, h ] + x (cid:46) h − y (cid:46) g, [ x, y ] + x (cid:47) h − y (cid:47) g (cid:1) for all g, h ∈ g and x, y ∈ h , called the bicrossed product associated to the matched pair ( g , h , (cid:47), (cid:46) ).Any bicrossed product g (cid:46)(cid:47) h factorizes through g = g × { } and h = { } × h ; the converse alsoholds [17, Proposition 8.3.2]: if a Lie algebra L factorizes through g and h , then there exist anisomorphism of Lie algebras L ∼ = g (cid:46)(cid:47) h , where g (cid:46)(cid:47) h is the bicrossed product associated to thematched pair ( g , h , (cid:47), (cid:46) ) whose actions are constructed from the unique decomposition[ x, g ] = x (cid:46) g + x (cid:47) g ∈ g + h (2.1)for all x ∈ h and g ∈ g . The matched pair ( g , h , (cid:47), (cid:46) ) defined by (2.1) is called the canonicalmatched pair associated to the factorization L = g + h . Remark 2.1.
Over the complex numbers C , an equivalent description for the factorization ofa Lie algebra L through two Lie subalgebras is given in [5, Definition 2.1] and in [6, Proposi-tion 2.2], in terms of complex product structures of L , i.e. linear maps f : L → L such that f (cid:54) = ± Id, f = f satisfying the integrability conditions f ([ x, y ]) = [ f ( x ) , y ] + [ x, f ( y )] − f (cid:0) [ f ( x ) , f ( y )] (cid:1) for all x, y ∈ L . The linear map f : g (cid:46)(cid:47) h → g (cid:46)(cid:47) h , f ( g, h ) := ( g, − h ) is a complex productstructure on any bicrossed product g (cid:46)(cid:47) h . Conversely, if f is a complex product structure on L ,then L factorizes through two Lie subalgebras L = L + + L − , where L ± denotes the eigenspacecorresponding to the eigenvalue ± f , that is L ∼ = L + (cid:46)(cid:47) L − .Let ( g , h , (cid:47), (cid:46) ) be a matched pair of Lie algebras. A linear map r : h → g is called a deformationmap [3, Definition 4.1] of the matched pair ( g , h , (cid:46), (cid:47) ) if the following compatibility holds for any x, y ∈ h r (cid:0) [ x, y ] (cid:1) − (cid:2) r ( x ) , r ( y ) (cid:3) = r (cid:0) y (cid:47) r ( x ) − x (cid:47) r ( y ) (cid:1) + x (cid:46) r ( y ) − y (cid:46) r ( x ) . (2.2) A.L. Agore and G. MilitaruWe denote by DM ( h , g | ( (cid:46), (cid:47) )) the set of all deformation maps of the matched pair ( g , h , (cid:46), (cid:47) ). If r ∈ DM ( h , g | ( (cid:46), (cid:47) )) then h r := h , as a vector space, with the new bracket defined for any x, y ∈ h by [ x, y ] r := [ x, y ] + x (cid:47) r ( y ) − y (cid:47) r ( x ) (2.3)is a Lie algebra called the r -deformation of h . A Lie algebra h is a complement of g ∼ = g × { } inthe bicrossed product g (cid:46)(cid:47) h if and only if h ∼ = h r , for some deformation map r ∈ DM ( h , g | ( (cid:46), (cid:47) ))[3, Theorem 4.3]. Computing all matched pairs between two given Lie algebras g and h and classifying all associatedbicrossed products g (cid:46)(cid:47) h is a challenging problem. In the case when g := k = k , the Abelian Liealgebra of dimension 1, they are parameterized by the set TwDer( h ) of all twisted derivationsof the Lie algebra h as defined in [4, Definition 4.2]: a twisted derivation of h is a pair ( λ, ∆)consisting of two linear maps λ : h → k and ∆ : h → h such that for any g, h ∈ h λ ([ g, h ]) = 0 , ∆([ g, h ]) = [∆( g ) , h ] + [ g, ∆( h )] + λ ( g )∆( h ) − λ ( h )∆( g ) . (3.1)TwDer( h ) contains the usual space of derivations Der( h ) via the canonical embedding Der( h ) (cid:44) → TwDer( h ), D (cid:55)→ (0 , D ), which is an isomorphism if h is a perfect Lie algebra (i.e. h = [ h , h ]). Asa special case of [4, Proposition 4.4 and Remark 4.5] we have: Proposition 3.1.
Let h be a Lie algebra. Then there exists a bijection between the set of allmatched pairs ( k , h , (cid:47), (cid:46) ) and the space TwDer( h ) of all twisted derivations of h given such thatthe matched pair ( k , h , (cid:47), (cid:46) ) corresponding to ( λ, ∆) ∈ TwDer( h ) is defined by h (cid:46) a = aλ ( h ) , h (cid:47) a = a ∆( h ) (3.2) for all h ∈ h and a ∈ k = k . The bicrossed product k (cid:46)(cid:47) h associated to the matched pair (3.2) is denoted by h ( λ, ∆) and has the bracket given for any a, b ∈ k and x, y ∈ h by { ( a, x ) , ( b, y ) } := (cid:0) bλ ( x ) − aλ ( y ) , [ x, y ] + b ∆( x ) − a ∆( y ) (cid:1) . (3.3) A Lie algebra L contains h as a subalgebra of codimension if and only if L is isomorphicto h ( λ, ∆) , for some ( λ, ∆) ∈ TwDer( h ) . Suppose { e i | i ∈ I } is a basis for the Lie algebra h . Then, h ( λ, ∆) has { F, e i | i ∈ I } as a basisand the bracket given for any i ∈ I by[ e i , F ] = λ ( e i ) F + ∆( e i ) , [ e i , e j ] = [ e i , e j ] h , where [ − , − ] h is the bracket on h . Above we identify e i = (0 , e i ) and denote F = (1 ,
0) in thebicrossed product k (cid:46)(cid:47) h . Classifying the Lie algebras h ( λ, ∆) is a difficult task. In what followswe deal with this problem for a perfect Lie algebra h : in this case TwDer( h ) = { } × Der( h ) andwe denote by h (∆) = h (0 , ∆) , for any ∆ ∈ Der( h ). Theorem 3.2.
Let h be a perfect Lie algebra and ∆ , ∆ (cid:48) ∈ Der( h ) . Then there exists a bijectionbetween the set of all morphisms of Lie algebras ϕ : h (∆) → h (∆ (cid:48) ) and the set of all triples ( α, h, v ) ∈ k × h × Hom
Lie ( h , h ) satisfying the following compatibility condition for all x ∈ h v (cid:0) ∆( x ) (cid:1) − α ∆ (cid:48) (cid:0) v ( x ) (cid:1) = [ v ( x ) , h ] . (3.4)atched Pair Deformations and the Factorization Index 5 The bijection is given such that the Lie algebra map ϕ = ϕ ( α,h,v ) corresponding to ( α, h, v ) isgiven by the formula ϕ : h (∆) → h (∆ (cid:48) ) , ϕ ( a, x ) = ( aα, ah + v ( x )) for all ( a, x ) ∈ h (∆) = k (cid:46)(cid:47) h . Furthermore, ϕ = ϕ ( α,h,v ) is an isomorphism of Lie algebras ifand only if α (cid:54) = 0 and v ∈ Aut
Lie ( h ) . Proof .
Any linear map ϕ : k × h → k × h is uniquely determined by a quadruple ( α, h, β, v ),where α ∈ k , h ∈ h and β : h → k , v : h → h are k -linear maps such that ϕ ( a, x ) = ϕ ( α,h,β,v ) = ( aα + β ( x ) , ah + v ( x )) . We will prove that ϕ defined above is a Lie algebra map if and only if β is the trivial map, v isa Lie algebra map and (3.4) holds. It is enough to test the compatibility ϕ (cid:0) [( a, x ) , ( b, y )] (cid:1) = [ ϕ ( a, x ) , ϕ ( b, y )] (3.5)for all generators of h (∆) = k × h , i.e. elements of the form (1 ,
0) and (0 , x ), for all x ∈ h .Moreover, since h is perfect (i.e. λ = 0) the bracket on h (∆) given by (3.3) takes the form: { ( a, x ) , ( b, y ) } = (0 , [ x, y ] + b ∆( x ) − a ∆( y )). Using this formula we obtain that (3.5) holds for(0 , x ) and (0 , y ) if and only if β (cid:0) [ x, y ] (cid:1) = 0 , v (cid:0) [ x, y ] (cid:1) = [ v ( x ) , v ( y )] + β ( y )∆( v ( x )) − β ( x )∆( v ( y )) . As h is perfect these two conditions are equivalent to the fact that β = 0 and v is a Lie algebramap. Finally, as β = 0, we can easily show that (3.5) holds in (1 ,
0) and (0 , x ) if and onlyif (3.4) holds. Thus, we have obtained that ϕ is a Lie algebra map if and only if v is a Liealgebra map, β = 0 and (3.4) holds. In what follows we denote by ϕ ( α,h,v ) the Lie algebra mapcorresponding to a quadruple ( α, h, β, v ) with β = 0. Suppose first that ϕ := ϕ ( α,h,v ) is a Liealgebra isomorphism. Then, there exists a Lie algebra map ϕ := ϕ ( γ,g,w ) : h (∆ (cid:48) ) → h (∆) suchthat ϕ ◦ ϕ ( a, x ) = ϕ ◦ ϕ ( a, x ) = ( a, x ) for all a ∈ k , x ∈ h . Thus, for all a ∈ k and x ∈ h , we have aαγ = a, aγ + v ( ag ) + v (cid:0) w ( x ) (cid:1) = x = aαg + w ( ah ) + w (cid:0) v ( x ) (cid:1) . (3.6)By the first part of (3.6) for a = 1 we obtain αγ = 1 and thus α (cid:54) = 0 while the second partof (3.6) for a = 0 implies v bijective. To end with, assume that α (cid:54) = 0 and v ∈ Aut
Lie ( h ).Then, it is straightforward to see that ϕ = ϕ ( α,h,v ) is an isomorphism with the inverse given by ϕ − := ϕ ( α − , − α − v − ( h ) ,v − ) . (cid:4) Let k ∗ be the units group of k and ( h , +) the underlying Abelian group of the Lie algebra h .Then the map given for any α ∈ k ∗ , v ∈ Aut
Lie ( h ) and h ∈ h by ϕ : k ∗ × Aut
Lie ( h ) → Aut Gr ( h , +) , ϕ ( α, v )( h ) := α − v ( h )is a morphism of groups. Thus, we can construct the semidirect product of groups h (cid:110) ϕ ( k ∗ × Aut
Lie ( h )) associated to ϕ . The next result shows that Aut Lie ( h (∆) ) is isomorphic to a certainsubgroup of the semidirect product of groups h (cid:110) ϕ ( k ∗ × Aut
Lie ( h )). Corollary 3.3.
Let h be a perfect Lie algebra and ∆ , ∆ (cid:48) ∈ Der( h ) . Then the Lie algebras h (∆) and h (∆ (cid:48) ) are isomorphic if and only if there exists a triple ( α, h, v ) ∈ k ∗ × h × Aut
Lie ( h ) suchthat v ◦ ∆ − α ∆ (cid:48) ◦ v = [ v ( − ) , h ] . Furthermore, there exists an isomorphism of groups Aut
Lie ( h (∆) ) ∼ = G ( h , ∆) := { ( α, h, v ) ∈ k ∗ × h × Aut
Lie ( h ) | v ◦ ∆ − α ∆ ◦ v = [ v ( − ) , h ] } , A.L. Agore and G. Militaru where G ( h , ∆) is a group with respect to the following multiplication ( α, h, v ) · ( β, g, w ) := ( αβ, βh + v ( g ) , v ◦ w ) (3.7) for all ( α, h, v ) , ( β, g, w ) ∈ G ( h , ∆) . Moreover, the canonical map G ( h , ∆) −→ h (cid:110) ϕ (cid:0) k ∗ × Aut
Lie ( h ) (cid:1) , ( α, h, v ) (cid:55)→ (cid:0) α − h, ( α, v ) (cid:1) is an injective morphism of groups. Proof .
The first part follows trivially from Theorem 3.2. Consider now γ, ψ ∈ Aut
Lie ( h (∆) ).Using again Theorem 3.2, we can find ( α, h, v ) , ( β, g, w ) ∈ k ∗ × h × Aut
Lie ( h ) such that γ = ϕ ( α,h,v ) and ψ = ϕ ( β,g,w ) . Then, for all a ∈ k , x ∈ h we have ϕ ( α,h,v ) ◦ ϕ ( β,g,w ) ( a, x ) = ϕ ( α,h,v ) (cid:0) aβ, ag + w ( x ) (cid:1) = (cid:0) αβa, aβh + av ( g ) + v ◦ w ( x ) (cid:1) = ϕ ( αβ,βh + v ( g ) ,v ◦ w ) ( a, x ) . Thus, Aut
Lie ( h (∆) ) is isomorphic to G ( h , ∆) with the multiplication given by (3.7). The lastassertion follows by a routine computation. (cid:4) Remark 3.4.
Let ∆ = [ x , − ] be an inner derivation of a perfect Lie algebra h . Then the groupAut Lie ( h ([ x , − ]) ) admits a simpler description as follows G ( h , [ x , − ]) = { ( α, h, v ) ∈ k ∗ × h × Aut
Lie ( h ) | v ( x ) − αx + h ∈ Z( h ) } , where Z( h ) is the center of h . Assume in addition that h has trivial center, i.e. Z( h ) = { } ; itfollows that there exists an isomorphism of groupsAut Lie ( h ([ x , − ]) ) ∼ = k ∗ × Aut
Lie ( h ) , since in this case any element h from a triple ( α, h, v ) ∈ G ( h , [ x , − ]) must be equal to αx − v ( x ).Moreover, in this context, the multiplication given by (3.7) is precisely that of a direct productof groups.A Lie algebra h is called complete (see [14, 22] for examples and structural results on thisclass of Lie algebras) if h has trivial center and any derivation is inner. A complete and perfectLie algebra is called sympathetic [7]: semisimple Lie algebras over a field of characteristic zeroare sympathetic and there exists a sympathetic non-semisimple Lie algebra in dimension 25. Forsympathetic Lie algebras, Theorem 3.2 takes the following form which considerably improves [4,Corollary 4.10], where the classification is made only up to an isomorphism of Lie algebras whichacts as identity on h . Corollary 3.5.
Let h be a sympathetic Lie algebra. Then up to an isomorphism of Lie algebrasthere exists only one Lie algebra that contains h as a Lie subalgebra of codimension one, namelythe direct product k × h of Lie algebras. Furthermore, there exists an isomorphism of groups Aut
Lie ( k × h ) ∼ = k ∗ × Aut
Lie ( h ) . Proof .
Since h is perfect any Lie algebra that contains h as a Lie subalgebra of codimension 1is isomorphic to h ( D ) , for some D ∈ Der( h ). As h is also complete, any derivation is inner. Foran arbitrary derivation D = [ d, − ] we can prove that h ( D ) ∼ = h (0) , where 0 = [0 , − ] is the trivialderivation and moreover h (0) is just the direct product of Lie algebras k × h . Indeed, by taking( α, h, v ) := (1 , − d, Id h ) one can see that relation (3.4) holds for D = [ d, − ] and D (cid:48) = [0 , − ], thatis h ( D ) ∼ = h (0) . The final part follows from Remark 3.4. (cid:4) atched Pair Deformations and the Factorization Index 7 Remark 3.6.
Let h be a perfect Lie algebra with a basis { e i | i ∈ I } , ∆ ∈ Der( h ) a given deriva-tion and consider the extension k ⊆ h (∆) = k (cid:46)(cid:47) h (∆) . In order to determine all complementsof k in h (∆) we have to describe the set of all deformation maps r : h → k of the matchedpair (3.2). A deformation map is completely determined by a family of scalars ( a ) i ∈ I satisfyingthe following compatibility condition for any i, j ∈ Ir (cid:0) [ e i , e j ] h (cid:1) = r (cid:0) a i ∆( e j ) − a j ∆( e i ) (cid:1) via the relation r ( e i ) = a i . For such an r = ( a i ) i ∈ I , the r -deformation of h is the Lie algebra h r having { e i | i ∈ I } as a basis and the bracket defined for any i, j ∈ I by[ e i , e j ] r = [ e i , e j ] h + a j ∆( e i ) − a i ∆( e j ) . Any complement of k in h (∆) is isomorphic to such an h r . An explicit example in dimension 5is given below. Example 3.7.
Let k be a field of characteristic (cid:54) = 2 and h the perfect 5-dimensional Lie algebrawith a basis { e , e , e , e , e } and bracket given by[ e , e ] = e , [ e , e ] = − e , [ e , e ] = [ e , e ] = e , [ e , e ] = 2 e , [ e , e ] = e , [ e , e ] = − e . By a straightforward computation it can be proved that the space of derivations Der( h ) coincideswith the space of all matrices from M ( k ) of the form A = a − a − a − a a a a a a a a − a − a ( a − a ) for all a , . . . , a ∈ k . Thus h is not complete since Der( h ) has dimension 6. One can show easilythat the derivation ∆ := e − e − e + e − e − e is not inner, where e ij ∈ M n ( k ) is thematrix having 1 in the ( i, j ) th position and zeros elsewhere. For the derivation ∆ we considerthe extension k ⊆ k (cid:46)(cid:47) h = h (∆) and we will describe all the complements of k in h (∆) . Bya routine computation it can be seen that r : h → k is a deformation map of the matchedpair (3.2) if and only if r := 0 (the trivial map) or r is given by r ( e ) := a, r ( e ) := − a − , r ( e ) = 2 , r ( e ) = r ( e ) = 0for some a ∈ k ∗ . Thus a Lie algebra C is a complement of k in h (∆) if and only if C ∼ = h or C ∼ = h a , where h a is the 5-dimensional Lie algebra with basis { e , e , e , e , e } and bracketgiven by[ e , e ] a := − a − e + ae + e + a − e , [ e , e ] a := − e − ae , [ e , e ] a := ae , [ e , e ] a := e + 2 ae , [ e , e ] a := a − e , [ e , e ] a := e − a − e , [ e , e ] a := − a − e , [ e , e ] a := 3 e , [ e , e ] a := 3 e for any a ∈ k ∗ . Remark that none of the matched pair deformations h a of the Lie algebra h isperfect since the dimension of the derived algebra [ h a , h a ] is equal to 3. A.L. Agore and G. Militaru In Section 3 we have described and classified all bicrossed products k (cid:46)(cid:47) h for a perfect Liealgebra h ; furthermore, Remark 3.6 and Example 3.7 describe all complements of k in a givenbicrossed product k (cid:46)(cid:47) h . In this section we approach the same questions for a given non-perfectLie algebra h := l (2 n + 1 , k ), where l (2 n + 1 , k ) is the (2 n + 1)-dimensional Lie algebra with basis { E i , F i , G | i = 1 , . . . , n } and bracket given for any i = 1 , . . . , n by[ E i , G ] := E i , [ G, F i ] := F i . First, we shall describe all bicrossed products k (cid:46)(cid:47) l (2 n + 1 , k ): they will explicitly describeall Lie algebras which contain l (2 n + 1 , k ) as a subalgebra of codimension 1. Then, as thesecond step, we shall find all r -deformations of the Lie algebra l (2 n + 1 , k ), for two givenextensions k ⊆ k (cid:46)(cid:47) l (2 n + 1 , k ). Based on Proposition 3.1 we have to compute first the spaceTwDer( l (2 n + 1 , k )) of all twisted derivations. Proposition 4.1.
There exists a bijection between
TwDer( l (2 n +1 , k )) and the set of all matrices ( A, B, C, D, λ , δ ) ∈ M n ( k ) × k × k n +1 satisfying the following conditions λ A = − δ n +1 I n , (2 + λ ) B = 0 , (2 − λ ) C = 0 , λ D = δ n +1 I n , (4.1) where δ = ( δ , . . . , δ n +1 ) ∈ k n +1 . The bijection is given such that the twisted derivation ( λ, ∆) ∈ TwDer( l (2 n + 1 , k )) associated to ( A, B, C, D, λ , δ ) is given by λ ( E i ) = λ ( F i ) := 0 , λ ( G ) := λ , (4.2)∆ := A B δ C D :0 0 δ n +1 . (4.3) T ( n ) denotes the set of all ( A, B, C, D, λ , δ ) ∈ M n ( k ) × k × k n +1 satisfying (4.1) . Proof .
The first compatibility condition (3.1) shows that a linear map λ : l (2 n + 1 , k ) → k ofa twisted derivation ( λ, D ) must have the form given by (4.2), for some λ ∈ k . We shall fixsuch a map for a given λ ∈ k . We write down the linear map ∆ : l (2 n + 1 , k ) → l (2 n + 1 , k ) asa matrix associated to the basis { E , . . . , E n , F , . . . , F n , G } of l (2 n + 1 , k ), as follows∆ = A B d , n +1 C D : d n +1 , ..d n +1 , n +1 for some matrices A, B, C, D ∈ M n ( k ) and some scalars d i,j ∈ k , for all i, j = 1 , . . . , n + 1.We denote A = ( a ij ), B = ( b ij ), C = ( c ij ), D = ( d ij ). It remains to check the compatibilitycondition (3.1) for ∆, i.e.∆([ g, h ]) = [∆( g ) , h ] + [ g, ∆( h )] + λ ( g )∆( h ) − λ ( h )∆( g )for all g (cid:54) = h ∈ { E , . . . , E n , F , . . . , F n , G } . As this is a routinely straightforward computationwe will only indicate the main steps of the proof. We can easily see that the compatibilitycondition (3.1) holds for ( g, h ) = ( E i , E j ) if and only if d n +1 ,i = 0, for all i = 1 , . . . , n . Inthe same way (3.1) holds for ( g, h ) = ( F i , F j ) if and only if d n +1 ,n + i = 0, for all i = 1 , . . . , n .This shows that ∆ has the form (4.3), that is the first 2 n entries from the last row of thematrix ∆ are all zeros and we will denote the last column of D by ( d , n +1 , . . . , d n +1 , n +1 ) = δ = ( δ , . . . , δ n +1 ). It follows from here that (3.1) holds trivially for the pair ( g, h ) = ( E i , F j ).atched Pair Deformations and the Factorization Index 9An easy computation shows that (3.1) holds for ( g, h ) = ( E i , G ) if and only if the followingequation holds(1 − λ ) n (cid:88) j =1 a j,i E j + n (cid:88) j =1 c j,i F j = n (cid:88) j =1 a j,i E j − n (cid:88) j =1 c j,i F j + δ n +1 E i , which is equivalent to − λ A = δ n +1 I n and (2 − λ ) C = 0, i.e. the first and the third equationsfrom (4.1). A similar computation shows that (3.1) holds for ( g, h ) = ( G, F i ) if and only if(2 + λ ) B = 0 and λ D = δ n +1 I n and the proof is finished. (cid:4) Let l (2 n + 1 , k ) ( A,B,C,D,λ ,δ ) be the bicrossed product k (cid:46)(cid:47) l (2 n + 1 , k ) associated to thematched pair given by the twisted derivation (cid:0) A = ( a ji ) , B = ( b ji ) , C = ( c ji ) , D = ( d ji ) , λ , δ = ( δ j ) (cid:1) ∈ T ( n ). From now on we will use the following convention: if one of the elementsof the 6-tuple ( A , B , C , D , λ , δ ) is equal to 0 then we will omit it when writing down theLie algebra l (2 n + 1 , k ) ( A,B,C,D,λ ,δ ) . A basis of l (2 n + 1 , k ) ( A,B,C,D,λ ,δ ) will be denoted by { E i , F i , G, H | i = 1 , . . . , n } : these Lie algebras can be explicitly described by first computingthe set T ( n ) and then using Proposition 3.1. Considering the equations (4.1) which define T ( n )a discussion involving the field k and the scalar λ is mandatory. For two sets X and Y we shalldenote by X (cid:116) Y the disjoint union of X and Y . As a conclusion of the above results we obtain: Theorem 4.2. (1) If k is a field such that char( k ) (cid:54) = 2 then T ( n ) ∼ = (cid:0) ( k \ { , ± } ) × k n +1 (cid:1) (cid:116) (cid:0) M n ( k ) × k n (cid:1) (cid:116) (cid:0) M n ( k ) × k n +1 (cid:1) (cid:116) (cid:0) M n ( k ) × k n +1 (cid:1) and the four families of Lie algebras containing l (2 n + 1 , k ) as a subalgebra of codimension arethe following: • the Lie algebra l (2 n + 1 , k ) ( λ ,δ ) with the bracket given for any i = 1 , . . . , n by [ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = − λ − δ n +1 E i , [ F i , H ] = λ − δ n +1 F i , [ G, H ] = λ H + n (cid:88) j =1 δ j E j + n (cid:88) j =1 δ n + j F j + δ n +1 G for all ( λ , δ ) ∈ ( k \ { , ± } ) × k n +1 . • the Lie algebra l (2 n + 1 , k ) ( A,D,δ ) with the bracket given for any i = 1 , . . . , n by [ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = n (cid:88) j =1 a ji E j , [ F i , H ] = n (cid:88) j =1 d ji F j , [ G, H ] = n (cid:88) j =1 δ j E j + n (cid:88) j =1 δ n + j F j for all ( A = ( a ij ) , D = ( d ij ) , δ ) ∈ M n ( k ) × M n ( k ) × k n . • the Lie algebra l (2 n + 1 , k ) ( C,δ ) with the bracket given for any i = 1 , . . . , n by [ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = − − δ n +1 E i + n (cid:88) j =1 c ji F j , [ F i , H ] = 2 − δ n +1 F i , [ G, H ] = 2 H + n (cid:88) j =1 δ j E j + n (cid:88) j =1 δ n + j F j + δ n +1 G for all ( C = ( c ij ) , δ ) ∈ M n ( k ) × k n +1 . • the Lie algebra l (2 n + 1 , k ) ( B,δ ) with the bracket given for any i = 1 , . . . , n by [ E i , G ] = E i , [ G, F i ] = F i , [ F i , H ] = n (cid:88) j =1 b ji E j − − δ n +1 F i , [ E i , H ] = 2 − δ n +1 E i , [ G, H ] = − H + n (cid:88) j =1 δ j E j + n (cid:88) j =1 δ n + j F j + δ n +1 G for all ( B = ( b ij ) , δ ) ∈ M n ( k ) × k n +1 . (2) If char( k ) = 2 then T ( n ) ∼ = (cid:0) M n ( k ) × k n (cid:1) (cid:116) (cid:0) k ∗ × k n +1 (cid:1) and the two families of Lie algebras containing l (2 n + 1 , k ) as a subalgebra of codimension arethe following: • the Lie algebra l (2 n + 1 , k ) ( A,B,C,D,δ ) with the bracket given for any i = 1 , . . . , n by [ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = n (cid:88) j =1 (cid:0) a ji E j + c ji F j (cid:1) , [ F i , H ] = n (cid:88) j =1 (cid:0) b ji E j + d ji F j (cid:1) , [ G, H ] = n (cid:88) j =1 δ j E j + n (cid:88) j =1 δ n + j F j for all ( A, B, C, D, δ ) ∈ M n ( k ) × k n . • the Lie algebra l (2 n + 1 , k ) ( λ ,δ ) with the bracket given for any i = 1 , . . . , n by [ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = − λ − δ n +1 E i , [ F i , H ] = λ − δ n +1 F i , [ G, H ] = λ H + n (cid:88) j =1 δ j E j + n (cid:88) j =1 δ n + j F j + δ n +1 G for all ( λ , δ ) ∈ k ∗ × k n +1 . Proof .
The proof relies on the use of Propositions 3.1 and 4.1 as well as the equations (4.1)defining T ( n ). Besides the discussion on the characteristic of k it is also necessary to considerwhether λ belongs to the set { , , − } . In the case that char( k ) (cid:54) = 2, the first Lie algebralisted is the bicrossed product which corresponds to the case when λ / ∈ { , , − } . In thiscase, we can easily see that (cid:0) A, B, C, D, λ , δ = ( δ j ) (cid:1) ∈ T ( n ) if and only if B = C = 0, A = − λ − δ n +1 I n and D = λ − δ n +1 I n . The Lie algebra l (2 n + 1 , k ) ( λ ,δ ) is exactly thebicrossed product k (cid:46)(cid:47) l (2 n + 1 , k ) corresponding to this twisted derivation. The Lie algebra l (2 n + 1 , k ) ( A,D,δ ) is the bicrossed product k (cid:46)(cid:47) l (2 n + 1 , k ) corresponding to the case λ = 0while the last two Lie algebras are the bicrossed products k (cid:46)(cid:47) l (2 n + 1 , k ) associated to thecase when λ = 2 and respectively λ = − k is equal to 2 we distinguish the following two possibilities: the Liealgebra l (2 n +1 , k ) ( A,B,C,D,δ ) is the bicrossed product k (cid:46)(cid:47) l (2 n +1 , k ) associated to λ = 0 whilethe Lie algebra l (2 n + 1 , k ) ( λ ,δ ) is the same bicrossed product but associated to λ (cid:54) = 0. (cid:4) Let k be a field of characteristic (cid:54) = 2 and l (2 n + 1 , k ) ( λ ,δ ) the Lie algebra of Theorem 4.2.In order to keep the computations efficient we will consider λ := 1 and δ := (0 , . . . , ,
1) andwe denote by L (2 n + 2 , k ) := l (2 n + 1 , k ) (1 , (0 ,..., , , the (2 n + 2)-dimensional Lie algebra havinga basis { E i , F i , G, H | i = 1 , . . . , n } and the bracket defined for any i = 1 , . . . , n by[ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = − E i , [ F i , H ] = F i , [ G, H ] = H + G. atched Pair Deformations and the Factorization Index 11We consider the Lie algebra extension kH ⊂ L (2 n + 2 , k ), where kH ∼ = k is the AbelianLie algebra of dimension 1. Of course, L (2 n + 2 , k ) factorizes through kH and l (2 n + 1 , k ),i.e. L (2 n + 2 , k ) = kH (cid:46)(cid:47) l (2 n + 1 , k ) – the actions (cid:47) : l (2 n + 1 , k ) × kH → l (2 n + 1 , k ) and (cid:46) : l (2 n + 1 , k ) × kH → kH of the canonical matched pair are given by E i (cid:47) H := − E i , F i (cid:47) H := F i , G (cid:47) H := G, G (cid:46) H := H (4.4)and all undefined actions are zero. Next we compute the set DM ( l (2 n + 1 , k ) , kH | ( (cid:46), (cid:47) )) of alldeformation maps of the matched pair ( kH, l (2 n + 1 , k ) , (cid:46), (cid:47) ) given by (4.4). Lemma 4.3.
Let k be a field of characteristic (cid:54) = 2 . Then there exists a bijection DM (cid:0) l (2 n + 1 , k ) , kH | ( (cid:46), (cid:47) ) (cid:1) ∼ = (cid:0) k n \ { } (cid:1) (cid:116) (cid:0) k n × k (cid:1) . The bijection is given such that the deformation map r = r a : l (2 n + 1 , k ) → kH associated to a = ( a i ) ∈ k n \ { } is given by r ( E i ) := a i H, r ( F i ) := 0 , r ( G ) := H, (4.5) while the deformation map r = r ( b,c ) : l (2 n + 1 , k ) → kH associated to ( b = ( b i ) , c ) ∈ k n × k isgiven as follows r ( E i ) := 0 , r ( F i ) := b i H, r ( G ) := cH (4.6) for all i = 1 , . . . , n . Proof .
Any linear map r : l (2 n + 1 , k ) → kH is uniquely determined by a triple ( a = ( a i ) , b =( b i ) , c ) ∈ k n × k n × k via: r ( E i ) := a i H , r ( F i ) := b i H and r ( G ) := cH , for all i = 1 , . . . , n . Weneed to check under what conditions such a map r = r ( a,b,c ) is a deformation map. Since kH isAbelian, equation (2.2) comes down to r ([ x, y ]) = r (cid:0) y (cid:47) r ( x ) − x (cid:47) r ( y ) (cid:1) + x (cid:46) r ( y ) − y (cid:46) r ( x ) , (4.7)which needs to be checked for all x, y ∈ { E i , F i , G | i = 1 , . . . , n } . Notice that (4.7) is symmetricali.e. if (4.7) is fulfilled for ( x, y ) then (4.7) is also fulfilled for ( y, x ). By a routinely computationit can be seen that r = r ( a,b,c ) is a deformation map if and only if a i b j = 0 , (1 − c ) a i = 0 (4.8)for all i, j = 1 , . . . , n . Indeed, (4.7) holds for ( x, y ) = ( E i , F j ) if and only if a i b j = 0 and it holdsfor ( x, y ) = ( E i , G ) if and only if a i = a i c . The other cases left to study are either automaticallyfulfilled or equivalent to one of the two conditions above. The first condition of (4.8) divides thedescription of deformation maps into two cases: the first one corresponds to a = ( a i ) (cid:54) = 0 andwe automatically have b = 0 and c = 1. The second case corresponds to a := 0 which impliesthat (4.8) holds for any ( b, c ) ∈ k n × k . (cid:4) The next result describes all deformations of l (2 n + 1 , k ) associated to the canonical matchedpair ( kH, l (2 n + 1 , k ) , (cid:46), (cid:47) ) given by (4.4). Proposition 4.4.
Let k be a field of characteristic (cid:54) = 2 and the extension of Lie algebras kH ⊂ L (2 n + 2 , k ) . Then a Lie algebra C is a complement of kH in L (2 n + 2 , k ) if and only if C is isomorphic to one of the Lie algebras from the three families defined below: • the Lie algebra l ( a ) (2 n + 1 , k ) having the bracket def ined for any i = 1 , . . . , n by [ E i , E j ] a := a i E j − a j E i , [ E i , F j ] a := − a i F j , [ E i , G ] a := − a i G (4.9) for all a = ( a i ) ∈ k n \ { } . • the Lie algebra l (cid:48) ( b ) (2 n + 1 , k ) having the bracket def ined for any i = 1 , . . . , n by [ E i , F j ] b := − b j E i , [ E i , G ] b := − E i , [ F i , F j ] b := b j F i − b i F j , [ F i , G ] b := F i − b i G for all b = ( b i ) ∈ k n . • the Lie algebra l (cid:48)(cid:48) ( b ) (2 n + 1 , k ) having the bracket def ined for any i = 1 , . . . , n by [ E i , F j ] b := − b j E i , [ F i , F j ] b := b j F i − b i F j , [ F i , G ] b := − b i G for all b = ( b i ) ∈ k n .Thus the factorization index [ L (2 n + 2 , k ) : kH ] f is equal to the number of types of isomor-phisms of Lie algebras of the set { l ( a ) (2 n + 1 , k ) , l (cid:48) ( b ) (2 n + 1 , k ) , l (cid:48)(cid:48) ( b ) (2 n + 1 , k ) | a ∈ k n \ { } , b ∈ k n } . Proof . l (2 n + 1 , k ) is a complement of kH in L (2 n + 2 , k ) and we can write L (2 n + 2 , k ) = kH (cid:46)(cid:47) l (2 n + 1 , k ), where the bicrossed product is associated to the matched pair given in (4.4).Hence, by [3, Theorem 4.3] any other complement C of kH in L (2 n + 2 , k ) is isomorphic toan r -deformation of l (2 n + 1 , k ), for some deformation map r : l (2 n + 1 , k ) → kH of the matchedpair (4.4). These are described in Lemma 4.3. The Lie algebra l ( a ) (2 n + 1 , k ) is precisely the r a -deformation of l (2 n + 1 , k ), where r a is given by (4.5). On the other hand the r ( b,c ) -deformationof l (2 n + 1 , k ), where r ( b,c ) is given by (4.6) for some ( b = ( b i ) , c ) ∈ k n × k , is the Lie algebradenoted by l ( b,c ) (2 n + 1 , k ) having the bracket given for any i = 1 , . . . , n by[ E i , F j ] ( b,c ) := − b j E i , [ E i , G ] ( b,c ) := (1 − c ) E i , [ F i , F j ] ( b,c ) := b j F i − b i F j , [ F i , G ] ( b,c ) := ( c − F i − b i G for all ( b = ( b i ) , c ) ∈ k n × k . Now, for c (cid:54) = 1 we can see that l ( b,c ) (2 n + 1 , k ) ∼ = l (cid:48) ( b ) (2 n + 1 , k ) (bysending G to ( c − − G ) while l ( b, (2 n + 1 , k ) = l (cid:48)(cid:48) ( b ) (2 n + 1 , k ) and we are done. (cid:4) Remark 4.5.
An attempt to compute [ L (2 n + 2 , k ) : kH ] f for an arbitrary integer n is hopeless.However, one can easily see that l (cid:48) (0) (2 n + 1 , k ) = l (2 n + 1 , k ) and l (cid:48)(cid:48) (0) (2 n + 1 , k ) = k n +10 , theAbelian Lie algebra of dimension 2 n + 1. Thus, [ L (2 n + 2 , k ) : kH ] f ≥
2. The case n = 1 ispresented below. Example 4.6.
Let k be a field of characteristic (cid:54) = 2 and consider { E, F, G } the basis of l (3 , k )with the bracket given by [ E, G ] = E and [ G, F ] = F . Then, the factorization index [ L (4 , k ) : kH ] f = 3. More precisely, the isomorphism classes of all complements of kH in L (4 , k ) arerepresented by the following three Lie algebras: l (3 , k ), k and the Lie algebra L − having { E, F, G } as a basis and the bracket given by[ F, E ] = F, [ E, G ] = − G. Since char( k ) (cid:54) = 2 the Lie algebras l (3 , k ) and L − are not isomorphic [9, Exercise 3.2]. For a ∈ k ∗ the Lie algebra l ( a ) (3 , k ) has the bracket given by [ E, F ] = − aF and [ E, G ] = − aG .Thus, l ( a ) (3 , k ) ∼ = l (1) (3 , k ), and the latter is isomorphic to the Lie algebra L − . On the otherhand we have: l (cid:48)(cid:48) (0) (3 , k ) = k and for b (cid:54) = 0 we can easily see that l (cid:48)(cid:48) ( b ) (3 , k ) ∼ = l (cid:48)(cid:48) (1) (3 , k ) ∼ = l (3 , k ).Finally, l (cid:48) (0) (3 , k ) = l (3 , k ) and for b (cid:54) = 0 we have that l (cid:48) ( b ) (3 , k ) ∼ = l (cid:48) (1) (3 , k ) – the latter is the Liealgebra having { f , f , f } as a basis and the bracket given by [ f , f ] = − f , [ f , f ] = f and[ f , f ] = f + f . This Lie algebra is also isomorphic to l (3 , k ), via the isomorphism which sends f to E , f to G and f to F − G .atched Pair Deformations and the Factorization Index 13Let k be a field of characteristic (cid:54) = 2 and l (2 n + 1 , k ) ( A,D,δ ) the Lie algebra of Theorem 4.2.In order to simplify computations we will assume A = D := I n and δ := (1 , , . . . , , m (2 n + 2 , k ) := l (2 n + 1 , k ) ( I n ,I n , (1 , ,..., , be the (2 n + 2)-dimensional Lie algebra having { E i , F i , G, H | i = 1 , . . . , n } as a basis and the bracket defined for any i = 1 , . . . , n by[ E i , G ] = E i , [ G, F i ] = F i , [ E i , H ] = E i , [ F i , H ] = F i , [ G, H ] = E + F n . We consider the Lie algebra extension kH ⊂ m (2 n + 2 , k ), where kH ∼ = k is the AbelianLie algebra of dimension 1. Of course, m (2 n + 2 , k ) factorizes through kH and l (2 n + 1 , k ),i.e. m (2 n +2 , k ) = kH (cid:46)(cid:47) l (2 n +1 , k ). Moreover, the canonical matched pair (cid:47) : l (2 n +1 , k ) × kH → l (2 n + 1 , k ) and (cid:46) : l (2 n + 1 , k ) × kH → kH associated to this factorization is given as follows: E i (cid:47) H := E i , F i (cid:47) H := F i , G (cid:47) H := E + F n (4.10)and all undefined actions are zero. In particular, we should notice that the left action (cid:46) : l (2 n + 1 , k ) × kH → kH is trivial. Next, we describe the set DM ( l (2 n + 1 , k ) , kH | ( (cid:46), (cid:47) )) of alldeformation maps of the matched pair ( kH, l (2 n + 1 , k ) , (cid:46), (cid:47) ) given by (4.10). Lemma 4.7.
Let k be a field of characteristic (cid:54) = 2 . Then there exists a bijection DM (cid:0) l (2 n + 1 , k ) , kH (cid:12)(cid:12) ( (cid:46), (cid:47) ) (cid:1) ∼ = (cid:0) k n \ { } (cid:1) (cid:116) (cid:0) k n \ { } (cid:1) (cid:116) k. The bijection is given such that the deformation map r = r a : l (2 n + 1 , k ) → kH associated to a = ( a i ) ∈ k n \ { } is given by r ( E i ) := a i H, r ( F i ) := 0 , r ( G ) := ( a − H (4.11) the deformation map r = r b : l (2 n + 1 , k ) → kH associated to another b = ( b i ) ∈ k n \ { } isgiven by r ( E i ) := 0 , r ( F i ) := b i H, r ( G ) := ( b n + 1) H, (4.12) while the deformation map r = r c : l (2 n + 1 , k ) → kH associated to c ∈ k is given by r ( E i ) := 0 , r ( F i ) := 0 , r ( G ) := cH (4.13) for all i = 1 , . . . , n . Proof .
Any linear map r : l (2 n + 1 , k ) → kH is uniquely determined by a triple ( a = ( a i ), b = ( b i ) , c ) ∈ k n × k n × k via: r ( E i ) := a i H , r ( F i ) := b i H and r ( G ) := cH , for all i = 1 , . . . , n .We only need to check when such a map r = r ( a,b,c ) is a deformation map. Since kH is theAbelian Lie algebra and the left action (cid:46) : l (2 n + 1 , k ) × kH → kH is trivial, equation (2.2)comes down to r ([ x, y ]) = r (cid:0) y (cid:47) r ( x ) − x (cid:47) r ( y ) (cid:1) . (4.14)Since (4.14) is symmetrical it is enough to check it only for pairs of the form ( E i , E j ), ( F i , F j ),( E i , F j ), ( E i , G ), and ( F i , G ), for all i, j = 1 , . . . , n . It is straightforward to see that (4.14) istrivially fulfilled for the pairs ( E i , E j ), ( F i , F j ) and ( E i , F j ). Moreover, (4.14) evaluated for( E i , G ) and respectively ( F i , G ) yields a i ( a + b n − c −
1) = 0 and b i ( a + b n − c + 1) = 0 forall i = 1 , . . . , n . Therefore, keeping in mind that we work over a field of characteristic (cid:54) = 2, thetriples ( a = ( a i ) , b = ( b i ) , c ) ∈ k n × k n × k for which r ( a,b,c ) becomes a deformation map are givenas follows: ( a = ( a i ) ∈ k n \ { } , b = 0 , c = a − a = 0 , b = ( b i ) ∈ k n \ { } , c = b n + 1) and( a = 0 , b = 0 , c ∈ k ). The corresponding deformation maps are exactly those listed above. (cid:4) l (2 n + 1 , k ) associated to the canonical matchedpair ( kH, l (2 n + 1 , k ) , (cid:46), (cid:47) ) given by (4.10). Proposition 4.8.
Let k be a field of characteristic (cid:54) = 2 and the extension of Lie algebras kH ⊂ m (2 n + 2 , k ) . Then a Lie algebra C is a complement of kH in m (2 n + 2 , k ) if and only if C is isomorphic to one of the Lie algebras from the three families defined below: • the Lie algebra l ( a ) (2 n + 1 , k ) having the bracket def ined for any i = 1 , . . . , n by [ E i , E j ] a := a j E i − a i E j , [ E i , F j ] a := − a i F j , [ E i , G ] a := a E i − a i ( E + F n ) , [ G, F i ] a := (2 − a ) F i for all a = ( a i ) ∈ k n \ { } . • the Lie algebra l (cid:48) ( b ) (2 n + 1 , k ) having the bracket def ined for any i = 1 , . . . , n by [ F i , F j ] b := b j F i − b i F j , [ E i , F j ] b := b j E i , [ E i , G ] b := (2 + b n ) E i , [ G, F i ] b := b i ( E + F n ) − b n F i for all b = ( b i ) ∈ k n \ { } . • the Lie algebra l (cid:48)(cid:48) ( c ) (2 n + 1 , k ) having the bracket def ined for any i = 1 , . . . , n by [ E i , G ] c := (1 + c ) E i , [ G, F i ] c := (1 − c ) F i for all c ∈ k .Thus the factorization index [ m (2 n + 2 , k ) : kH ] f is equal to the number of types of isomor-phisms of Lie algebras of the set (cid:8) l ( a ) (2 n + 1 , k ) , l (cid:48) ( b ) (2 n + 1 , k ) , l (cid:48)(cid:48) ( c ) (2 n + 1 , k ) | a, b ∈ k n \ { } , c ∈ k (cid:9) . Proof .
As in the proof of Proposition 4.4 we make use of [3, Theorem 4.3]. More precisely,this implies that all complements C of kH in m (2 n + 2 , k ) are isomorphic to an r -deformation of l (2 n + 1 , k ), for some deformation map r : l (2 n + 1 , k ) → kH of the matched pair (4.10). Theseare described in Lemma 4.7. By a straightforward computation it can be seen that l ( a ) (2 n + 1 , k )is exactly the complement corresponding to the deformation map given by (4.11), l (cid:48) ( b ) (2 n + 1 , k )corresponds to the deformation map given by (4.12) while l (cid:48)(cid:48) ( c ) (2 n + 1 , k ) is implemented by thedeformation map given by (4.13). (cid:4) Example 4.9.
Let k be a field of characteristic (cid:54) = 2. Then, the factorization index [ m (4 , k ) : kH ] f depends essentially on the field k . We will prove that all complements of kH in m (4 , k )are isomorphic to a Lie algebra of the form: L α : [ x, z ] = x, [ y, z ] = αy, with α ∈ k. Hence, [ m (4 , k ) : kH ] f = ∞ , if | k | = ∞ and [ m (4 , k ) : kH ] f = (1 + p n ) /
2, if | k | = p n , where p ≥ n = 1, the Lie algebras described in Proposition 4.8 become l ( a ) (3 , k ) : [ E, F ] a := − aF, [ E, G ] a := − aF, [ G, F ] a := (2 − a ) F, l (cid:48) ( b ) (3 , k ) : [ E, F ] b := bE, [ E, G ] b := (2 + b ) E, [ G, F ] b := bE, l (cid:48)(cid:48) ( c ) (3 , k ) : [ E, G ] c := (1 + c ) E, [ G, F ] c := (1 − c ) F,a, b ∈ k ∗ , c ∈ k . To start with, we should notice that the first two Lie algebras l ( a ) and l (cid:48) ( b ) areisomorphic for all a, b ∈ k ∗ . The isomorphism γ : l ( a ) → l (cid:48) ( b ) is given as follows γ ( E ) := 2 − ( b − a ) E + 2 − ( b − a + 2) F + 2 − ( a − b ) G, γ ( F ) := E,γ ( G ) := 2 − ( b − a + 4) E + 2 − ( b − a + 4) F + 2 − ( a − b − G. atched Pair Deformations and the Factorization Index 15Moreover, the map ϕ : l ( a ) → L given by ϕ ( E ) := y + az, ϕ ( F ) := x, ϕ ( G ) := x + y + ( a − z is an isomorphism of Lie algebras for all a ∈ k ∗ . Therefore, the first two Lie algebras are bothisomorphic to L for all a, b ∈ k ∗ . We are left to study the family l (cid:48)(cid:48) ( c ) . If c = − l (cid:48)(cid:48) ( − is again isomorphic to L . Suppose now that c (cid:54) = −
1. Then the map ψ : l (cid:48)(cid:48) ( c ) → L ( c − c +1) − given by ψ ( E ) := x, ψ ( F ) := y, ψ ( G ) := ( c + 1) z is an isomorphism of Lie algebras. Finally, we point out here that if α / ∈ { β, β − } then L α isnot isomorphic to L β (see, for instance [9, Exercise 3.2]) and the conclusion follows. Remark 4.10.
We end this section with two more applications. The deformation of a given Liealgebra h associated to a matched pair ( g , h , (cid:46), (cid:47) ) of Lie algebras and to a deformation map r as defined by (2.3) is a very general method of constructing new Lie algebras out of a given Liealgebra. It is therefore natural to ask if the properties of a Lie algebra are preserved by thisnew type of deformation. We will see that in general the answer is negative. First of all weremark that the Lie algebra h := l (2 n + 1 , k ) is metabelian, that is [[ h , h ] , [ h , h ]] = 0. Now, if welook at the matched pair deformation h r = l ( a ) (2 n + 1 , k ) of h given by (4.9) of Proposition 4.4,for a = ( a i ) ∈ k n \ { } we can easily see that l ( a ) (2 n + 1 , k ) is not a metabelian Lie algebra,but a 3-step solvable Lie algebra. Thus the property of being metabelian is not preserved bythe r -deformation of a Lie algebra.Next we consider an example of a somewhat different nature. First recall [18] that a Liealgebra h is called self-dual (or metric ) if there exists a non-degenerate invariant bilinear form B : h × h → k , i.e. B ([ a, b ] , c ) = B ( a, [ b, c ]), for all a, b, c ∈ h . Self-dual Lie algebras generalizefinite-dimensional complex semisimple Lie algebras (the second Cartan’s criterion shows thatany finite-dimensional complex semisimple Lie algebra is self-dual since its Killing form is non-degenerate and invariant). Besides the mathematical interest in studying self-dual Lie algebras,they are also important and have been intensively studied in physics [10, 20]. Now, h := l (2 n + 1 , k ) is not a self-dual Lie algebra since if B : l (2 n + 1 , k ) × l (2 n + 1 , k ) → k is an arbitraryinvariant bilinear form then we can easily prove that B ( E i , − ) = 0 and thus any invariant formis degenerate. On the other hand, the r -deformation of l (2 n + 1 , k ) denoted by l (cid:48)(cid:48) (0) (2 n + 1 , k ) inRemark 4.5 is self-dual since it is just the (2 n + 1)-dimensional Abelian Lie algebra. The paper is devoted to the factorization problem and its converse, the classifying complementsproblem, at the level of Lie algebras. Both problems are very difficult ones; even the caseconsidered in this paper, namely g = k , illustrates the complexity of the two problems. We endthe paper with the following two open questions: Question 1.
Let n ≥ . Does there exist a Lie algebra h and a matched pair of Lie algebras ( gl ( n, k ) , h , (cid:47), (cid:46) ) such that gl ( n, k ) (cid:46)(cid:47) h ∼ = gl ( n + 1 , k ) ? A more restricted version of this question is the following: does the canonical inclusion gl ( n, k ) (cid:44) → gl ( n + 1 , k ) have a complement that is a Lie subalgebra of gl ( n + 1 , k )? Although itseems unlikely for such a complement to exist we could not find any proof or reference to thisproblem in the literature.Secondly, having [2, Corollary 3.2] as a source of inspiration we ask: Question 2.
Let n ≥ . Does there exist a matched pair of Lie algebras ( g , h , (cid:47), (cid:46) ) suchthat any n -dimensional Lie algebra L is isomorphic to an r -deformation of h associated to thismatched pair? S n +1 by S n and the cyclic group C n . Acknowledgements
We would like to thank the referees for their comments and suggestions that substantiallyimproved the first version of this paper. A.L. Agore is research fellow ‘Aspirant’ of FWO-Vlaanderen. This work was supported by a grant of the Romanian National Authority forScientific Research, CNCS-UEFISCDI, grant no. 88/05.10.2011.
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