Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-perfect case the classification of these Lie algebras is a difficult task. Let $\mathfrak{l} (2n+1, k)$ be the Lie algebra with the bracket $[E_i, G] = E_i$, $[G, F_i] = F_i$, for all $i = 1, \dots, n$. We explicitly describe all Lie algebras containing $\mathfrak{l} (2n+1, k)$ as a subalgebra of codimension $1$ by computing all possible bicrossed products $k \bowtie \mathfrak{l} (2n+1, k)$. They are parameterized by a set of matrices ${\rm M}_n (k)^4 \times k^{2n+2}$ which are explicitly determined. Several matched pair deformations of $\mathfrak{l} (2n+1, k)$ are described in order to compute the factorization index of some extensions of the type $k \subset k \bowtie \mathfrak{l} (2n+1, k)$. We provide an example of such extension having an infinite factorization index.


Introduction
The theory of Lie algebras is among the most developed fields in algebra due to his broad applicability in differential geometry, theoretical physics, quantum field theory, classical or quantum mechanics and others. Besides the purely algebraic interest in this problem, the classification of Lie algebras of a given dimension is a central theme of study in modern group analysis of differential equations -for further explanations and an historical background see [21]. The Levi-Malcev theorem reduces the classification of all finite-dimensional Lie algebras over a field of characteristic zero to the following three subsequent problems: (1) the classification of all semi-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 are direct sums of semi-simple Lie algebras and solvable Lie algebras.
Surprisingly, among these three problems, the last one is the least studied and the most difficult. 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 h called the arXiv:1312.4018v2 [math.RA] 16 Jun 2014 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 be equivalently 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, , ) and classify up to an isomorphism all associated bicrossed products g h. Leaving aside the semi-simple/solvable case this is just the factorization problem for Lie algebras -we refer to [1] for more details and additional 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 its general form, namely the factorization problem and its converse, called the classifying complement 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 = {0}), describe explicitly, classify all complements and compute the cardinal of the isomorphism classes of all complements (which will be called the factorization index [L : g] f of g in L). Our starting point 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 is parameterized by the space TwDer(h) of twisted derivations of h. The pioneer work on this subject was performed by K.H. Hofmann: [12, Theorem I] describes the structure of n-dimensional real Lie algebras containing a given subalgebra of dimension n − 1. Equivalently, this proves that the set of all matched pairs of Lie algebras (k 0 , h, , ) (by k 0 we will denote the Abelian Lie algebra of dimension 1) and the space TwDer(h) of all twisted derivations of h are in one-to-one correspondence; moreover, any Lie algebra L containing h as a subalgebra of codimension 1 is isomorphic to a bicrossed product k 0 h = h (λ,∆) , for some (λ, ∆) ∈ TwDer(h). The classification 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 automorphisms of such Lie algebras is fully described in Corollary 3.3: it appears as a subgroup of a certain semidirect product h (k * × Aut Lie (h)) of groups. At this point we mention that the classification of automorphisms groups of all indecomposable real Lie algebras of dimension up to five was obtained recently in [11] where the importance of this subject in mathematical physics is 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 subalgebra of codimension one, namely the direct product k 0 × h and Aut Lie (k 0 × h) ∼ = k * × Aut Lie (h). Now, k 0 is a subalgebra of k 0 h = h (λ,∆) having h as a complement: for a 5-dimensional perfect Lie algebra all complements of k 0 in h (λ,∆) are described in Example 3.7 as matched pair deformations of h. Section 4 treats the same problem for a given (2n + 1)-dimensional non-perfect Lie algebra h := l(2n + 1, k). Theorem 4.2 describes explicitly all Lie algebras containing l(2n + 1, k) as a subalgebra of codimension 1. They are parameterized by a set T (n) of matrices (A, B, C, D, λ 0 , δ) ∈ M n (k) 4 × k × k 2n+1 : there are four such families of Lie algebras if the characteristic of k is = 2 and two families in characteristic 2. All complements of k 0 in two such bicrossed products k 0 l(2n + 1, k) are described by computing all matched pair deformations of the Lie algebra l(2n + 1, k) in Propositions 4.4 and 4. 8. In particular, in Example 4.6 we construct an example where the factorization index of k 0 in the 4-dimensional Lie algebra m(4, k) is infinite: that is k 0 has an infinite family of non-isomorphic complements in m(4, k). To conclude, there are three reasons for which we considered the Lie algebra l(2n+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(2n + 1, k) serves for constructing two counterexamples in Remark 4.10 which show that some properties of Lie algebras are not preserves by the matched pair deformation. Finally, having [2, Corollary 3.2] as a source of inspiration we believe that any (2n + 1)-dimensional Lie algebra is isomorphic to an r-deformation of l(2n + 1, k) associated to a given matched pair: a more general open question is stated at the end of the paper.

Preliminaries
All vector spaces, Lie algebras, linear or bilinear maps are over an arbitrary field k. The Abelian Lie algebra of dimension n will be denoted by k n 0 . For two given Lie algebras g and h we denote by Aut Lie (g) the group of automorphisms of g and by Hom Lie (g, h) the space of all Lie algebra maps between g and h. A Lie algebra L factorizes through g and h if g and h are Lie subalgebras of L such that L = g + h and g ∩ h = {0}. In this case h is called a complement of g in L; if g is an ideal of L, then a complement h, if it exists, is unique being isomorphic to the quotient Lie algebra L/g. In general, if g is only a subalgebra of L, then we are very far from having unique complements; for a given extension g ⊂ L of Lie algebras, the number of types of isomorphisms of all complements of g in L is called the factorization index of g in L and is denoted by [L : g] fa theoretical formula for computing [L : g] f is given in [3,Theorem 4.5]. For basic concepts and unexplained notions on Lie algebras we refer to [9,13].
A matched pair of Lie algebras [15,17] is a system (g, h, , ) consisting of two Lie algebras g and h and two bilinear maps : h × g → g, : h × g → h such that (g, ) is a left h-module, (h, ) is a right g-module and the following compatibilities hold for all g, h ∈ g and x, y ∈ h Let (g, h, , ) be a matched pair of Lie algebras. Then g h := g × h, as a vector space, is a Lie algebra with the bracket defined by for all g, h ∈ g and x, y ∈ h, called the bicrossed product associated to the matched pair (g, h, , ). Any bicrossed product g h factorizes through g = g × {0} and h = {0} × h; the converse also holds [17, Proposition 8. 3.2]: if a Lie algebra L factorizes through g and h, then there exist an isomorphism of Lie algebras L ∼ = g h, where g h is the bicrossed product associated to the matched pair (g, h, , ) whose actions are constructed from the unique decomposition for all x ∈ h and g ∈ g. The matched pair (g, h, , ) defined by (2.1) is called the canonical matched pair associated to the factorization L = g + h.
for all x, y ∈ L. The linear map f : g h → g h, f (g, h) := (g, −h) is a complex product structure on any bicrossed product g 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 eigenspace corresponding to the eigenvalue ±1 of f , that is L ∼ = L + L − .
Let (g, h, , ) be a matched pair of Lie algebras. A linear map r : h → g is called a deformation map [3, Definition 4.1] of the matched pair (g, h, , ) if the following compatibility holds for any We denote by DM(h, g|( , )) the set of all deformation maps of the matched pair (g, h, , for all h ∈ h and a ∈ k = k 0 . The bicrossed product k 0 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 Lie algebra L contains h as a subalgebra of codimension 1 if and only if L is isomorphic to 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 basis and the bracket given for any i ∈ I by The bijection is given such that the Lie algebra map ϕ = ϕ (α,h,v) corresponding to (α, h, v) is given by the formula is an isomorphism of Lie algebras if and only if α = 0 and v ∈ Aut Lie (h).

Proof . Any linear map
We will prove that ϕ defined above is a Lie algebra map if and only if β is the trivial map, v is a Lie algebra map and (3.4) holds. It is enough to test the compatibility . Using this formula we obtain that (3.5) holds for (0, x) and (0, y) if and only if As h is perfect these two conditions are equivalent to the fact that β = 0 and v is a Lie algebra map. Finally, as β = 0, we can easily show that (3.5) Thus, for all a ∈ k and x ∈ h, we have By the first part of (3.6) for a = 1 we obtain αγ = 1 and thus α = 0 while the second part of (3.6) for a = 0 implies v bijective. To end with, assume that α = 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 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 is a morphism of groups. Thus, we can construct the semidirect product of groups h ϕ (k * × Aut Lie (h)) associated to ϕ. The next result shows that Aut Lie (h (∆) ) is isomorphic to a certain subgroup of the semidirect product of groups h ϕ (k * × Aut Lie (h)).
Furthermore, there exists an isomorphism of groups where G(h, ∆) is a group with respect to the following multiplication for all (α, h, v), (β, g, w) ∈ G(h, ∆). Moreover, the canonical map 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 Thus, Aut Lie (h (∆) ) is isomorphic to G(h, ∆) with the multiplication given by (3.7). The last assertion follows by a routine computation.
) admits a simpler description as follows where Z(h) is the center of h. Assume in addition that h has trivial center, i.e. Z(h) = {0}; it follows that there exists an isomorphism of groups Moreover, in this context, the multiplication given by (3.7) is precisely that of a direct product of groups.
A Lie algebra h is called complete (see [14,22] for examples and structural results on this class of Lie algebras) if h has trivial center and any derivation is inner. A complete and perfect Lie algebra is called sympathetic [7]: semisimple Lie algebras over a field of characteristic zero are sympathetic and there exists a sympathetic non-semisimple Lie algebra in dimension 25. For sympathetic 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 which acts as identity on h.
Corollary 3. 5. Let h be a sympathetic Lie algebra. Then up to an isomorphism of Lie algebras there exists only one Lie algebra that contains h as a Lie subalgebra of codimension one, namely the direct product k 0 × h of Lie algebras. Furthermore, there exists an isomorphism of groups Proof . Since h is perfect any Lie algebra that contains h as a Lie subalgebra of codimension 1 is isomorphic to h (D) , for some D ∈ Der(h). As h is also complete, any derivation is inner. Remark 3. 6. Let h be a perfect Lie algebra with a basis {e i | i ∈ I}, ∆ ∈ Der(h) a given derivation and consider the extension k 0 ⊆ h (∆) = k 0 h (∆) . In order to determine all complements of k 0 in h (∆) we have to describe the set of all deformation maps r : h → k 0 of the matched pair (3.2). A deformation map is completely determined by a family of scalars (a) i∈I satisfying the following compatibility condition for any i, j ∈ I r [e i , e j ] h = r a i ∆(e j ) − a j ∆(e i ) 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 Any complement of k 0 in h (∆) is isomorphic to such an h r . An explicit example in dimension 5 is given below.

The non-perfect case
In Section 3 we have described and classified all bicrossed products k 0 h for a perfect Lie algebra h; furthermore, Remark 3.6 and Example 3.7 describe all complements of k 0 in a given bicrossed product k 0 h. In this section we approach the same questions for a given non-perfect Lie algebra h := l(2n + 1, k), where l(2n + 1, k) is the (2n + 1)-dimensional Lie algebra with basis {E i , F i , G | i = 1, . . . , n} and bracket given for any i = 1, . . . , n by First, we shall describe all bicrossed products k 0 l(2n + 1, k): they will explicitly describe all Lie algebras which contain l(2n + 1, k) as a subalgebra of codimension 1. Then, as the second step, we shall find all r-deformations of the Lie algebra l(2n + 1, k), for two given extensions k 0 ⊆ k 0 l(2n + 1, k). Based on Proposition 3.1 we have to compute first the space TwDer(l(2n + 1, k)) of all twisted derivations.
. It remains to check the compatibility condition (3.1) for ∆, i.e.
An easy computation shows that (3.1) holds for (g, h) = (E i , G) if and only if the following equation holds which is equivalent to −λ 0 A = δ 2n+1 I n and (2 − λ 0 )C = 0, i.e. the first and the third equations from (4.1). A similar computation shows that (3.1) holds for (g, h) = (G, F i ) if and only if (2 + λ 0 )B = 0 and λ 0 D = δ 2n+1 I n and the proof is finished.
Let l(2n + 1, k) (A,B,C,D,λ 0 ,δ) be the bicrossed product k 0 l(2n + 1, k) associated to the matched pair given by the twisted derivation A = (a ji ), B = (b ji ), C = (c ji ), D = (d ji ), λ 0 , δ = (δ j ) ∈ T (n). From now on we will use the following convention: if one of the elements of the 6-tuple (A, B, C, D, λ 0 , δ) is equal to 0 then we will omit it when writing down the Lie algebra l(2n + 1, k) (A,B,C,D,λ 0 ,δ) . A basis of l(2n + 1, k) (A,B,C,D,λ 0 ,δ) will be denoted by . . , n}: these Lie algebras can be explicitly described by first computing the 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 0 is mandatory. For two sets X and Y we shall denote by X Y the disjoint union of X and Y . As a conclusion of the above results we obtain: and the four families of Lie algebras containing l(2n + 1, k) as a subalgebra of codimension 1 are the following: • the Lie algebra l 1 (2n + 1, k) (λ 0 ,δ) with the bracket given for any i = 1, . . . , n by • the Lie algebra l 2 (2n + 1, k) (A,D,δ) with the bracket given for any i = 1, . . . , n by • the Lie algebra l 3 (2n + 1, k) (C,δ) with the bracket given for any i = 1, . . . , n by • the Lie algebra l 4 (2n + 1, k) (B,δ) with the bracket given for any i = 1, . . . , n by and the two families of Lie algebras containing l(2n + 1, k) as a subalgebra of codimension 1 are the following: • the Lie algebra l 1 (2n + 1, k) (A,B,C,D,δ) with the bracket given for any i = 1, . . . , n by • the Lie algebra l 2 (2n + 1, k) (λ 0 ,δ) with the bracket given for any i = 1, . . . , n by 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 consider whether λ 0 belongs to the set {0, 2, −2}. In the case that char(k) = 2, the first Lie algebra listed is the bicrossed product which corresponds to the case when λ 0 / ∈ {0, 2, −2}. In this case, we can easily see that A, B, C, D, λ 0 , δ = (δ j ) ∈ T (n) if and only if B = C = 0, A = −λ −1 0 δ 2n+1 I n and D = λ −1 0 δ 2n+1 I n . The Lie algebra l 1 (2n + 1, k) (λ 0 ,δ) is exactly the bicrossed product k 0 l(2n + 1, k) corresponding to this twisted derivation. The Lie algebra l 2 (2n + 1, k) (A,D,δ) is the bicrossed product k 0 l(2n + 1, k) corresponding to the case λ 0 = 0 while the last two Lie algebras are the bicrossed products k 0 l(2n + 1, k) associated to the case when λ 0 = 2 and respectively λ 0 = −2.
The bijection is given such that the deformation map r = r a : l(2n + 1, k) → kH associated to a = (a i ) ∈ k n \ {0} is given by The next result describes all deformations of l(2n + 1, k) associated to the canonical matched pair (kH, l(2n + 1, k), , ) given by (4.4).
Lemma 4. 7. Let k be a field of characteristic = 2. Then there exists a bijection The bijection is given such that the deformation map r = r a : l(2n + 1, k) → kH associated to a = (a i ) ∈ k n \ {0} is given by  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) is trivially 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 1 + b n − c − 1) = 0 and b i (a 1 + b n − c + 1) = 0 for all i = 1, . . . , n. Therefore, keeping in mind that we work over a field of characteristic = 2, the triples (a = (a i ), b = (b i ), c) ∈ k n × k n × k for which r (a,b,c) becomes a deformation map are given as follows: (a = (a i ) ∈ k n \ {0}, b = 0, c = a 1 − 1), (a = 0, b = (b i ) ∈ k n \ {0}, c = b n + 1) and (a = 0, b = 0, c ∈ k). The corresponding deformation maps are exactly those listed above.
Remark 4. 10. We end this section with two more applications. The deformation of a given Lie algebra h associated to a matched pair (g, h, , ) 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 Lie algebra. It is therefore natural to ask if the properties of a Lie algebra are preserved by this new type of deformation. We will see that in general the answer is negative. First of all we remark that the Lie algebra h := l(2n + 1, k) is metabelian, that is [[h, h], [h, h]] = 0. Now, if we look at the matched pair deformation h r = l (a) (2n + 1, k) of h given by (4.9) of Proposition 4.4, for a = (a i ) ∈ k n \ {0} we can easily see that l (a) (2n + 1, k) is not a metabelian Lie algebra, but a 3-step solvable Lie algebra. Thus the property of being metabelian is not preserved by the r-deformation of a Lie algebra.
Next we consider an example of a somewhat different nature. First recall [18] that a Lie algebra 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 generalize finite-dimensional complex semisimple Lie algebras (the second Cartan's criterion shows that any finite-dimensional complex semisimple Lie algebra is self-dual since its Killing form is nondegenerate 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(2n + 1, k) is not a self-dual Lie algebra since if B : l(2n + 1, k) × l(2n + 1, k) → k is an arbitrary invariant bilinear form then we can easily prove that B(E i , −) = 0 and thus any invariant form is degenerate. On the other hand, the r-deformation of l(2n + 1, k) denoted by l (0) (2n + 1, k) in Remark 4.5 is self-dual since it is just the (2n + 1)-dimensional Abelian Lie algebra.

Two open questions
The paper is devoted to the factorization problem and its converse, the classifying complements problem, at the level of Lie algebras. Both problems are very difficult ones; even the case considered in this paper, namely g = k 0 , illustrates the complexity of the two problems. We end the paper with the following two open questions: Question 1. Let n ≥ 2. Does there exist a Lie algebra h and a matched pair of Lie algebras (gl(n, k), h, , ) such that gl(n, k) h ∼ = gl(n + 1, k)?
A more restricted version of this question is the following: does the canonical inclusion gl(n, k) → gl(n + 1, k) have a complement that is a Lie subalgebra of gl(n + 1, k)? Although it seems unlikely for such a complement to exist we could not find any proof or reference to this problem in the literature.
Secondly, having [2, Corollary 3.2] as a source of inspiration we ask: Question 2. Let n ≥ 2. Does there exist a matched pair of Lie algebras (g, h, , ) such that any n-dimensional Lie algebra L is isomorphic to an r-deformation of h associated to this matched pair?
At the level of groups, question 2 has a positive answer by considering the canonical matched pair associated to the factorization of S n+1 by S n and the cyclic group C n .