Bass-Serre theory for Lie algebras: a homological approach
aa r X i v : . [ m a t h . G R ] J a n BASS-SERRE THEORY FOR LIE ALGEBRAS: A HOMOLOGICALAPPROACH
D. H. KOCHLOUKOVA AND C. MARTÍNEZ-PÉREZA
BSTRACT . We develop a version of the Bass-Serre theory for Lie algebras(over a field k ) via a homological approach. We define the notion of fundamentalLie algebra of a graph of Lie algebras and show that this construction yieldsMayer-Vietoris sequences. We extend some well known results in group theoryto N -graded Lie algebras: for example, we show that one relator N -graded Liealgebras are iterated HNN extensions with free bases which can be used forcohomology computations and apply the Mayer-Vietoris sequence to give someresults about coherence of Lie algebras.
1. I
NTRODUCTION
The Bass-Serre Theory is one of the corner-stones of modern group theory. Ithas definitely contributed to the flourishing of geometric group theory, lead to manyapplications and it has also proven to be useful to find shorter and more elegantproofs of known results. From the point of view of group cohomology, in the coreof the Bass-Serre Theory lies the fact that associated to a group action on a treeone can build a short exact sequence that yields Mayer-Vitoris exact sequences. Inthis paper, see Section 2, we develop a Bass-Serre theory of Lie algebras which,if not otherwise stated are defined over a field k of arbitrary characteristic. In thecase of Lie algebras, the two basic constructions which are the building blocks ofthe Bass-Serre theory, i.e. free products with amalgamation and HNN-extensions,have been already considered in the literature ( [4, Chapter 4], [14], [23], [29]) andwe use them to define the fundamental Lie algebra of a graph of Lie algebras. Liealgebras do not act on trees but we show that associated to these constructions onealso has short exact sequences: Theorem A.
Let ∆ be a graph of Lie algebras with fundamental Lie algebra L = L ∆ . Then there is a short exact sequence of right U ( L ) -modules → ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L ) → ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L ) → k → , Here, U ( L ) is the universal enveloping algebra of the Lie algebra L , see Subsec-tion 2.3 for the rest of notation. As in the group case, this leads to Mayer-Vietorisexact sequences (Corollary 2.7) and allows us to extend to Lie algebras some classi-cal results for groups. The main feature of our Lie algebra version of the Bass-Serretheory is the use of homological language. However, there is no complete analoguefor Lie algebras of the Bass-Serre theory. Indeed in [26] Shirshov showed the ex-istence of Lie algebras such that their free Lie product has a Lie subalgebra that isnot free, is not isomorphic to any subalgebra of any of the factors and cannot bedecomposed as the free Lie product of any of its subalgebras. Note that Shirshov Mathematics Subject Classification. does not use homological language. Moreover, as Mikhalev, Umirbaev and Zolo-tykh showed in [28] there exist non-free Lie algebras (over a field of characteristic p >
2) of cohomological dimension 1.One of the classical applications of the Bass-Serre theory for groups is the studyof one-relator groups. One-relator Lie algebras are known to share many of the(co)homological properties of one-relator groups, for example it is known thatthey are of type FP ∞ and have cohomological dimension at most two [4, Theorem3.10.6, Corollary 3.10.7]. In the case of groups, both results can be shown by em-bedding one relator groups in iterated HNN extensions. We denote N = { , , . . . } .An N -graded Lie algebra is a Lie algebra L with a decomposition as vector space L = ⊕ i ∈ N L i , where [ L i , L j ] ⊆ L i + j . In Section 4 we show that in the N -graded case,one relator Lie algebras are iterated HNN-extensions. Theorem B.
Let L = h X | r i be a one relator N -graded Lie algebra with X finite,where the N -grading is induced by some weight function ω : X → Z + , i.e. r ∈ F ( X ) is homogeneous with respect to ω . Then L is an iterated HNN-extension of Liealgebras L = ( . . . ( A ∗ h n ) ∗ h n − ) . . . ) ∗ h such that A and all the associated Lie subalgebras in each of the HNN-extensionsare free. This result combined with Theorem A can be used for explicit computations incohomology.Coherence is another group theoretical property deeply related to Bass-Serretheory. For example, fundamental groups of graphs of groups where the edgegroups have the property that all its subgroups are finitely generated and all itsvertex groups are coherent are known to be coherent [31, Lemma 4.8, page 41] andan analogous result holds for the corresponding group rings [1], [19]. Droms [9]has characterised the right angled Artin groups A Γ which are coherent. Here weconsider similar results for Lie algebras and their universal enveloping algebras,using as a main tool the Mayer-Vietoris sequences constructed in Section 2. Theorem C.
Assume that the Lie algebra L is the fundamental Lie algebra of agraph of Lie algebras such that for the vertex Lie algebras L v the universal envelop-ing algebra U ( L v ) is coherent and for each edge algebra L e , U ( L e ) is Noetherian.Then U ( L ) is coherent. Let Γ be a finite simple graph with vertex set V ( Γ ) and edge set E ( Γ ) ⊂ V ( Γ ) × V ( Γ ) , we assume that there are no loops or double edges. Associated to Γ thereis a group called the right angled Artin group G Γ given by the presentation in thecategory of groups G Γ = h V ( Γ ) | [ u , v ] = { u , v } ∈ E ( Γ ) i and also a k -Lie algebra, we call the right angled Artin Lie algebra L Γ , given by thepresentation in the category of Lie algebras L Γ = h V ( Γ ) | [ u , v ] = { u , v } ∈ E ( Γ ) i . The algebra L Γ is naturally graded and generated in degree 1 by the elements of V ( Γ ) . Theorem D.
Let L Γ be a right angled Artin Lie algebra. Then the following con-ditions are equivalent :1) U ( L Γ ) is coherent;2) the graph Γ is chordal, i.e., it has no n-cycle embedded as a full subgraph forn ≥ ;3) every finitely generated N -graded subalgebra of L Γ is FP ∞ . The main examples of Lie algebras considered in the first sections of paper (one-relator and right angled Artin Lie algebras) are defined using presentations in termsof generators and relations in the category of Lie algebras and are precisely the Liealgebras associated to the descending central series of the analogous groups aftertensoring with ⊗ Z k . In all these cases one sees a very close relation between the(co)homological finiteness properties of the Lie algebra and those of the group.Theorem D shows that with respect to coherence N -graded Lie algebras resem-ble discrete groups. In section 6 we will see an example that shows that rightangled Artin Lie algebras do not always have the same properties as right angledArtin groups and it is linked with the fact that for N -graded Lie algebras Kuroshtype result does not hold. The fact that a Kurosh type result does not hold for gen-eral Lie algebras was observed earlier by Shirshov [26]. As well we will show anexample of N -graded Lie algebra that has infinitely many ends but does not splitas a free product of N -graded Lie algebras. It was shown by Feldman in [15] thatthe Shirshov example is a Lie algebra with infinitely many ends that is not a freeproduct but Shirshov’s example is not N -graded.Unless otherwise stated, all along the paper by module we mean rigth module. Acknowledgements
During the preparation of this work the first named au-thor was partially supported by CNPq grant 301779/2017-1 and by FAPESP grant2018/23690-6. The second named author was partially supported by PGC2018-101179-B-I00 and by Grupo Álgebra y Geometría, Gobierno de Aragón and Feder2014-2020 “Construyendo Europa desde Aragón".2. B
ASS -S ERRE THEORY FOR L IE ALGEBRAS
In this section we develop a version of Bass-Serre theory for Lie algebras. Inthe context of Lie algebras we can not talk about actions but we do have all the(co)homological consequences that in the case of groups one can obtain by usingactions on trees. More precisely, from the action of a group on a tree one can derivethe existence of certain exact sequences that can be used to prove (co)homologicalresults. Here, we will also derive similar sequences using different methods.We begin by considering the Lie algebra version of the two constructions whichare the basis for Bass-Serre theory: amalgamated free products and HNN exten-sions.Throughout the paper all Lie algebras are over a field k , for a Lie algebra L wedenote by U ( L ) the universal enveloping algebra of L and ε : U ( L ) → k denotes the augmentation map that sends L to 0 and is the identity on k . We saythat L has a presentation (in terms of generators and relations) h X | R i if L ≃ F / N , D. H. KOCHLOUKOVA AND C. MARTÍNEZ-PÉREZ where F is the free Lie algebra with a free basis X , R is a subset of F and N isthe ideal in F generated by R . Sometimes instead of R we write a set of relations { α j = β j | j ∈ J } that should be interpreted as R = { α j − β j | j ∈ J } . If M i = h X i | R i i is a Lie algebra for i ∈ I then h∪ i X i | ( ∪ i ∈ I R i ) ∪ { α j − β j } j ∈ J i is denoted by h∪ i ∈ I M i | α j = β j for j ∈ J i or by h M , . . . , M n | α j = β j for j ∈ J i if I = { , . . . , n } .2.1. Amalgamated Lie algebra products and the associated short exact se-quence.
Let L , L , L be Lie algebras and assume that we have monomorphisms σ : L → L , τ : L → L . The free amalgamated product of the Lie algebras L and L with amalgam L is the Lie algebra given by the presentation in terms ofgenerators and relations L = L ∗ L L = h L , L | σ ( a ) = τ ( a ) , a ∈ L i . Proposition 2.1.
Let L = L ∗ L L be a free amalgamated product of Lie algebras.Then there is an exact complex of right U ( L ) -modules → k ⊗ U ( L ) U ( L ) α −→ ( k ⊗ U ( L ) U ( L )) ⊕ ( k ⊗ U ( L ) U ( L )) β −→ k → , where α ( ⊗ λ ) = ( ⊗ λ , − ⊗ λ ) and β ( ⊗ λ , ⊗ λ ) = ε ( λ ) + ε ( λ ) .Proof. Note that by [4, Def. 4.2.1, Thm. 4.4.2], the canonical maps L , L → L ∗ L L are injective. By [14, Prop. 3] there is a short exact sequence of left U ( L ) -modules 0 → U ( L ) L ∂ −→ U ( L ) L ⊕ U ( L ) L ∂ −→ U ( L ) L → ∂ ( λ ) = ( λ , − λ ) and ∂ ( λ , λ ) = λ + λ for λ i ∈ U ( L ) L i , ≤ i ≤ , where we consider L as a Lie subalgebra of both L and L and consider L and L as Lie subalgebras of L . Similarly there is a short exact sequence of right U ( L ) -modules C : 0 → L U ( L ) → L U ( L ) ⊕ L U ( L ) → LU ( L ) → . The short exact sequence C can be embedded in a short exact sequence of right U ( L ) -modules C : 0 → U ( L ) → U ( L ) ⊕ U ( L ) → U ( L ) → λ ∈ U ( L ) to ( λ , − λ ) , and ( λ , λ ) ∈ U ( L ) ⊕ U ( L ) to λ + λ . Thus wehave a short exact sequence of complexes of right U ( L ) -modules0 → C → C → C / C → × C and C ) we deduce that the third complex C / C : 0 → k ⊗ U ( L ) U ( L ) → ( k ⊗ U ( L ) U ( L )) ⊕ ( k ⊗ U ( L ) U ( L )) → k → (cid:3) HNN-extensions of Lie algebras and the associated short exact sequence.
HNN extensions of Lie algebras were considered by Lichtman-Shirvani in [23] andindependently by Wasserman in [29].Let L be a Lie algebra and take A ≤ L a Lie subalgebra. A derivation d : A → L is a k -linear map such that d ([ a , b ]) = [ a , d ( b )] + [ d ( a ) , b ] for any a , b ∈ A . Given a derivation d : A → L we set(1) W = h L , t | [ t , a ] = d ( a ) for a ∈ A i . We call W a Lie algebra HNN-extension with base
Lie subalgebra L , associated Lie subalgebra A and stable letter t . By [29] the canonical map L → W is injective. Proposition 2.2.
Let W be an HNN extension Lie algebra W = h L , t | [ t , a ] = d ( a ) for a ∈ A i . Then there is a short exact sequence of right U ( W ) -modules → k ⊗ U ( A ) U ( W ) α −→ k ⊗ U ( L ) U ( W ) β −→ k → given by α ( ⊗ λ ) = ⊗ t λ and β ( ⊗ λ ) = ε ( λ ) , where ε : U ( W ) → k is theaugmentation map.Proof. The fact that β is surjective is obvious. It is also obvious that Im α ⊆ Ker β .Consider the augmentation map ε : U ( W ) → k . Its kernel is the augmentation ideal WU ( W ) = U ( W ) W , so we have a short exact sequence0 → WU ( W ) → U ( W ) ε −→ k → . Applying the right exact functor k ⊗ U ( L ) − to this sequence yields an exact se-quence k ⊗ U ( L ) ( WU ( W )) → k ⊗ U ( L ) U ( W ) β → k → β is the image of k ⊗ U ( L ) ( WU ( W )) . Take any element0 = b ∈ Ker ( β ) . It must be in the image of k ⊗ U ( L ) ( WU ( W )) , so it is expressibleas a sum of monomials of the form 1 ⊗ λ , where λ is a word written associativelyin the generators of W . As the generators of W are the generators of L togetherwith the element t , we see that for each 1 ⊗ λ = k ⊗ U ( L ) U ( W ) we may assume λ = t λ for some other monomial λ in the generators of W . This means that theelement b is a sum of monomials of the form 1 ⊗ t λ so it lies in Im ( α ) .We have to show that α is a monomorphism. Note that this will follow if weprove this: Claim.
For any λ ∈ U ( W ) such that t λ ∈ LU ( W ) , we have λ ∈ AU ( W ) . Let Γ be a (possibly infinite) generating set of the Lie algebra L containing agenerating set Γ of the Lie algebra A . Set Γ = Γ ∪ { t } . Let F be a free Lie algebra(over the field k ) with free basis Γ . Then F subjects onto W via a map that is theidentity on Γ . Let R be the kernel of this surjection, hence R is an ideal of F suchthat W = F / R . Consider the standard resolution of the trivial right U ( F ) -module k (2) 0 → ⊕ Γ U ( F ) → U ( F ) → k → − ⊗ U ( R ) k . Observe that U ( F ) is free as U ( R ) -module (viamultiplication) and that U ( W ) = U ( F ) ⊗ U ( R ) k . Therefore by the long exact se-quence in homology associated to (2) we get an exact sequence0 = Tor U ( R ) ( U ( F ) , k ) → Tor U ( R ) ( k , k ) → ⊕ Γ U ( W ) → U ( W ) → k → D. H. KOCHLOUKOVA AND C. MARTÍNEZ-PÉREZ and as Tor U ( R ) ( k , k ) ≃ R / [ R , R ] = R ab the exact sequence above is(3) 0 → R ab δ → ⊕ Γ U ( W ) δ → U ( W ) → k → . We analyze the map δ . We denote ⊕ Γ U ( W ) = ⊕ γ ∈ Γ e γ U ( W ) ⊕ e t U ( W ) , then δ ( e γ ) = γ for γ ∈ Γ and δ ( e t ) = t . Now, let λ ∈ U ( W ) be such that t λ ∈ LU ( W ) . This means that t λ = ∑ γ ∈ Γ γλ γ with λ γ ∈ U ( W ) . Then δ ( e t λ − ∑ γ ∈ Γ e γ λ γ ) = t λ − ∑ γ ∈ Γ γλ γ = , so by the exactness of (3) e t λ − ∑ γ ∈ Γ e γ λ γ = δ ( ω ) for some ω ∈ R ab . Note that any r ∈ R can be writen as a linear combination of Lie monomials on Γ and then in U ( W ) we can write r as ∑ i ,..., i k k i ,..., i k γ i . . . γ i k . Therefore δ ( r + [ R , R ]) = ∑ i ,..., i k k i ,..., i k e γ i γ i . . . γ i k Recall that the ideal R is generated by the relators of L and by relators of the form [ t , a ] − d ( a ) for a ∈ Γ . By the above description δ sends the images of the relatorsof L to elements of ∑ γ ∈ Γ e γ U ( W ) . Since [ t , a ] − d ( a ) = ta − at − d ( a ) we deducethat for r = ([ t , a ] − d ( a )) ◦ v where ◦ denotes the adjoint action, v ∈ U ( W ) and a ∈ Γ we have δ ( r + [ R , R ]) ∈ e t av − e a tv + ∑ γ ∈ Γ e γ U ( W ) . Hence e t λ − ∑ γ ∈ Γ e γ λ γ = δ ( ω ) ∈ ∑ a ∈ Γ ( e a t − e t a ) U ( W ) + ∑ γ ∈ Γ e γ U ( W ) ⊆ e t AU ( W ) + ∑ γ ∈ Γ e γ U ( W ) = e t AU ( W ) ⊕ ( ⊕ γ ∈ Γ e γ U ( W )) So λ ∈ AU ( W ) as we wanted to prove. (cid:3) Fundamental Lie algebra of a graph of Lie algebras and the associatedshort exact sequence. A finite graph of Lie algebras ∆ is a set ∆ = ∆ ( V ( ∆ ) , E ( ∆ ) , Γ , T , { L v } v ∈ V , { L e } e ∈ E ( Γ ) , { σ e } e ∈ E ( Γ ) , { τ e } e ∈ E ( T ) , { d e } e ∈ E ( Γ ) r E ( T ) ) with the following data. We have finite sets V ( ∆ ) (vertices) and E ( ∆ ) (edges) sothat elements of E ( ∆ ) are ordered pairs e = ( v , w ) in V ( ∆ ) × V ( ∆ ) . For an edge e ∈ E ( ∆ ) we denote by σ ( e ) (resp. τ ( e ) ) the beginning (resp. the end of e ). Asusual e is the inverse edge so that τ ( e ) = σ ( e ) and σ ( e ) = τ ( e ) and whenever e ∈ E ( ∆ ) , then also ¯ e ∈ E ( ∆ ) . A loop is an edge such that ¯ e = e . Γ is an underlyingoriented finite graph. It is a graph with vertex set V ( Γ ) = V ( ∆ ) and as edge set E ( Γ ) we choose exactly one of each pair { e , ¯ e } so that all loops are in Γ . We fix a maximal forest T in Γ i.e. T is a subgraph of Γ which is a disjoint union of treessuch that V ( T ) = V ( Γ ) and which is maximal under these conditions. For everyvertex v ∈ V ( Γ ) we have a Lie algebra L v and for every edge e ∈ E ( Γ ) we have aLie algebra L e and a monomorphism of Lie algebras σ e : L e → L σ ( e ) . Moreover, if e ∈ E ( T ) we also have a monomorphism τ e : L e → L τ ( e ) and if e ∈ E ( Γ ) r E ( T ) we have a derivation d e : L e → L τ ( e ) . Associated to a graph of Lie algebras as before, consider the Lie algebra givenby the presentation (in terms of generators and relators) L T = h∪ v ∈ V ( Γ ) L v | σ e ( a ) = τ e ( a ) for e ∈ E ( T ) , a ∈ L e i i.e. we make a free product with amalgamation for every edge e of T . We definethe fundamental Lie algebra of the graph of Lie algebras ∆ as the Lie algebra givenby the presentation L ∆ = h L T , { e } e ∈ A | [ e , σ e ( a )] = d e ( a ) for e ∈ A , a ∈ L e i Note that in the definition of the corresponding notion for groups the choice ofthe maximal forest is not important as changing the maximal tree produces anisomorphic group. In fact, there is more symmetry in the group case since insteadof derivations d e we have monomorphisms. In the Lie algebra case the situationis not symmetric since derivations are not homomorphisms of Lie algebras and inthe original graph of Lie algebras ∆ the maximal forest T and the orientation of theedges in E ( Γ ) r E ( T ) has to be fixed explicitly in order to define the derivations d e . The precise orientation of the edges in T is not important, though. Lemma 2.3.
Let ∆ be a finite graph of Lie algebras with fundamental Lie algebraL ∆ . Then for any vertex v the map L v → L ∆ is a monomorphism.Proof. This is known if Γ is a single edge because in that case the fundamentalLie algebra is either a free product with amalgamation or, in the case when theedge is a loop, an HNN extension. For HNN extensions it follows by the work ofWasserman that the base of the HNN extension embeds in the HNN extension [29].In the case of an amalgamated product of two Lie algebras each one of the two Liealgebras embeds in the amalgamated product by [4, Thm. 4.4.2].In the general case we can induct on the size of Γ which is | V ( Γ ) | + | E ( Γ ) | . Fixan edge e of T with vertices v and v . Consider a new graph e Γ obtained from Γ by squashing the edge e to a point v and let e T be the maximal forest of e Γ that is obtained from T by squashing the edge e to v . Then there is a bijectionbetween E ( Γ ) r E ( T ) and its image E ( e Γ ) r E ( e T ) and there is an obvious way toconstruct a new graph of Lie algebras e ∆ with underlying oriented graph e Γ wherefor v ∈ V ( Γ ) r { v , v } the associated Lie algebra in e ∆ is the old Lie algebra L v and L v = L v ∗ L e L v . The set of edges e E of e ∆ is in bijection with E ( ∆ ) r { e , e } andthe edge Lie algebras in e ∆ are defined to be the old ones i.e. { L e | e ∈ E r { e , e }} and the monomorphisms and the derivations from the edge Lie algebras in e ∆ tothe corresponding vertex Lie algebras in e ∆ are the old ones from ∆ combined if D. H. KOCHLOUKOVA AND C. MARTÍNEZ-PÉREZ necessary with the embeddings of L v and L v in L v = L v ∗ L e L v . Thus L ∆ = L e ∆ and, as e Γ has smaller size than Γ , by induction the canonical map from a Lie algebraassociated to a vertex in e ∆ to L e ∆ is injective. In particular this is the case for theLie algebra L v and as L v , L v also embed in L v we get the result.Finally if E ( T ) = ∅ we deduce that E ( ∆ ) contains only loops. Thus Γ = ∪ i Γ i isa disjoint union of graphs where each V ( Γ i ) contains precisely one vertex, say v i ,and we have the corresponding decomposition ∆ = ∪ i ∆ i of graphs of Lie algebras.Then L ∆ is the free product of the Lie algebras L ∆ i and each L ∆ i is obtained from L v i by applying several times HNN-extensions with stable letters corresponding toloops from E ( Γ i ) . Finally L v i embeds in L ∆ i and L ∆ i embeds in L ∆ . (cid:3) We can prove now Theorem A from the introduction.
Theorem 2.4.
Let ∆ be a finite graph of Lie algebras with fundamental Lie algebraL = L ∆ . Then there is a short exact sequence of right U ( L ) -modules → ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L ) → ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L ) → k → , Proof.
It suffices to prove the result when the underlying graph is just one edge(i.e. consider two cases: free product with amalgamation and HNN extension),then use the definition with the maximal forest and induction on the number ofvertices. The two basic cases free product with amalgamation and HNN extensionwere proved in Subsections 2.1 and 2.2. We explain the details below.
Case 1 . Suppose Γ = T . We induct on the size of T . The induction starts with T a vertex and the result is obvious in this case. Assume now that T is obtainedfrom a forest T by adding an edge e such that T and T have the same number ofconnected components. Let ∆ be the graph of Lie algebras with underlying graph Γ = T and vertex, edge Lie algebras as in ∆ and the same monomorphisms as in ∆ . Then for L = L ∆ and L = L ∆ we have L = L ∗ L e L v , where v is the vertex of e that is not a vertex of T . This gives a short exact sequence of U ( L ) -modules(4) 0 → k ⊗ U ( L e ) U ( L ) → k ⊗ U ( L v ) U ( L ) ⊕ k ⊗ U ( L ) U ( L ) → k → ∆ i.e. there is ashort exact sequence of U ( L ) -modules0 → ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L ) → ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L ) → k → − ⊗ U ( L ) U ( L ) we get the short exact sequence(5) 0 → ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L ) → ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L ) → k ⊗ U ( L ) U ( L ) → k ⊗ U ( L ) U ( L ) we obtain a short exact sequenceof U ( L ) -modules0 → k ⊗ U ( L e ) U ( L ) ⊕ ( ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L )) → k ⊗ U ( L v ) U ( L ) ⊕ ( ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L )) → k → Case 2 . We induct on the size of E ( Γ ) r E ( T ) , the case Γ = T being the startingpoint of the induction. Assume for the inductive step that Γ is a subgraph of Γ that contains T such that Γ is obtained from Γ by adding an edge e . We candefine as before ∆ to be the graph of Lie algebras with underlying graph Γ andall structural data: edge and vertex Lie algebras, the structure monomorphisms and derivations as in ∆ . Then for L = L ∆ and L = L ∆ we have that L is an HNNextension of L with stable letter e i.e. L = h L , e | [ e , σ e ( b )] = d e ( b ) for b ∈ L e i . Thus we have a short exact sequence of U ( L ) -modules(6) 0 → k ⊗ U ( L e ) U ( L ) → k ⊗ U ( L ) U ( L ) → k → . By induction the result holds for ∆ , hence there is a short exact sequence of U ( L ) -modules that after applying the exact functor − ⊗ U ( L ) U ( L ) yields the short exactsequence(7) 0 → ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L ) → ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L ) → k ⊗ U ( L ) U ( L ) → k ⊗ U ( L ) U ( L ) gives the exact se-quence0 → k ⊗ U ( L e ) U ( L ) ⊕ ( ⊕ e ∈ E ( Γ ) k ⊗ U ( L e ) U ( L )) → ⊕ v ∈ V ( Γ ) k ⊗ U ( L v ) U ( L ) → k → , which is precisely the result we want to prove. (cid:3) If L is a Lie algebra then U ( L ) is a Hopf algebra with comultiplication ∆ : U ( L ) → U ( L ) ⊗ U ( L ) given by ∆ ( g ) = ⊗ g + g ⊗ g ∈ L . This fact canbe used to define a U ( L ) -module structure in the k -tensor product of two right U ( L ) -modules U and V via ( u ⊗ v ) g : = ug ⊗ v + u ⊗ vg for g ∈ L , u ∈ U and v ∈ V . Moreover this combined with induction for a U ( T ) -module W and T ≤ L a Lie subalgebra yields an isomorphism of U ( L ) -modules(8) ( W ⊗ k V ) ⊗ U ( T ) U ( L ) ∼ = ( W ⊗ U ( T ) U ( L )) ⊗ k V . For arbitrary elements we use the comultiplication ∆ to define the isomorphism in(8), i.e. if for λ ∈ U ( L ) we have ∆ ( λ ) = ∑ i λ i , ⊗ λ i , , where λ i , j ∈ U ( L ) , the isomorphism from (8) is defined by ( w ⊗ v ) ⊗ λ ∑ i ( w ⊗ λ i , ) ⊗ v λ i , . In particular, (8) implies for W = k that(9) V ⊗ U ( T ) U ( L ) ∼ = ( k ⊗ U ( T ) U ( L )) ⊗ k V . Theorem 2.4 implies the following corollaries.
Corollary 2.5.
Let L be a Lie algebra and V a right U ( L ) -module. Then there is ashort exact sequence of right U ( L ) -modules (10) 0 → ⊕ e ∈ E ( Γ ) V ⊗ U ( L e ) U ( L ) → ⊕ v ∈ V ( Γ ) V ⊗ U ( L v ) U ( L ) → V → , and we get, for any left U ( L ) -module A, a long exact sequence → ⊕ e ∈ E ( Γ ) Tor U ( L e ) i ( V , A ) → ⊕ v ∈ V ( Γ ) Tor U ( L v ) i ( V , A ) → Tor Li ( V , A ) → ⊕ e ∈ E ( Γ ) Tor U ( L e ) i − ( V , A ) → . . . → Tor L ( V , A ) → ⊕ e ∈ E ( Γ ) Tor U ( L e ) ( V , A ) → ⊕ v ∈ V ( Γ ) Tor L v ( V , A ) → Tor L ( V , A ) → , and an analogous long exact sequence of Ext functors. Proof.
Applying the exact functor − ⊗ k V to the short exact sequence of Theorem2.4 together with (9) yields the short exact sequence (10). This induces, for anyleft U ( L ) -module A , a long exact sequence in Tor . . . → Tor U ( L ) i ( ⊕ e ∈ E ( Γ ) V ⊗ U ( L e ) U ( L ) , A ) → Tor U ( L ) i ( ⊕ v ∈ V ( Γ ) V ⊗ U ( L v ) U ( L ) , A ) → Tor U ( L ) i ( L , A ) → Tor U ( L ) i − ( ⊕ e ∈ E ( Γ ) V ⊗ U ( L e ) U ( L ) , A ) → . . . Note that Tor U ( L ) i ( ⊕ e ∈ E ( Γ ) V ⊗ U ( L e ) U ( L ) , A ) ≃ ⊕ e ∈ E ( Γ ) Tor U ( L ) i ( V ⊗ U ( L e ) U ( L ) , A ) and by a version of Shapiro Lemma for Lie algebras Tor U ( L ) i ( V ⊗ U ( L e ) U ( L ) , A ) ≃ Tor U ( L e ) i ( V , A ) . Similarly Tor U ( L ) i ( ⊕ v ∈ V ( Γ ) V ⊗ U ( L v ) U ( L ) , A ) ≃ ⊕ v ∈ V ( Γ ) Tor U ( L v ) i ( V , A ) .and we also get a long exact sequence → ⊕ e ∈ E ( Γ ) Tor U ( L e ) i ( V , A ) → ⊕ v ∈ V ( Γ ) Tor U ( L v ) i ( V , A ) → Tor U ( L ) i ( V , A ) →⊕ e ∈ E ( Γ ) Tor U ( L e ) i − ( V , A ) → . . . → Tor U ( L ) ( V , A ) → ⊕ e ∈ E ( Γ ) Tor U ( L e ) ( V , A ) →⊕ v ∈ V ( Γ ) Tor U ( L v ) ( V , A ) → Tor L ( V , A ) → . (cid:3) Remark . In the particular case when L = L v ∗ L e is an HNN-extension and the U ( L ) -module V is U ( L ) , (10) is0 → U ( L ) ⊗ U ( L e ) U ( L ) f → U ( L ) ⊗ U ( L v ) U ( L ) → U ( L ) → f is given by f ( ⊗ ) = ⊗ t − t ⊗
1. This mapcan be described as the linearization of the map that yields a similar short exactsequence for a ring HNN extension in [7, (1) pg. 438], where 1 ⊗ ⊗ − t ⊗ t − . Corollary 2.7.
Let ∆ be a graph of Lie algebras with fundamental Lie algebraL = L ∆ . Then for any right U ( L ) -module B and any left U ( L ) -module A there isa) a long exact sequence in homology → ⊕ e ∈ E ( Γ ) H i ( L e , A ) → ⊕ v ∈ V ( Γ ) H i ( L v , A ) → H i ( L , A ) → ⊕ e ∈ E ( Γ ) H i − ( L e , A ) → . . . → H ( L , A ) → ⊕ e ∈ E ( Γ ) H ( L e , A ) → ⊕ v ∈ V ( Γ ) H ( L v , A ) → H ( L , A ) → b) a long exact sequence in cohomology → H ( L , B ) → ⊕ v ∈ V ( Γ ) H ( L v , B ) → ⊕ e ∈ E ( Γ ) H ( L e , B ) → H ( L , B ) → . . . → H i ( L , B ) → ⊕ v ∈ V ( Γ ) H i ( L v , B ) → ⊕ e ∈ E ( Γ ) H i ( L e , B ) → ⊕ H i + ( L , B ) → . . . A Lie algebra L is of type FP n if the trivial U ( L ) -module k is FP n i.e. there isa projective resolution of the module where all projectives are finitely generated indimensions ≤ n . We note that type FP is equivalent to L being finitely generatedas a Lie algebra. If L is finitely presented in terms of generators and relations then L is FP . Whether the converse holds is an open problem. Lemma 2.8.
Let S ≤ L be a Lie subalgebra of an arbitrary Lie algebra L. Then Sis of type FP n if and only if the induced U ( L ) -module k ⊗ U ( S ) U ( L ) is of type FP n . Proof. If S is FP n there is a resolution P of the trivial U ( S ) -module k with finitelygenerated modules in dimensions ≤ n . Then P ⊗ U ( S ) U ( L ) is a resolution thatshows that k ⊗ U ( S ) U ( L ) is FP n as U ( L ) -module.For the converse suppose we use induction on n ≥
1, the case n = n > FP n − and suppose that k ⊗ U ( S ) U ( L ) is FP n . By induction S is FP n − . Then wehave an exact complex of U ( S ) -modules P : 0 → Ker ( d n ) → P n − d n −→ P n − → . . . → P → k → k is the trivial U ( S ) -module and P i is projective and finitely generated for i ≤ n −
1. Then we have an exact complex P ⊗ U ( S ) U ( L ) : 0 → Ker ( d n ) ⊗ U ( S ) U ( L ) → P n − ⊗ U ( S ) U ( L ) d n −→ P n − ⊗ U ( S ) U ( L ) → . . . → P ⊗ U ( S ) U ( L ) → k ⊗ U ( S ) U ( L ) → P i ⊗ U ( S ) U ( L ) is a finitely generated projective module and k ⊗ U ( S ) U ( L ) is FP n as U ( L ) -module we deduce by [6, Prop. 4.3] that Ker ( d n ) ⊗ U ( S ) U ( L ) is a fin-tely generated U ( L ) -module, hence Ker ( d n ) is finitely generated as U ( S ) -moduleand so S is FP n . (cid:3) Corollary 2.9.
Let ∆ be a graph of Lie algebras with fundamental Lie algebraL = L ∆ . Thena) if L e is FP m − for every e ∈ E ( Γ ) and L v is FP m for every v ∈ V ( Γ ) then L is FP m ;b) if L is FP m and L e is FP m for every e ∈ E ( Γ ) then L v is FP m for every v ∈ V ( Γ ) ;c) if L is FP m and L v is FP m for every v ∈ V ( Γ ) then L e is FP m − for everye ∈ E ( Γ ) .Proof. Let R be any associative ring with 1 and 0 → A → B → C → R -modules. Then by [5, Proposition 1.4]1) if A and C are FP m then B is FP m ;2) if A is FP m − and B is FP m then C is FP m ;3) if B and C are FP m then A is FP m − .The Corollary is a consequence of the above statements applied for R = U ( L ) together with Lemma 2.8 and the short exact sequence given by Theorem 2.4. (cid:3) The following two results are well known for groups and extend also to Liealgebras.
Proposition 2.10. a) [23] Let L be a Lie algebra HNN extension with a base Liesubalgebra L , associated Lie subalgebra L and stable letter t. Let H be a Liesubalgebra of L such that H ∩ L = and H ∩ L is a free Lie algebra. Then H isa free Lie algebra.b) [18] Let L = L ∗ L L be an amalgameted product of Lie algebras and H bea Lie subalgebra of L such that H ∩ L = and H ∩ L i is free for i = , . Then His a free Lie algebra. As a consequence we have
Theorem 2.11.
Let ∆ be a graph of Lie algebras with a fundamental Lie algebraL = L ∆ . If H intersects every edge Lie algebra L e trivially and every vertex Liealgebra in a free Lie subalgebra then H is a free Lie algebra.Proof. It follows by induction on the number of vertices of the underlying graphusing for the inductive step the previous proposition. (cid:3)
Note that the above theorem cannot be proved by cohomological reasoning sinceby [28] if char ( k ) = p ≥ U ( L ) is a free associative k -algebra, hence the projective dimension of the trivial U ( L ) -module k is 1 but L isnot free. Furthermore in [26] Shirshov showed the existence of Lie algebras suchthat their free Lie product has a Lie subalgebra that is not free, is not isomorphicto any subalgebra of any of the factors and cannot be decomposed as the free Lieproduct of any of its subalgebras. Thus we cannot have a very general Kurosh typedecomposition theorem generalising Theorem 2.11.2.4. An application: a resolution of the trivial module for right angled ArtinLie algebras.
Let Γ be a finite simple graph with vertex set V ( Γ ) and edge set E ( Γ ) , here simple means that there are no loops or double edges and we see eachedge as an unordered pair { v , w } with v , w ∈ V ( Γ ) . Associated to Γ there is a groupcalled the right angled Artin group G Γ given by the presentation by generators andrelations in the category of groups G Γ = h V ( Γ ) | [ u , v ] = { u , v } ∈ E ( Γ ) i and also a k -Lie algebra called the right angled Artin Lie algebra L Γ given by thepresentation by generators and relations in the category of Lie algebras L Γ = h V ( Γ ) | [ u , v ] = { u , v } ∈ E ( Γ ) i . Recall that the flag complex ∆ Γ associated to Γ is the simplicial complex withsimplices the (non-empty) ordered subsets of vertices of Γ that span a completesubgraph. The cone C ∆ Γ is the complex that one gets allowing the subsets ofvertices to be empty. We can linearly order the vertices in V ( Γ ) . Consider thefollowing complex(11) P Γ : . . . ∂ n + −→ P n ∂ n −→ P n − ∂ n − −→ . . . ∂ −→ P ∂ −→ k → , where P n = ⊕ w = { v i ,..., v in } c w U ( L ) and the direct sum is over the simplices of C ∆ Γ , i.e., over all (possibly empty)subsets w = { v i , . . . , v i n } of V ( Γ ) such that the subgraph of Γ that they span iscomplete. Every c w U ( L ) is isomorphic to the free right U ( L ) -module and we havefor the differential after assuming v i < . . . < v i n that ∂ n ( c w ) = ∑ r ( − ) r − c w r { v ir } v i r , c ∅ = K , P = c ∅ U ( L ) = U ( L ) and ∂ is the augmentation map. So (11) is a Liealgebra version of the projective resolution associated to the universal cover of theSalvetti complex for right angled Artin groups. This complex (11) is the minimalresolution of the Lie algebra L Γ and in particular is exact (see [2, proof of Theorem1.2, pages 11 and 12]). This fact can be understood as a Lie algebra version ofthe fact that for groups the Salvetti complex is a K ( G Γ , ) . In the next result we derive it using the short exact sequence for a free product with amalgamation of Liealgebras, note that our argument is different than the one in [2] and was obtainedindependently. Proposition 2.12. P Γ is a free resolution of the trivial U ( L ) -module K.Proof. We induct on the number of vertices in Γ . We consider first the case when Γ is a full graph. Then L = L Γ is abelian and P is the standard Koszul complex,so is exact.Assume now that Γ is not complete. Then we can find full proper subcomplexes Γ , Γ , Γ such that Γ ∩ Γ = Γ and Γ = Γ ∪ Γ . The presentation of L i = L Γ i , i = , , L = L ∗ L L . Then by induction P Γ i are all exact complexes for i = , , U ( L ) -modules0 → k ⊗ U ( L ) U ( L ) → ( k ⊗ U ( L ) U ( L )) ⊕ ( k ⊗ U ( L ) U ( L )) → k → k ⊗ λ to ( k ⊗ λ ) + ( k ⊗ − λ ) and the second sends k ⊗ λ + k ⊗ λ to k ε ( λ ) + k ε ( λ ) , where ε i : U ( L i ) → k is the augmentationmap. This short exact sequence extends to the short exact sequence of complexes(12)0 → P Γ ⊗ U ( L ) U ( L ) → ( P Γ ⊗ U ( L ) U ( L )) ⊕ ( P Γ ⊗ U ( L ) U ( L )) → P Γ → c w ⊗ c w ⊗ + ( c w ⊗ − ) and c w ⊗ λ + c w ⊗ λ goes to c w λ − c w λ i.e. it is induced by the natural embeddings of P Γ in P Γ i and of P Γ i in P Γ for i = , P Γ i are exact for i = , , P Γ i ⊗ U ( L i ) U ( L ) are exact for i = , ,
2. Thus (12) is a short exact sequence ofcomplexes, where the first two of the complexes are exact hence the last one, P Γ ,is exact too. (cid:3)
3. G
RADED L IE ALGEBRAS
Here we recall some results on graded Lie algebras and their homological prop-erties.We denote N = { , , . . . } and N = N ∪ { } . An N -graded Lie algebra is a Liealgebra L with a decomposition as vector space L = ⊕ i ∈ N L i , where [ L i , L j ] ⊆ L i + j . L is called an N -graded Lie algebra of finite type if each L i is finite dimensionalover k . Note that every finitely generated N -graded Lie algebra L is of finite type.It is easy to check whether an N -graded Lie algebra L is finitely generated, as it isequivalent to L / [ L , L ] ≃ H ( L , k ) is finite dimensional.The N -grading on L induces a natural N -grading on the universal envelopingalgebra R = U ( L ) of L defined by R = ⊕ i ∈ N R i , where R = K . R i = ∑ i + ... + i j = i , j ≥ L i . . . L i j An N -graded R -module V = ⊕ i ∈ N V i is defined by the property V i R j ⊆ V i + j . Ahomomorphism of R -modules ϕ : V = ⊕ i ≥ V i → W = ⊕ i ≥ W i between N -graded R -modules is called graded if ϕ ( V i ) ⊆ W i for every i .We state the version of the Nakayama Lemma for an arbitrary N -graded R -modules V : V = V ⊗ R k = V is finitely gen-erated as R -module if and only if V ⊗ R k is finite dimensional over k [30, Section2]. Lemma 3.1. [17] , [30] Let L be an N -graded Lie algebra. Then L is of type FP ifand only if L is finitely presented (in terms of generators and relations). Let L be an N -graded Lie algebra. We say that L is graded FP m if there is agraded projective resolution ( i.e. of graded U ( L ) -modules with graded homomor-phisms) of the trivial U ( L ) -module k , such that the projective modules in dimensionsmaller or equal to m are all finitely generated.Let L = ⊕ i ≥ L i be an N -graded Lie algebra. An ideal N of L is a graded ideal if N = ⊕ i ≥ N ∩ L i .The following result shows that homologically graded Lie algebras behave aspro- p groups i.e. it is a Lie graded algebra version of the pro- p -groups result [16,Thm. A]. Proposition 3.2. [17]
Let L be an N -graded Lie algebra over a field k and N be agraded ideal such that U ( L / N ) is left and right Noetherian. Then L is graded FP m if and only if H i ( N , k ) is finitely generated as U ( L / N ) -module for every i ≤ m. Thisimplies that both graded FP m and ordinary FP m are the same property. Applying the above result for N = L we obtain that L is FP m if and only ifH i ( L , k ) is finite dimensional for every i ≤ m .For completeness we include two more results, that probably are well known. Inboth cases the results show how similar N -graded Lie algebras are to pro- p groups,since the same type of results hold for pro- p groups with k substituted with F p . Lemma 3.3.
Let L be a N -graded Lie algebra. Then L is a free Lie algebra if andonly if H ( L , k ) = .Proof. Let X be a minimal set of generators of L . As L is N -graded we have that L = [ L , L ] + kX , where kX is the k -vector space spanned by X . Since L is N -gradedwe can choose X to be a graded subset of L i.e. X is a disjoint union ∪ i ≥ X i where X i ⊆ L i .Let F ( X ) be the free Lie algebra on X and F ( X ) / R ≃ L . Note that we have anatural N -grading of F ( X ) , where the elements of X i have degree i , thus F ( X ) / R ≃ L is an isomorphism of N -graded Lie algebras. Since L is N -graded and X isminimal, we have that dim k L / [ L , L ] = | X | = dim k F ( X ) / [ F ( X ) , F ( X )] , hence R ⊆ [ F , F ] . By Hopf formula0 = H ( L , k ) ≃ ( R ∩ [ F , F ]) / [ R , F ] = R / [ R , F ] ≃ R / [ R , R ] ⊗ U ( L ) k , where we view R / [ R , R ] as a right U ( L ) -module via the adjoint action of F ( X ) that factors through F ( X ) / R ≃ L . By the Nakayama lemma for the graded U ( L ) -module R / [ R , R ] we get that R / [ R , R ] =
0. Since every subalgebra of a free Lie al-gebra is free ([4, Theorem 2.8.3]) we conclude that R is free, hence dim k ( R / [ R , R ]) is the minimal number of generators of R . Thus R = L ≃ F ( X ) is free. (cid:3) Let L be an N -graded Lie algebra. An N -graded presentation h X | R i is apresentation, where X ⊆ L is an N -graded subset of L and R is N -graded subsetof the N -graded free Lie algebra F ( X ) with a free basis X . Note that if L is an N -graded Lie algebra with an N -graded generating set X then L has an N -gradedpresentation h X | R i . Lemma 3.4.
Let L be a N -graded Lie algebra with N -graded presentation h X | R i ,with X minimal and R minimal possible once X is fixed. Then | X | = dim k H ( L , k ) , | R | = dim k H ( L , k ) Proof.
The first follows immediately from the fact that by the minimality of X theimage of X in H ( L , k ) ≃ L / [ L , L ] is a basis as a k -vector space. Set R the kernelof the epimorphism F ( X ) → G that is the identity on X , i.e. we have L ≃ F ( X ) / R and H ( L , k ) ≃ ( R ∩ [ F , F ]) / [ R , F ] = R / [ R , F ] ≃ R / [ R , R ] ⊗ U ( L ) k . Note that R generates R as an ideal of F ( X ) , hence the image of R is a generatingset of R / [ R , R ] as U ( L ) -module, where we view R / [ R , R ] as U ( L ) via the adjointaction. Then the image of R is a generating set of R / [ R , R ] ⊗ U ( L ) k as a k -vectorspace. Let R be a subset of R that is a basis of R / [ R , R ] ⊗ U ( L ) k as a vector spaceand let I be the ideal of F ( X ) generated by R . Then R = I + [ R , R ] and both I and R are N -graded ideals ( hence graded Lie subalgebras of F ( X ) ). Hence for the N -graded Lie algebra S = R / I we have S = [ S , S ] , hence S =
0, i.e. R = I . (cid:3)
4. G
RADED ONE RELATOR L IE ALGEBRAS AS ITERATED
HNN
EXTENSIONS
A 1-relator Lie algebra is a Lie algebra admitting a presentation (in terms ofgenerators and relations) of the form L = h X | r i where X is an arbitrary set and r is an element of the free Lie algebra F ( X ) on X . Inother words, L is the quotient of F ( X ) with the ideal generated by r . Assume thatthere is some weight function ω : X → Z + . This induces an N -grading on F ( X ) . If r is homogeneous when seen as element of F ( X ) then the ideal that it generates isalso homogeneous so the quotient algebra L is N -graded.In this section we use the notation A ∗ h for a Lie algebra HNN extension witha base subalgebra A and stable letter t . This notation does not specify the associ-ated Lie subalgebra. We prove the following result which is Theorem B from theintroduction. Theorem 4.1.
Let L = h X | r i be a one relator N -graded Lie algebra with X finite,where the N -grading is induced by some weight function ω : X → Z + , i.e. r ∈ F ( X ) is homogeneous with respect to ω . Then L is an iterated HNN-extension of Liealgebras L = ( . . . ( A ∗ h n ) ∗ h n − ) . . . ) ∗ h such that A and all the associated Lie subalgebras in each of the HNN-extensionsare free. As a consequence one can prove that for L as in the hypothesis cd ( L ) ≤ Theorem 4.2. (Labute, [22, Proof of Theorem 1, page 182] ) LetL = h X | r i be a 1-relator Lie algebra with X finite. Assume that there is some weight function ω : X → Z + such that r is homogeneous with respect to the grading induced by ω on F ( X ) . Then there is some ideal H of L of finite codimension such that H isfree as Lie algebra. Note first that we may assume | X | ≥
1. The weight function ω can be extendedto F ( X ) via ω ([ u , v ]) = ω ( u ) + ω ( v ) . Fix an order < in X which is compatible with ω (i.e., such that x < y implies ω ( x ) ≤ ω ( y ) ). Following [22] a w eighted Hall setwith respect to X and ω is a basis H of F ( X ) consisting of Lie monomials togetherwith a well-ordering < such that1) X is an ordered subset of H ,2) for u , v ∈ H , ω ( u ) < ω ( v ) implies u < v ,3) for [ a , b ] ∈ F ( X ) , [ a , b ] ∈ H if and only if the following conditions hold: a , b ∈ H , a < b and if b = [ c , d ] , c , d ∈ H then a ≥ c ,4) if [ a , b ] , [ c , d ] ∈ H are such that ω ([ a , b ]) = ω ([ c , d ]) , then [ a , b ] < [ c , d ] ifand only if either b < d or b = d and a < c .From this it can be deduced that every v ∈ H admits a canonical decomposition which is an expression v = [ u , . . . , u m , z ] (commutators are right normed) with u ≥ . . . ≥ u m < z , u . . . , u m ∈ H , z ∈ X . Proof of Theorem 4.2 (Sketch)
Consider H with the total order induced by < : H = { h < h < . . . } and define H = H , X = X andH i = H i − r { h i } = H r { h , . . . , h i } , X i = { [ h i , . . . , h i | {z } n -times , z ] | n ≥ , h i = z ∈ X i − } . Then X i is the set of indecomposable elements of H i and h i + ∈ X i . Also, X i − r { h i } ⊆ X i . Moreover, H i is a weighted Hall set with respect to X i and the weight induced by ω . In particular, H i is a basis for the free Lie algebra F ( X i ) . We also have F ( X ) = F ( X ) ⊇ F ( X ) ⊇ . . . ⊇ F ( X i ) ⊇ . . . and each F ( X i + ) is an ideal of F ( X i ) of codimension 1 such that F ( X i ) / F ( X i + ) = Kh i + . Finally, ∩ F ( X i ) = k is a field, we may assume that r = a + α where a ∈ H and α is a linear combination of elements α j ∈ H with α j < a . Let I denote the ideal of F ( X ) generated by r . Labute shows that for any i , if I ⊆ F ( X i ) then • either I ⊆ F ( X i + ) and a and each α j ∈ H i + , • or a ∈ X i , the family F below generates the ideal I as an ideal of F ( X i ) and F can be extended to a basis of F ( X i ) . In this case, F ( X i ) / I is free. F = { [ h i , j i . . ., h i , h i − , j i − . . ., h i − , . . . , h , j . . ., h , r ] | j i , j i − , . . . , j ≥ } . As ∩ F ( X i ) = t with I ⊆ F ( X t ) and H : = F ( X t ) / I is free. Moreover H is an ideal of L and L / H = F ( X ) / F ( X t ) has finite dimension. (cid:3) Remark . If L = B ∗ h is an HNN extension of k -Lie algebras, there is an epimor-phism π : L → k with π ( h ) =
1. The kernel of π is the ideal of L generated by theLie subalgebra B . Proof of Theorem 4.1
We use the same notation as in the proof of Theorem 4.2. Webegin by choosing, for any i ≤ t , a finite subset Y i ⊆ X i such that r ∈ F ( Y i ) , h i ∈ Y i − and Y i − r { h i } ⊆ Y i . To do that, let Y = X = X andproceed inductively: if r ∈ F ( Y i − ) we can choose a non-negative integer j ( i ) suchthat r lies in the free Lie subalgebra generated by Y i = { [ h i , j . . ., h i , z ] | z ∈ Y i − r { h i } , ≤ j ≤ j ( i ) } . Observe that at each step we could be choosing the integer j ( i ) to be smallestpossible, however we prefer not to do so at this point and allow j ( i ) to be arbitrarilybig. For each i let Z i = { [ h i , j . . ., h i , z ] | z ∈ Y i − r { h i } , ≤ j < j ( i ) } . The adjoint action of h i induces a map Z i → Y i b [ h i , b ] Put B i = h Y i | r i and let A i be the Lie subalgebra of B i generated by Z i . The mapabove induces a derivation d i : A i → B i b [ h i , b ] and we may form the Lie algebra HNN extension B i ∗ h i with associated subalgebra A i . We have B i ∗ h i = h Y i , h i | r , [ h i , b ] = d ( b ) for b ∈ A i i = h Y i − | r i = B i − . Therefore L = F ( X ) / I = ( . . . (( B t ∗ h t ) ∗ h t − ) . . . ∗ h and B t = h Y t | r i . Here t is the number from the scketch of the proof of Theorem4.2 and B t is a Lie subalgebra of the free Lie algebra H , hence is free itself.At this point we have shown that L is an iterated HNN-extension of one relatoralgebras so that the first one is free and we still have to prove that the associatedalgebras A i can be chosen to be all free. To do that it suffices to chose all the values j ( i ) to be minimal possible. Then we have that r F ( Z i ) for any i which, using theFreiheitssatz for Lie algebras implies that the Lie algebras A i are free. (cid:3)
5. C
OHERENCE FOR L IE ALGEBRAS AND UNIVERSAL ENVELOPINGALGEBRAS
Let R be an associative ring with 1. Recall that an R -module M is called coherentif every finitely generated R -submodule of M is finitely presented. The ring R iscalled coherent (meaning right coherent with our convention) if it has the propertythat any finitely generated R -submodule of R , i.e. right ideal, is finitely presented.This is equivalent to every finitely presented (right) R -module is coherent. Note thata finitely generated coherent R -module M is of homological type FP ∞ , in particularis finitely presented and that every finitely presented R -module over a coherent ring R is a coherent R -module, hence is FP ∞ . For example, free non-abelian polynomialrings over a field are coherent [27, Cor. 3.3]. And abelian polynomial rings over afiled and on infinitely many variables are coherent but not noetherian. And a group G is coherent if any finitely generated subgroup H ≤ G is also finitely presented.Again, free groups are coherent. By analogy we set Definition 5.1.
A Lie algebra is coherent if any finitely generated subalgebra S ≤ L is also finitely presented. If L is N -graded, then we say that it is graded-coherent ifany finitely generated N -graded subalgebra S ≤ L is also finitely presented.Assume that for a group G the group ring kG is coherent and H is a finitelygenerated subgroup of G . As H is finitely generated, the induced module U = k ⊗ kH kG is a finitely presented (right) kG -module, hence is FP ∞ and in particularis FP . This implies that the group H is F P by [5, Prop. 1.4], see Lemma 2.8. Butin general this does not imply that H is finitely presented, so we cannot claim that G is a coherent group. The problem is that, for groups, being of type FP does notimply being finitely presented as the examples constructed by Bestvina and Bradyin [3] show. However, things change if we work with N -graded Lie algebras, asin that case both properties are equivalent (see [30] and also [17] where we extendthis fact to higher degrees). Lemma 5.2.
Let L be an N -graded Lie algebra. Assume that the universal en-veloping algebra U ( L ) is coherent. Then L is graded-coherent.Proof. Let S ≤ L be a finitely generated N -graded subalgebra. Consider the in-duced U ( L ) -module M = k ⊗ U ( S ) U ( L ) . By Lemma 2.8, M is FP . As U ( L ) iscoherent M is also of type FP and so is S again by Lemma 2.8. As S is N -graded, S is finitely presented (see above). (cid:3) For rings there is a construction of free amalgamated products and also a no-tion of HNN extension. In both cases, if the rings that play the role of the edgegroups are assumed to be Noetherian and the rings that play the role of the vertexgroups are assumed to be coherent, the resulting ring is coherent (see [1] for freeamalgamated products, [7] for HNN extensions).At this point, we can prove Theorem C of the introduction.
Theorem 5.3.
Assume that the Lie algebra L is the fundamental Lie algebra of agraph of Lie algebras such that for the vertex Lie algebras L v the universal envelop-ing algebra U ( L v ) is coherent and for each edge algebra L e , U ( L e ) is Noetherian.Then U ( L ) is coherent.Proof. The result follows by an adaptation of the main argument of [1, Corol-lary 13] using the long exact sequence of Tor functors above. For the reader’s convenience, we summarize the idea of the proof and shift from left coherenceconsidered in [1] to right coherence considered here. The main ingredients are:i) By a right-module version of Lemma 7 in [1] (the result goes back toChase), a ring R is right coherent if and only if ∏ I R is flat as left R -module, where I is a set of cardinality card R . And this is equivalent toany module of the form ∏ i ∈ J E i being flat, where J is an arbitrary indexset and the E i are flat left R -modules.ii) If a ring R is Noetherian, then it is coherent and moreover the natural map ( ∏ i ∈ J B i ) ⊗ R A → ∏ i ∈ J ( B i ⊗ R A ) is an injection where J is any index setand B i , A are arbitrary R -modules.Now, let V be any right R -module for R = U ( L ) and I an index set of cardinalitycard R . By the long exact sequence of Tor functors Corollary 2.5 for any s ≥ Ls ( V , ∏ I R ) is sandwiched as follows: → ⊕ v ∈ V ( Γ ) Tor U ( L v ) s ( V , ∏ I R ) → Tor Rs ( V , ∏ I R ) → ⊕ e ∈ E ( Γ ) Tor U ( L e ) s − ( V , ∏ I R ) → Now, as each U ( L v ) is coherent, i) implies ⊕ v ∈ V ( Γ ) Tor U ( L v ) s ( V , ∏ I R ) = s ≥ s >
1, i) implies that also ⊕ e ∈ E ( Γ ) Tor U ( L e ) s − ( V , ∏ I R ) =
0. This implies Tor Rs ( V , ∏ I R ) = s >
1. In the case when s = R ( V , ∏ I R ) ⊕ e ∈ E ( Γ ) V ⊗ U ( L e ) ( ∏ I R ) ⊕ v ∈ V ( Γ ) V ⊗ U ( L v ) ( ∏ I R ) = ∏ I Tor R ( V , R ) ⊕ e ∈ E ( Γ ) ∏ I ( V ⊗ L e R ) ⊕ v ∈ V ( Γ ) ∏ I ( V ⊗ L v R ) τ µ As τ is a monomorphism by ii), we deduce that also µ is mono thus Tor ( V , ∏ I R ) =
0. As this happens for any V , we deduce that ∏ I R is flat as left module so i) impliesthat R is coherent. (cid:3) Recall that groups of the form h X ∪ Y | u = v i where X and Y are disjoint, u isa word in X and v a word in Y are called one relator pinched and groups h X , t | t − ut = v i where t X , u and v are words in X are called one relator cyclicallypinched . By analogy, we set Definition 5.4.
A Lie algebra of the form h X ∪ Y | u = v i where X and Y are disjoint, u lies in the free Lie algebra generated by X and v lies in the free Lie algebragenerated by Y is called one relator pinched and a Lie algebra h X , t | [ u , t ] = v i where t X and u and v lie in the free Lie algebra generated by X are called onerelator cyclically pinched . Corollary 5.5.
One relator pinched and one relator cyclically pinched N -gradedLie algebras such that the corresponding relators are homogeneous are graded-coherent.Proof. In both cases, the corresponding Lie algebra is N -graded (because the re-lators are assumed to be homogeneous). Also, both constructions yield the fun-damental Lie algebra of a graph of Lie algebras with free Lie algebras as verticesand one dimensional Lie algebra as an edge. For a free Lie algebra F , U ( F ) is the free K -polynomial ring which is coherent and for a one-dimensional Lie algebra T , U ( T ) is the polynomial ring in one variable, which is Noetherian. It suffices touse Lemma 5.2 and Theorem 5.3. (cid:3) In the case of a right angled Artin group A Γ , Γ a finite connected graph, it wasproven by Droms [9] that A Γ is coherent if and only if the graph Γ is chordal ,meaning that there is no k -cycle, k ≥
4, embedded as a full subgraph of Γ . Usingthe previous results we can easily extend this to Lie algebras and we get CorollaryD from the introduction. Recall that right angled Artin Lie algebras L Γ are N -graded with grade components ( L Γ ) = kv + . . . + kv m where { v , . . . , v m } = V ( Γ ) and ( L Γ ) i = [( L Γ ) , ( L Γ ) i − ] for i ≥ [ L Γ , L Γ ] = ⊕ i ≥ ( L Γ ) i . Corollary 5.6.
Let L Γ be a right angled Artin Lie algebra. The following areequivalent: i) U ( L Γ ) is coherent, ii) L Γ is graded-coherent, iii) the graph Γ is chordal.Proof. The implication from i) to ii) is Lemma 5.2. Assume that L Γ is graded-coherent, we claim that Γ is chordal. Otherwise, we can find some cycle ∆ of, say, k vertices, k ≥
4, which is a full subgraph of Γ . As any graded subalgebra of L ∆ isgraded-coherent, L ∆ is graded-coherent. But this is a contradiction: take any linearmap χ : L ∆ / [ L ∆ , L ∆ ] → k such that χ ( v ) = v vertex of Γ . Let I χ ⊳ L ∆ be the ideal of L ∆ that projectsonto Ker χ . As ⊕ i ≥ ( L ∆ ) i = [ L ∆ , L ∆ ] ≤ I χ , I χ is also N -graded. Moreover, by [17, Corollary D] I χ is finitely generated since ∆ is connected but not finitely presented since ∆ is not 1-connected.Next, assume that Γ is chordal. We claim that U ( L Γ ) is coherent. As Γ ischordal, either Γ is complete or there are subgraphs Γ , Γ ⊆ Γ such that Γ = Γ ∪ Γ and Γ ∩ Γ is complete (see [9]). In the first case L Γ is finite dimensionalabelian so U ( L Γ ) is an abelian polynomial ring, so is Noetherian and coherent. Inthe second case we may assume by induction that U ( L Γ ) and U ( L Γ ) are bothcoherent and as U ( L Γ ∩ Γ ) is Noetherian we also get the result by Theorem 5.3. (cid:3) By [25] the commutator subgroup of a right angled Artin group is free if andonly if the the underlying graph is chordal. We show here a Lie algebra version ofthis fact.
Lemma 5.7.
Let L Γ be a right angled Artin Lie algebra, where Γ is a chordalgraph. Then [ L Γ , L Γ ] is a free Lie algebra.Proof. If Γ is a full graph there is nothing to prove as L Γ is abelian.By [8] since Γ is chordal there is a decomposition Γ ∪ Γ = Γ , where Γ ∩ Γ = Γ , where Γ is a full graph. This gives a decomposition L Γ = L Γ ∗ L Γ L Γ Note that by the defining relations of a right angled Lie algebra we have thatthe inclusion L Γ i → L Γ induces an inclusion L Γ i / [ L Γ i , L Γ i ] → L Γ / [ L Γ , L Γ ] . Hence [ L Γ , L Γ ] ∩ L Γ i = [ L Γ i , L Γ i ] for i = , , . Then by induction on the number of verticeswe can assume the results holds for L Γ and L Γ . Thus [ L Γ , L Γ ] ∩ L Γ i = [ L Γ i , L Γ i ] is a free Lie algebra for i = , . Then by Proposition 2.10 [ L Γ , L Γ ] is free. (cid:3) Proposition 5.8.
Let L be an N -graded Lie algebra such that [ L , L ] is a free Liealgebra. Then every finitely generated graded Lie subalgebra S of L is of homolog-ical type FP ∞ .Proof. Consider the short exact sequence of graded Lie algebras 0 → F → S → Q →
0, where F = S ∩ [ L , L ] , then Q is finitely generated abelian. Consider the Liealgebra version of the LHS spectral sequence E i , j = H i ( Q , H j ( F , k )) that converges to H i + j ( S , k ) . Since F is free E i , j = j ≥
2, hence the shortexact sequence is concentrated in two lines. Hence E ∞ i , j = E i , j is a subquotient of E i , j , so dim E i , j ≤ dim E i , j and there is a short exact sequence0 → E n − , → H n ( S , k ) → E n , → . Since S is finitely generated we have that H ( S , k ) is finite dimensional, in particular E , = H ( Q , H ( F , k )) is finite dimensional. Since V = H ( F , k ) ≃ F / [ F , F ] is agraded U ( Q ) -module via the adjoint action we know that V is finitely generated as U ( Q ) -module if and only if H ( Q , V ) is finite dimensional. Note that U ( Q ) is aNoetherian ring, so once we have that V is a finitely generated as U ( Q ) -module, itis of type FP ∞ over U ( Q ) and hence H i ( Q , V ) is finite dimensional for every i ≥ E i , = H i ( Q , V ) is finite dimensional for every i ≥ E i , = H i ( Q , k ) is finite dimensional since Q is finite dimensional abelianLie algebra. Thus dim k ( H i ( S , k )) < ∞ for every i and by Proposition 3.2 this impliesthat S is of homological type FP ∞ . (cid:3) The following result follows directly from Proposition 5.8. Note it strengthensCorollary 5.6.
Corollary 5.9.
Let L Γ be a right angled Artin Lie algebra, where Γ is a chordalgraph. Let S ≤ L be a finitely generated N -graded Lie subalgebra. Then S is oftype FP ∞ . It is known that an ascending HNN-extension of a free group is coherent [13].As a corollary of Proposition 5.8 we show that the same holds for Lie algebras. Wecall an HNN-extension Lie algebra W = h L , t | [ t , a ] = d ( a ) for a ∈ A i ascending if A = L . Lemma 5.10.
An ascending HNN-extension of a free Lie algebra is graded-coherent.Proof.
By assumption W = h L , t | [ t , a ] = d ( a ) for a ∈ A i with L = A free Lie alge-bra. Thus A is an ideal of codimension 1 in W , hence [ W , W ] ⊆ A is free. Then wecan apply Proposition 5.8. (cid:3) By an analogy with the case of groups we may ask:
Question 5.11.
Assume that L is the fundamental Lie algebra of a graph of Liealgebras such that the vertex Lie algebras L v are coherent and the edge algebrasL e have the property that every subalgebra is finitely generated. Is then L coherent?(again, this is true for groups, see [31] ).
6. A
N EXAMPLE
Let L = M ∗ N , where M = ka ⊕ kb , N = kx are abelian Lie algebrasThus there is a presentation ( in terms of generators and relations) L = h a , b , x | [ a , b ] i . Note that L is a N -graded Lie algebra, where a , b , x all have degree 1. Let S be the N -graded Lie subalgebra S = h a , b , z = [ x , a ] , t = [ x , b ] i ≤ L Lemma 6.1. a) a , b , z , t is a minimal set of generators of S and dim k H ( S , k ) = .b) S has a presentation h a , b , z , t | [ z , b ] − [ t , a ] , [ a , b ] i .Proof. a) Consider first the ideal J of L generated by x . Note J is a graded ideal of L that as a Lie algebra is generated by the homogeneous elements T = { ad ( a ) i ad ( b ) j ( x ) | i , j ≥ } where ad ( y )( s ) = [ s , y ] . Furthermore by Proposition 2.10 J is a free Lie algebraand if T are linearly independent in J / [ J , J ] we can conclude that T is a free basisof J . To show this consider the Lie algebra M = U ( M ) ⋋ M , where the adjointaction of M on the ideal U ( M ) of M is given by right multiplication. Note that M is a quotient of L via the homomorphism ϕ that is identity on a and b and sends x to 1 ∈ U ( M ) . Note that ϕ ( J ) = U ( M ) is abelian hence J / [ J , J ] maps surjectivelyto U ( M ) and ϕ ( T ) is a basis of U ( M ) as a k -vector space. Hence T is linearlyindependent in J / [ J , J ] as required.Consider J = J ∩ S . Thus we have a split short exact sequence of Lie algebras(13) 0 → J → S → M → J as a Lie algebra is generated by T = T r { x } , hence J is a free Liealgebra with a free basis T . Then J / [ J , J ] is a k -vector space with basis T . Notethat ad ( a ) i ad ( b ) j − ( t ) = ad ( a ) i ad ( b ) j ( x ) = ad ( a ) i − ad ( b ) j ( z ) , hence J / [ J , J ] is a right U ( M ) -module generated by z and t subject to the relation ad ( a )( t ) = ad ( b )( z ) . Furthermore J / [ J , S ] ≃ ( J / [ J , J ]) ⊗ U ( M ) k ≃ k ⊕ k . Since S is a split extension of J by M and M is abelian we conclude that S / [ S , S ] ≃ J / [ J , S ] ⊕ M has dimension 4 . Since a , b , z , t is a generating set of S we deduce that a , b , z , t is a minimal generatingset of S .Consider the spectral sequence E i , j = H i ( M , H j ( J , k )) converging to H i + j ( S , k ) . Since J is free H j ( J , k ) = j ≥
2. Hence E ∞ i , j = j ≥
2. Since (13) splits the map µ : H ( S , k ) → H ( M , k ) = E , is a split epimorphism. By the convergence of the spectral sequence there is anexact sequence 0 → E ∞ , → H ( S , k ) → E ∞ , → E ∞ , = E , is precisely the image of µ . Thus E ∞ , = E , = E , .By the bi-degrees of the differentials and since E ∞ , = E ∞ , = E , = H ( M , H ( J , k )) ≃ H ( M , J / [ J , J ]) . Let V = J / [ J , J ] . Recall that V is is a right U ( M ) -module generated by z and t subject to the relation ad ( a )( t ) = ad ( b )( z ) . Hence there is a short exact sequenceof right U ( M ) -modules0 → U ( M ) → U ( M ) ⊕ U ( M ) → V → . . . → = H ( M , U ( M ) ⊕ U ( M )) → H ( M , V ) → U ( M ) ⊗ U ( M ) k → ( U ( M ) ⊕ U ( M )) ⊗ U ( M ) k → V ⊗ U ( M ) k → . . . → = H ( M , U ( M ) ⊕ U ( M )) → H ( M , V ) → k → k → k → H ( M , V ) ≃ k anddim k H ( S , k ) = dim E , + dim E , = dim k H ( M , V ) + dim k H ( M , k ) = + = a , b , z , t is a minimal generating set by Lemma 3.4 we conclude that S has a presentation h a , b , z , t | R i where | R | = R should be of minimal possible degrees (as we work with the N -graded Lie algebra S ). On other hand it is easy to verify by hand that [ z , b ] − [ t , a ] , [ a , b ] are the relations of the smallest degree ( recall that a , b have degree 1 and z , t have degree 2). Hence we get that R can be chosen { [ z , b ] − [ t , a ] , [ a , b ] } . (cid:3) Lemma 6.2.
Let M , N , L , S be as before. Then S is a Lie subalgebra of L = M ∗ Nthat contains M but S = M ∗ Q for any Lie algebra Q.Proof.
Suppose S = M ∗ Q and let I be the ideal of S generated by M . Then usingLemma 6.1 we have Q ≃ S / I = h a , b , z , t | [ z , b ] − [ t , a ] , [ a , b ] , a , b i = h z , t | ∅ i is a free Lie algebra. Then2 = dim k H ( S , k ) = dim k H ( M ∗ Q , k ) = dim k ( H ( M , k ) ⊕ H ( Q , k )) = dim k H ( M , k ) =
1a contradiction. (cid:3)
Lemma 6.3.
Consider the Lie algebras S = h x , x , x , x | [ x , x ] , [ x , x ] + [ x , x ] i , e S = h x , x , x , x | [ x , x ] , [ x , x ] i and b S = h x , x , x , x | [ x , x ] , [ x , x ] i . Then thenilpotent Lie algebras E = S / [ S , S , S ] , e E = e S / [ e S , e S , e S ] and b E = b S / [ b S , b S , b S ] are pair-wise non-isomorphic. In particular the Lie algebras S, e S and b S are pairwise non-isomorphic.
Proof.
We write E = V ⊕ [ E , E ] , e E = V ⊕ [ e E , e E ] and b E = V ⊕ [ b E , b E ] , where V is thevector space spanned by x , x , x , x .1) It is easy to see that if there is an isomorphism ϕ between two of the Liealgebras E , e E and b E then there is a N -graded isomorphism ϕ between the sameLie algebras i.e. is induced by an automorphism of V as a vector space. Forexample if ϕ ( x i ) = v i + w i , where v i ∈ V and w i is an element of the correspondingderived Lie subalgebra then we can define ϕ ( x i ) = v i .2) Let I be the ideal of e E generated by x . Then e E / I = : F is a 3-generated freenilpotent N -graded Lie algebra of class 2.2.1) Suppose that e E and E are isomorphic as N -graded Lie algebras. Then thereis an ideal J of E generated by one element of V such that E / J ≃ F . Let π : E → E / J be the canonical epimorphism. Since [ x , x ] = [ π ( x ) , π ( x )] =
0. Since F is free nilpotent with π ( x ) , π ( x ) either zero or of degree 1 we deducethat either π ( x ) = λπ ( x ) for some λ ∈ k or π ( x ) =
0. Recall that [ x , x ] +[ x , x ] = E , hence(14) 0 = [ π ( x ) , π ( x )] + [ π ( x ) , π ( x )] . π ( x − λ x ) = J is generated as an ideal by x − λ x and F hasfree generators π ( x ) , π ( x ) , π ( x ) . By (14) 0 = [ π ( x ) , π ( x )] + [ π ( x ) , π ( x )] =[ π ( x ) , π ( x )] + [ λπ ( x ) , π ( x )] = [ π ( x ) , λπ ( x ) − π ( x )] . Then we get a contra-diction with F a free nilpotent Lie algebra of class 2 with free generators π ( x ) , π ( x ) , π ( x ) .2.1.2) If π ( x ) = J is generated as an ideal by x . And we can argue as incase 2.1.1.2.2) Suppose that e E and b E are isomorphic as N -graded Lie algebras. Then thereis an ideal b J of b E generated by one element of V such that b E / b J ≃ F . Let π : b E → b E / b J be the canonical epimorphism. Since [ x , x ] = b E we have that [ π ( x ) , π ( x )] = π ( x ) = λπ ( x ) forsome λ ∈ k or π ( x ) =
0. Thus for some a ∈ { x , x − λ x } ⊂ V we have that π ( a ) =
0. Similarly since [ x , x ] = E , there is some b ∈ { x , x − µ x } ⊂ V for some µ ∈ k such that π ( b ) =
0. Thus
Ker ( π ) ∩ V is a vector space over k ofdimension at least 2. This contradicts the fact that Ker ( π ) = b J is an ideal of b E generated by one element of V .2.3) Assume now that there is an N -graded isomorphism between E and b E . Thenin V ⊆ E there are elements y , y , y , y that are linearly independent and such that [ y , y ] = = [ y , y ] . Then there are elements v , v in E such that kv = kv ,either v or v does not belong to kx + kx and [ v , v ] = v = ∑ ≤ i ≤ a i x i , v = ∑ ≤ i ≤ b i x i , where all a i , b i ∈ k . Then 0 = [ v , v ] = ∑ ≤ i < j ≤ ( a i b j − a j b i )[ x i , x j ] . Using that [ x , x ] = , [ x , x ] + [ x , x ] = E wededuce that 0 = ∑ ≤ i < j ≤ ( a i b j − a j b i )[ x i , x j ] = ( a b − a b )[ x , x ]+( a b − a b + a b − a b )[ x , x ] + ( a b − a b )[ x , x ] + ( a b − a b )[ x , x ] . Since [ x , x ] , [ x , x ] , [ x , x ] , [ x , x ] are linearly independent in E we deduce that a b − a b = , a b − a b = , a b − a b = , a b − a b + a b − a b = . Recall that either v or v does not belong to kx + kx , say v . Then either b = b =
0. Without loss of generality b =
0. Then a = a b b , a = a b b , a b = a b = a b b b . If b = a = a b b b b = a b b , hence a i = a b b i and v = a b v , a contradiction.If b = a =
0, 0 = a b − a b + a b − a b = a b − a b hence a = a b b . Hence a i = a b b i and v = a b v , a contradiction. (cid:3) The following result shows that even in the cases of N -graded Lie algebrasKurosh type result does not hold. Lemma 6.4.
Let M , S be as before. Then S = C ∗ D for any non-zero N -graded Liesubalgebras C and D of S.Proof. Suppose S = C ∗ D . Recall that M ≤ S is abelian. If M is not a subalgebra of C or of D we have that dim k ( C ∩ M ) ≤
1, dim k ( D ∩ M ) ≤ M is a free Lie algebra, a contradiction.Suppose from now that M is a Lie subalgebra of C . Recall that by Lemma 6.1 S = h a , b , z , t | [ z , b ] − [ t , a ] , [ a , b ] i . If C is generated by 2 elements since C is N -graded i.e. C = ⊕ i ≥ C i , M = ka ⊕ kb ⊆ C we deduce that C = M , a contradictionwith Lemma 6.2.If C is generated by at least 3 elements, since S = C ∗ D is generated by precisely4 elements we conclude that C is generated by 3 elements and D by 1 element.Then D = kw and since S has 2 relators, C = h a , b , v | [ a , b ] , r i for some relation r on a , b , v . Since S has a presentation with quadratic relations the same holds for C ,so r = α [ a , v ] + β [ b , v ] = [ α a + β b , v ] for some α , β ∈ k not both zero. Suppose α =
0. Then for b a = α a + β b we have C = h b a , b , v | [ b a , b ] , [ b a , v ] i and C ∗ D isisomorphic to the Lie algebra e S from Lemma 6.3. By Lemma 6.3 S and e S are notisomorphic, a contradiction. (cid:3) In [15] it was shown an example of a Lie algebra with infinitely many ends thatis not a free product. The example from [15] is derived from the Shirshov exampleof a Lie algebra that is not a free product [26], thus it is not N -graded. Next weshow that there is a N -graded example. Corollary 6.5.
There exists a N -graded Lie algebra L that has infinitely manyends i.e. dim k H ( L , U ( L )) = ∞ but L is not a free product of N -graded Liesubalgebras.Proof. By [15, Thm. 1.4] for any Lie subalgebra L of L ∗ L such that L is not asubalgebra of L i for i = , k H ( L , U ( L )) = ∞ . We apply thisfor L = S , L = M , L = N . (cid:3) Lemma 6.6.
S is not a right angled Artin Lie algebra.Proof.
Suppose that S is a right angled Artin Lie algebra. Since dim k H ( S , k ) = k H ( S , k ) = S has exactly 4 vertices and 2 edges. Thus up to isomorphism there are two optionsfor S : e S = h x , x , x , x | [ x , x ] , [ x , x ] i or b S = h x , x , x , x | [ x , x ] , [ x , x ] i . By Lemma 6.3 any two of the Lie algebras S / [ S , S , S ] , e S / [ e S , e S , e S ] and b S / [ b S , b S , b S ] arenot isomorphic. (cid:3) Observe that L = M ∗ N is a right angled Artin Lie algebra whose underlyinggraph Γ is one edge and a separate vertex. Note that the square or the line with4 vertices do not embed in Γ . By [10] the right-angled Artin group G Γ has theproperty that every finitely generated subgroup is a RAAG. The Lie algebra versionof this results does not hold by Lemma 6.6. The reason this happens is that in thecase of Lie algebras there is no Kurosh type theorem.R EFERENCES [1] Åberg, H.; Coherence of amalgamations, J. of Algebra 78, (1982), 372-385.[2] Bartholdi, L; Runde, H; Schick, Th.; Right angled Artin groups and partial commutation, oldand new. arXiv:1904.12151.[3] Bestvina, M.; Brady, N.; Morse theory and finiteness properties of groups, Inventiones mathe-maticae (1997), 129, Iss. 3, pp 445 - 470[4] Bokut, L. A. ; Kukin, G. P.; Algorithmic and combinatorial algebra. Vol. 255. Springer Science& Business Media, 2012[5] Bieri, R.; Homological dimension of discrete groups, Queen Mary College MathematicalNotes, Queen Mary College Department of Pure Mathematics, London, second edition, 1981[6] Brown, K. S.; Cohomology of groups, 2nd edition, Springer-Verlag, 1994[7] Dicks, W. The HNN Construction for Rings, J. of Algebra 81, (1983), 434-487.[8] G. A. Dirac,On rigid circuit graphs, Abhandlungen aus dem Mathematischen Seminar der Uni-versit ät Hamburg,25(1961), 71–76[9] Droms, C.; Graph groups, coherence and 3-manifolds. J. of Algebra 106, (1987), 484-489.[10] Droms, C.; Subgroups of Graph Groups, Journal of Algebra110, 1987, 519 - 522.[11] Duchamp, G.; Krob, D.; The free partially commutative Lie algebra: bases and ranks, Adv.Math. 95, (1992), 92 - 126[12] Duchamp, G.; Krob, D.; The lower central series of the free partially commutative group,Semigroup Forum 45, (1992), 385 - 394[13] Feighn, M.; Handel, M.; Mapping tori of free group automorphisms are coherent. Ann. Math.(2), 149 (3), (1999), 1061-1077.[14] Feldman, G. L.; Exact Mayer-Vietoris sequences for free products of Lie algebras with anamalgamated subalgebra, Theory of mappings, its generalizations and applications, 212 - 220,"Naukova Dumka", Kiev, 1982[15] Feldman, G. L.; Ends of Lie algebras, Uspekhi Mat. Nauk, 1983, Vol. 38, Iss. 1 ( 229), 199 -200[16] King, J. D.; Homological finiteness conditions for pro-p groups. Comm. Algebra 27 (1999), no.10, 4969 - 4991.[17] Kochloukova, D.; Martínez-Pérez, C.; Coabelian ideals in right angled Artin Lie algebras.Preprint, arXiv 2006.02883[18] Kukin, G.P.; Subalgebras of the free Lie sum of Lie algebras with a joint sub- algebra. Algebraand Logic 11 (1972), 33-49.[19] Lam, K. Y.; Group rings of HNN extensions and the coherence property. Journal of Pure andApplied Algebra 11, 1-3 (1977), 9-13.[20] Labute, J. P. Algèbres de Lie et pro- p -groupes dèfinis par une seule relation. Invent. Math. 4,(1967), 142-158.[21] Labute, J. P. The determination of the Lie algebra associated to the lower central series of agroup. Trans. Amer. Math. Soc. 288 (1985), no. 1, 51 - 57.[22] Labute, J. P.; Free ideals of one-relator graded Lie algebras. Trans. Amer. Math. Soc. 347(1995), no. 1, 175-188.[23] Lichtman, A. I., Shirvani, M.; HNN-extensions for Lie algebras, Proc. AMS, 125, No. 12(1997), 3501-3508.[24] Rotman, J.; Introduction to Homological Algebra, 2nd Edition, Springer, 2009.[25] Servatius, H.; Droms C.; Servatius, B.; Surface subgroups of graph groups, Proc. Amer. Math.Soc.106(1989), no. 3, 573–578. [26] Širsov, A. I.; On a Hypothesis in the Theory of Lie Algebras, Sibirsk. Mat. Ž. 3 (1962), 297-301.[27] Swan, R.; K -Theory of coherent rings, Journal of Algebra and Its Applications. Vol. 18(2019),No. 09, 1950161, 16p.[28] Mikhalev, A. A.; Umirbaev, U. U.; Zolotykh, A. A. A Lie algebra with cohomological dimen-sion one over a field of prime characteristic is not necessarily free, First International Tainan-Moscow Algebra Workshop (Tainan, 1994), 257–264, de Gruyter, Berlin, 1996.[29] Wasserman, A.; A derivation HNN construction for Lie algebras. Israel J. of Math. 106 (1998),79-92.[30] Weigel, T.; Graded Lie algebras of type FP, Israel J. of Math. 205, (1) (2015), 185-209.[31] Wilton, H. Solutions to Bestvina & Feighn exercises in group theory. In Geometric and Coho-mological Methods in Group Theory.
Bridson, M; Kropholler, P; Leary, I. Vol. 358. CambridgeUniversity Press, 2009.D
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