Representation Type Of One Point Extensions Of Iterated Tubular Algebras

The purpose of this work is to show that if Λ a strongly simply connected semi-regular iterated tubular algebra and M is an indecomposable Λmodule then Λ[M ] is tame if and only if qΛ[M] is weakly non negative.


Introduction
Given an algebra one question of interest is to know its representation type.In particular it is not easy to know when a given algebra is of tame representation type.One possible approach to this problem is to consider the Tits form associated with the ordinary quiver of the algebra.It is known that, given a tame algebra Λ, the Tits form q Λ is weakly non negative.The converse has been shown for some families of algebras, as for instance tilted, quasi-tilted or iterated tubular algebras, but it is not true in general.More precisely, it holds that if Λ is tilted, quasi-tilted or iterated tubular algebra, then Λ is tame if and only if the Tits quadratic form is weakly non negative.
In order to investigate the representation type of a given triangular algebra Λ we assume that HH 1 (Λ) = 0, i.e. the first Hochschild Cohomology group vanishes.Then, up to duality, we get Λ = B[M ] with B a connected algebra and M an indecomposable module.Under these hypotheses we look at the following problem: if B has the property that the Tits form determines the tameness of its representation type, when the same property holds for Λ.In [16] de la Peña have shown that the result holds in case that B is a tame concealed algebra not of type Ã n .A similar result was obtained by Chalom and Merklen in [8] in case that B is a tilted algebra of euclidian type not of type Ã n .We consider the case that B is a strongly simply connected tame quasi-tilted algebra of canonical type, and the more general situation when B is a strongly simply connected iterated tubular algebra.Our main result in this note is the following: Theorem 1.1 Let B be strongly simply connected semi-regular iterated tubular algebra and M an indecomposable B-module, then the one point extension Λ = B[M ] is tame if and only if the Tits form q Λ is weakly non negative.
Note that tubular and tame quasi-tilted algebras are particular cases of iterated tubular algebras.
For several years have been stated the following well-known conjecture: If Λ is strongly simply connected then the Tits form determines the tameness of its representation type.Recently Brüstle and Skowroński had announced that this conjecture holds true.In [3] Assem and Castonguay gave necessary and sufficient conditions for the one point extension of a tree hereditary algebra to be again a strongly simply connected algebra.Observe that in our case, if B is a strongly simply connected algebra and M is an indecomposable module then Λ = B[M ] is a simply connected algebra with HH 1 (Λ) = 0, but Λ is not necessarily a strongly simply connected algebra.
We start in section 1 with preliminary results and useful definitions.In section 2, the case of one-point extensions of strongly simply connected tubular algebras by indecomposable modules is studied.In section 3, we prove the theorem for the case of one-point extensions of strongly simply connected tame quasi-tilted algebras of canonical type by indecomposable modules, and in section 4, the case of one-point extensions of strongly simply connected semi-regular iterated tubular algebras by indecomposable modules.

Preliminaries
Through the whole work, k denotes an algebraically closed field.By an algebra Λ we mean a finite-dimensional, basic and connected k-algebra of the form Λ ∼ = kQ/I where Q is a finite quiver and I an admissible ideal.Let Λ-mod denote the category of finite-dimensional left Λ-modules, and Λ-ind a full subcategory of Λ-mod consisting of a complete set of non-isomorphic indecomposable objects of Λ-mod.For each i ∈ Q 0 we denote by S i (resp.P i , I i ) the corresponding simple Λ-module (resp.the projective cover, injective envelope of S i ).
We begin now recalling the concepts and results that form a background for our work.A vector-space category (IK, | |) is a pair given by a Krull-Schmidt k-category IK and a faithful functor | | : IK → mod k (see [19]).Given a vector-space category (IK, | |), its objects (resp.the morphisms) are usually considered to be the objects (resp.the morphisms) of the image of | |, and its subspace category U(IK) is defined as follows: the objects are triples (X, U, ϕ) with X ∈ ObjIK, U a k-vector space and ϕ: U → |X|, k-linear.The morphisms (X, U, ϕ) → (X ′ , U ′ , ϕ ′ ) are the pairs (α, β) with β: X → X ′ in IK, α : U → U ′ k-linear and such that |β|ϕ = ϕ ′ α.Clearly, any object of U(IK) is isomorphic to a direct sum of a triple (X, U, ϕ) with ϕ : U → |X| injective and copies of (0, k, 0).Modules over a one point extension B[M ] can be identified with triples (X, U, ϕ) where X ∈ B-mod, U is a k-vectorspace and ϕ : U → Hom(M, X) is k-linear.It is known that the representation type of B[M ] depends on the representation type of B and of U(Hom(M, B −mod), see [19] for other notions and notations related to vector-space categories.
Associated to an algebra Λ of finite global dimension, there exist in the Grothendieck group of Λ, two quadratic forms.These forms are very important tools in the study of tame algebras.
Let C B be the Cartan matrix of B and let x and y vectors in K 0 (B).Then we have a bilinear form < x, y >= xC −T B y T where the corresponding quadratic form is called the Euler form of B, see [20].
By other hand, the Tits quadratic form is given by: x i .xj .dimk Ext 1 B (S i , S j ) + i,j∈Q0 x i .xj .dimk Ext 2 B (S i , S j ) see [6].
If B is such that gldimB ≤ 2 then for any B-module it follows that gldimB[M ] ≤ 3. Hence, using Bongartz result (see [6]) that is, if gldimB ≤ 2 then χ B = q B , it is possible to relate the Euler and the Tits form for Λ = B[M ].
Let X = (Y, k n , f ) be a Λ-module and let: where e is the extension vertex.
Comparing this two quadratic forms we get the following relation: Proposition 2.1 With the above notation: Proof: We give here a short argument: and, by the other hand A (S j , S e )).Comparing the dimensions of the Exts, we get the desired result.
We recall now some preliminary concepts that will be very useful in order to state our results.
Let Λ be a basic finite dimensional associative algebra (with unit) over an algebraically closed field k.Then Λ ≃ kQ/I for some finite quiver Q and some admissible ideal I of the path algebra kQ, and the pair (Q, I) is called a presentation for Λ.
Let (Q, I) be a connected bound quiver.
and, for any non empty proper subset J ⊂ {1, 2, ..., m}, we have j∈J λ j w j / ∈ I(x, y).A walk in Q from x to y is a path of the quiver formed by ..α εt t where α i are arrows in Q and ε i ∈ {1, −1} for all i, with source x and target y.We denote by e x the trivial path at x. Let ∼ be the least equivalence relation on the set of all walks in Q such that: λ i w i is a minimal relation, then w i ∼ w j for all i, j.
(c) If u ∼ v, then wuw ′ ∼ wvw ′ whenever these compositions make sense.Let x ∈ Q 0 be arbitrary.The set π 1 (Q, I, x) of equivalences classes u of closed paths u starting and ending at x has a group structure defined by the operation u.v = u.v.Since Q is connected then this group does not depend on the choice of x.We denote it π 1 (Q, I) and call it the fundamental group of (Q, I), see [15].
A triangular algebra Λ is simply connected if, for any presentation (Q, I) of Λ, the fundamental group π 1 (Q, I) is trivial.
An algebra B is a convex subcategory of Λ if there is a full and convex subquiver The algebra Λ is said to be strongly simply connected if any full convex subcategory of Λ is simply connected.(See [22]).
Given a directed component Γ of Γ A , its orbit graph O(Γ) has as points the τ -orbits O(M ) of the modules M in Γ.There exists an edge The number of such edges equals dim k Irr(τ m M, τ n N ) or dim k Irr(τ n N, τ m M ) re-spectively, where Irr(X,Y) denotes the space of irreducible morphisms from X to Y .A component Γ of Γ Λ is of tree type if its orbit graph O (Γ) is a tree.
It was shown in [16] that if B is a tame algebra, then q B is weakly non negative, this important result was also obtained by Drozd for matrix problems in [12] .It is known that the converse is not true in general, see example in section 2.4 of [17], but it is true for some families of algebras, as tubular algebras [20], quasitilted algebras [21] and iterated tubular algebras [18].The main motivation of our work was the result in [16] that if C is a tame concealed algebra, not of type Ãn , and M an indecomposable C-module, then the one point extension C[M ] is tame if and only if q C[M] is weakly non negative.This result was extended in [8] to the case of B be a strongly simply connected tilted algebra of euclidian type, i.e, if B is a strongly simply connected tilted algebra of euclidian type and M an indecomposable B-module, then the one point extension B[M ] is tame if and only if q B[M] is weakly non negative.Our objective is to generalize this result to the case when B is a strongly simply connected tame quasi-tilted algebras of canonical type, or B is a strongly simply connected iterated tubular algebra.We start considering the case when B is a tubular algebra.

One Point Extensions Of Tubular Algebras
In this section, we consider the class of tubular algebras considered by Ringel in [20].We recall that a tubular algebra B is a tubular extension of a tame concealed algebra B 0 with tubular type (2, 2, 2, 2) , (3, 3, 3) , (4, 4, 2) or (6, 3, 2).Any tubular algebra is also co-tubular, that is i ray modules of the separating tubular families of the corresponding algebras, and R i , R ′ i branches.We begin by proving the following lemma: Lemma 3.1 Let B 0 be a convex subcategory of B such that B is a iterated coextension or a branch coextension of B 0 and assume that Proof: The proof is done by induction in the number of the coextensions and the length of the branch.
We recall the structure of the Auslander-Reiten quiver of a tubular algebra, as in [20] ( pag.273).Let B be a tubular algebra, then ∞ with B 0 and B ∞ both tame concealed.We have the following pairwise disjoint modulo classes: are tubular families.Also the indecomposable projective modules belong to P 0 or T 0 and the indecomposable injective modules belong to T ∞ or Q ∞ .Now, we consider the case of B be a tubular algebra with directed components of tree type and we get the following result: Theorem 3.1 Let B be a tubular algebra with each directed component of tree type and M be an indecomposable B-module.If B[M ] is wild then q B[M] is strongly indefinite.
Proof: Observe that the pre-injective component and the pre-projective component of Γ B are of tree type.Then B 0 and B ∞ are not of type Ã n .
First consider the case that M is an indecomposable module in one of the following families : P 0 , T 0 , ∪ γ∈P 1 (k) T , then there exists an indecomposable injective module I such that Hom(M, I) = 0.It follows that M and I are separated by a separating tubular family, then this non zero morphism factor trough a orthogonal tubular family, in particular factors trough five orthogonal bricks, then by Nazarova theorem it follows that Hom B (M, modB) is wild and, see in prop.3.3 of [18], the corresponding quadratic form is strongly indefinite.Now, consider the case that M belongs to Q ∞ .Observe that this pre-injective component corresponds to the pre-injective component of the algebra B ∞ , that is, the pre-injective component of a tame concealed algebra not of type Ã n .Then, the situation is similar to the one consider in theorem 2.2 of [16] and the result follows with the same arguments.
It only remains to consider the case when M belongs to T ∞ .The analysis is analogous to the one in the case of a tilted algebra of euclidian type considered in theorem 2.3 of [8].For the convenience of the reader we repeat some of these arguments here.If M ∞ = M | B∞ is such that M ∞ = 0, then suppM is contained in a branch R and the vector-space category Hom(M, B − mod) is the same as Hom(M, R − mod).It follows from [14], that if Hom(M, R − mod) is wild then is tame, there are two possibilities: either M ∞ is a ray module or M ∞ is a module of regular length two in the tube of rank n − 2 and B ∞ is tame concealed of type Dn .
In case that M is a ray module over B, the same argument that in [8] shows that B[M ] is an algebra with acceptable projective modules.Also if M = M ∞ and therefore, M is a ray module over B ∞ , then again B[M ] = B[M ∞ ] is an algebra with acceptable projective modules.It follows by [18] that B[M ] is wild if and only if q B[M] is strongly indefinite.
Suppose M is not a ray module over B, M = M ∞ and M ∞ is a ray module.It is not difficult to show that category Hom(M, B − mod) has three pieces, that is, the ray of T e that starts in M ∞ , Hom(M ∞ , Q ∞ ) where Q ∞ is the pre-injective component of B ∞ and the subcategory given by the successors of M in the tube, that are not is given by some of the patterns given in [ [19], pag 254].Here, we are using the results given by Ringel, in [ [19], pag 254], theorem 3, and so, we follow the notations and definitions given there.The fundamental case that remains to consider is when M is not an injective module, since in case that M is an injective module B[M ] is a coil enlargement of B and so is tame.Now, consider the case that M is an injective module and there exists a sectional path M → Y 0 → . . .→ Y t with t ≥ 1.
In first place, we observe that Hom B (Y i , X) = 0 for and Hom(τ −1 M, X) = 0 for all pre-injective X.
In particular, Hom(Y i , X) = Hom(τ −1 M, X) = 0 for all X such that Hom(M ∞ , X) 0 = with X in the pre-injective component.
We compute the quadratic form for the case ( D5 , 2), the other cases are similar.Let L be the B-module Now, consider the case ( Dn , n − 2).The pattern is given by: is wild, again the quadratic form is strongly indefinite.On the other hand, the case t = 1 split in two possibilities with the same behavior that in [8] and the result holds.
It remains to look at the case that M ∞ is a module of regular length 2 in a tube of rank n − 2 and B ∞ is tame concealed algebra of type Dn .If M = M ∞ lies in a stable tube, then Hom(M, B − mod) = Hom(M ∞ , B ∞ − mod) and therefore both are tame or both wild.So, suppose that M belongs to a co-inserted tube.Since M ∞ has regular length 2, there exist E 1 and E 0 ray-modules over the ray-modules of the tube.
Observe that if M = M ∞ , then Hom(M, B − mod) has the same pattern that Hom(M ∞ , B ∞ − mod).If M is a B ∞ -module, then Hom B (M, N ) = 0 for modules N in the same tube that M or for modules N in the pre-injective component.Hence, since Hom(M, N ) = Hom(M ∞ , N ∞ ) it follows that the pattern is one of patterns given in [19], and then is tame.Considering the situations when the branch is co-inserted in E j for some j , in any cases the situation is similar to the one in [8] and the result holds.
The condition that B has each directed components of tree type, for tubular algebras is in fact equivalent to the condition of strongly simple connectedness, see tubular family, where some tubes contain projective modules ( with support in B − ) and some tubes contain injective modules (with support in B + ).Observe that,by [21] each component Γ of Γ B is contained in B + -mod or in B − -mod.If M belongs to a tube that contains projective modules, or a stable tube, then, the vectorspace categories Hom(M, B − mod) and Hom(M, B − − mod) are isomorphic.By other hand, B − [M ] is a convex subcategory of B[M ] by 3.1, and since B − has the pre-injective component of tree type, the result follows from [8].Now, consider the case where M belongs to a tube with injective modules (so, is wild and C is of tree type then also B + [M ] is wild, and q B[M] is strongly indefinite.
Then, assume that is also iterated tubular, and so is tame.If M = M 0 , and M is a ray-module, by similar arguments to the one in 3.1, B[M ] is an algebra with acceptable projective modules and so, by theorem 3.4 of [18], the representation type is determined by the quadratic form.And by the other hand, if the support of M is contained in the branch, the result follows from [14].So, assume that M is not a ray module, that M 0 = 0 and that C[M 0 ] is tame , so M 0 is a ray-module or is a module of level two in a tube of rank n − 2, and C is a Dn concealed algebra.The case that M 0 is a ray module is solved with a similar argument that in 3.1.Now, consider that C is a concealed algebra of Dn -type, and that C[M 0 ] is a 2-tubular algebra.Also, M = M 0 .Observe that the pre-injective C-modules can be immersed in the pre-injective component of B − , that is, the pre-injective component of B ( see [11]).It follows that there exist a faithful functor F : C − mod → B − mod such that F (X) is pre-injective if X is pre-injective.Moreover, if X is a pre-injective C-module such that dim Hom C (M 0 , X) = 2 then dim Hom B (M, F (X)) = 2. So, the vector-space category Hom(M, B − mod) contains the vector-space category given by the pattern Dn−2 n−2 as in [19], pag 253.Look at the vector-space category Hom(M, B − mod) whose objects are Hom(M, X) for X in the tube.Let E 0 , E 1 , • • • E n−3 be the ray-modules over C of the tube where M 0 lies.Assume that M 0 is the middle term of the almost split sequence 0 → E 1 → M 0 → E 0 → 0. Again, consider the possibilities that the branch is co-inserted in some of the ray modules E j these cases are analogous to the case of B tilted of euclidian type considered in [8], pag.8. Now, consider the non domestic case.There are two possibilities: i) If B + is tubular and B − is domestic, since B is tame, the Auslander-Reiten quiver of B + is given by P 0 , T 0 , ∪ γ∈P 1 (k) T , T ∞ , Q ∞ and the Auslander-Reiten quiver of B is given by P is the preinjective component of A and of B − and the new projective modules are inserted in stable tubes belonging to T ∞ .We denote T − ∞ the new family of tubes.In this case, considering all the possibilities for M , shortly saying, if suppM ⊂ B + or suppM ⊂ B − the result follows from the previous cases.
ii) If B − is tubular and B + is domestic, as B is tame, the Auslander-Reiten quiver of B is given by: P + 0 , T + 0 , ∪ γ∈P 1 (k) T γ , T ∞ , Q ∞ and the new injective modules, having support in B + are inserted in the T 0 .Let M be an indecomposable module in a tube in T + 0 , that is a tube containing injective modules, and so suppM ⊂ B + .The vector-space category Hom(M, −) is finite for those modules with the support contained in the branch, then the result follows from [14].If Hom(M, −) is infinite there exists an injective module I, outside of the tube, such that Hom(M, I) = 0, but again this morphism factors through infinite families of tubes and the result follows as in theorem 3.1,or [18].
iii) Finally, consider that B + and B − are tubular, in this case the Auslander-Reiten quiver of B is Since suppM ⊂ B + or suppM ⊂ B − all cases were already considered.
As in the case of tubular algebras, if B is a tame quasi-tilted algebra that is not tilted, then B is strongly simply connected if and only if B + and B − are strongly simply connected, if and only if each directed component of B + and B − is of tree type, see [4].
Remark 4.1 The statement of 4.1 remains true if we replace the hypotheses of strongly simply connectedness by the following one, B + , B − , and C have the preinjective components of tree type.

One Point Extensions Of Semi-Regular Iterated Tubular Algebras
In this section we consider one-point extensions of semi-regular iterated tubular algebras by indecomposable modules.We recall the structure of the Auslander-Reiten quiver of a semi-regular iterated tubular algebra B, see [18], in order to consider the vector-space categories that arise in the one point extension.For this purpose, consider the construction of B by steps.Assuming that B is a n-iterated tubular algebra ( with n ≥ 2) we consider: i ] that is quasi-tilted, because B is semi-regular n-iterated tubular.If we are going to extend one more step, we need that i ] that is also quasi-tilted, and if B is n-iterated, we have the n-quasi-tilted algebras . Observe that all B + i and B − i are tubular, except, maybe, the first one B + 1 and the last one B − n .We recall an example from [18] which will be useful for understand the general situation.
Let A given by the quiver , with all λ i , µ j different elements of the field k, and λ i = µ j we have four tame concealed algebras and two tame quasi-tilted algebras B 0 and B 1 .
with the induced relations.The Auslander-Reiten quiver of A is given by the pre-projective component of C 0 , a semi-regular tubular family T 0 of B 1 -modules, separating tubular families T γ with γ ∈ P 1 of B 1 -modules, a semi-regular tubular family T 1 containing semi-regular tubes, each of them formed of B 1 -modules or B 2 -modules, separating tubular families T δ with δ ∈ P 1 of B 2 -modules and a semiregular tubular family T 2 of B 2 -modules, and the pre-injective component of C 3 .
The following lemma state the general situation, and allows us to apply the results obtained in section 2 to the situation of semi-regular iterated tubular algebras.
Lemma 5.1 Let B be a semi-regular n-iterated tubular algebra and let M be an indecomposable A-module.Then there exists B i a tame quasi-tilted algebra such that a) B i is a full convex subcategory of B. b) suppM ⊂ B i .and the respective Tits quadratic forms are strongly indefinite.In case that B + 1 is tilted, observe that the pre-projective component of B[M ] is the same that the pre-projective component of B + 1 [M ].Since B 1 is quasi-tilted, the result follows from 4.1 and the fact that B + 1 [M ] is a full convex subcategory in B[M ].So, suppose now that M is in a tube.Then M is a B i -module, if i = n, then B − i is tubular, by 3.1 we only need to consider that M is in a tube that contains injective modules.Let T 1 , • • • , T ∞ be the tubular families containing injective modules.Consider first that M belongs to a tubular family T k with k = ∞.In case that the support of M is contained in the branch, it follows that Hom B (M, −) is finite and the result follows by [14].Consider M in a tube with support in B + i not in a branch, then there exists an injective in another tubular family T j , j ∈ B i such that Hom B (M, I j ) = 0. Since B − i is tubular this morphism factors through a orthogonal tubular family.It follows from the argument of the five orthogonal modules in [18] that Hom Bi (M, modB i ) is wild, and since Hom Bi (M, modB i ) ⊂ Hom B (M, modB) we get that B[M ] is wild.By other hand, since Hom B (M, I j ) = 0 this morphism factors through a B i -module X such that q Bi (dimX) = 0.By other way, B i is a full convex subcategory then q B (dimX) = q Bi (dimX) = 0. Note also that B i [M ] is a full convex subcategory of Λ = B[M ], then it follows that q Λ (2dimX + e s ) = q Bi[M] (2dimX + e s ) = 1 − 2Hom Bi (M, X) < 0. Now, consider the case that M belongs to T ∞ the last tubular family containing injective modules, it follows that M is a B n -module, with B n a quasi-tilted algebra and the result follows again from 4.1.