The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

We study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.


Introduction and the Main Results
Throughout the paper k will denote a fixed algebraically closed field. By an algebra is meant an associative finite-dimensional k-algebra with an identity, which we shall assume (without loss of generality) to be basic. Then such an algebra has a presentation A ∼ = kQ A /I , where Q A = (Q 0 , Q 1 ) is the ordinary quiver of A with the set of vertices Q 0 and the set of arrows Q 1 and I is an admissible ideal in the path algebra kQ A of Q A . If the quiver Q A has no oriented cycles, the algebra A is said to be triangular. For an algebra A, we denote by mod A the category of finitely generated right A-modules, and by ind A a full subcategory of mod A consisting of a complete set of representatives of the isomorphism classes of indecomposable modules. We shall denote by rad A the Jacobson radical of mod A, and by rad ∞ A the intersection of all powers rad i A , i ≥ 1, of rad A . Moreover, we denote by A the Auslander-Reiten quiver of A, and by τ A and τ − A the Auslander-Reiten translations D Tr and Tr D, respectively. We will not distinguish between a module in ind A and the vertex of A corresponding to it. Following [45], a family C of components is said to be generalized standard if rad ∞ A (X, Y ) = 0 for all modules X and Y in C . We note that different components in a generalized standard family C are orthogonal, and all but finitely many τ A -orbits in C are τ A -periodic (see [45, (2.3)]). We refer to [37] for the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of algebras.
Following Assem and Skowroński [7], a triangular algebra A is called simply connected if, for any presentation A ∼ = kQ A /I of A as a bound quiver algebra, the fundamental group π 1 (Q A , I ) of (Q A , I ) is trivial (see Section 2). The importance of these algebras follows from the fact that often we may reduce (using techniques of Galois coverings) the study of the module category of an algebra to that for the corresponding simply connected algebras. Let us note that to prove that an algebra is simply connected seems to be a difficult problem, because one has to check that various fundamental groups are trivial. Therefore, it is worth looking for a simpler characterization of simple connectedness. In [44,Problem 1] Skowroński has asked, whether it is true that a tame triangular algebra A is simply connected if and only if the first Hochschild cohomology space H 1 (A) of A vanishes. This equivalence is true for representation-finite algebras [3,Proposition 3.7] (see also [12] for the general case), for tilted algebras (see [5] for the tame case and [25] for the general case), for quasitilted algebras (see [3] for the tame case and [26] for the general case), for piecewise hereditary algebras of type any quiver [25], and for weakly shod algebras [4].
A prominent role in the representation theory of algebras is played by the algebras with separating families of Auslander-Reiten components. A concept of a separating family of tubes has been introduced by Ringel in [40,41] who proved that they occur in the Auslander-Reiten quivers of hereditary algebras of Euclidean type, tubular algebras, and canonical algebras. In order to deal with wider classes of algebras, the following more general concept of a separating family of Auslander-Reiten components was proposed by Assem, Skowroński and Tomé in [10] (see also [33]). A family C = (C i ) i∈I of components of the Auslander-Reiten quiver A of an algebra A is called separating in mod A if the components of A split into three disjoint families P A , C A = C and Q A such that: (S1) C A is a sincere generalized standard family of components; (S2) Hom A (Q A , P A ) = 0, Hom A (Q A , C A ) = 0, Hom A (C A , P A ) = 0; (S3) any homomorphism from P A to Q A in mod A factors through the additive category add(C A ) of C A . Then we say that C A separates P A from Q A and write A = P A ∪ C A ∪ Q A . We note that then P A and Q A are uniquely determined by C A (see [10, (2.1)] or [41, (3.1)]). Moreover, C A is called sincere if any simple A-module occurs as a composition factor of a module in C A . We note that if A is an algebra of finite representation type that C A = A is trivially a unique separating component of A , with P A and Q A being empty. Frequently, we may recover A completely from the shape and categorical behavior of the separating family C A of components of A . For example, the tilted algebras [24,41], or more generally double tilted algebras [39](the strict shod algebras in the sense of [15]), are determined by their (separating) connecting components. Further, it was proved in [28] that the class of algebras with a separating family of stable tubes coincides with the class of concealed canonical algebras. This was extended in [29] to a characterization of all quasitilted algebras of canonical type, for which the Auslander-Reiten quiver admits a separating family of semiregular tubes. Then, the latter has been extended in [33] to a characterization of algebras with a separating family of almost cyclic coherent Auslander-Reiten components. Recall that a component of an Auslander-Reiten quiver A is called almost cyclic if all but finitely many modules in lie on oriented cycles contained entirely in . Moreover, a component of A is said to be coherent if the following two conditions are satisfied: (C1) For each projective module P in there is an infinite sectional path P = X 1 → X 2 → · · · → X i → X i+1 → X i+2 → · · · (that is, X i = τ A X i+2 for any i ≥ 1) in ; (C2) For each injective module I in there is an infinite sectional path · · · → Y j +2 → Y j +1 → Y j → · · · → Y 2 → Y 1 = I (that is, Y j +2 = τ A Y j for any j ≥ 1) in . We are now in position to formulate the first main result of the paper, which answers positively the above mentioned question of Skowroński [44,Problem 1] for tame algebras with separating almost cyclic coherent Auslander-Reiten components. It has been proved in [33, Theorem A] that the Auslander-Reiten quiver A of an algebra A admits a separating family C A of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a finite product of concealed canonical algebras C 1 , . . . , C m by an iterated application of admissible algebra operations of types (ad 1)-(ad 5) and their duals. These algebras are called generalized multicoil algebras (see Section 3 for details). Note that for such an algebra A, we have that A is triangular, gl. dim A ≤ 3, and pd A M ≤ 2 or id A M ≤ 2 for any module M in ind A (see [33,Corollary B and Theorem E]). Moreover, let A = P A ∪ C A ∪ Q A be the induced decomposition of A . Then, by [33,Theorem C], there are uniquely determined quotient algebras A (l) = A (l) 1 × · · · × A (l) m and A (r) = A (r) 1 × · · · × A (r) m of A which are the quasitilted algebras of canonical type such that P A = P A (l) and Q A = Q A (r) .
Let A be a generalized multicoil algebra obtained from a concealed canonical algebra C = C 1 × · · · × C m and C = A 0 , A 1 , . . . , A n = A be an admissible sequence for A (see Section 3). In order to formulate the next result we need one more definition. Namely, if the sectional paths occurring in the definitions of the operations (ad 4), (fad 4), (ad 4 * ), (fad 4 * ) come from a component or two components of the same connected algebra A i , i ∈ {0, . . . , n − 1}, then we will say that A i+1 contains an exceptional configuration of modules.
The following theorem is the second main result of the paper.
m of A such that the following statements are equivalent: (i) A is simply connected. This paper is organized as follows. In Section 2 we recall some concepts and facts from representation theory, which are necessary for further considerations. Section 3 is devoted to describing some properties of almost cyclic coherent components of the Auslander-Reiten quivers of algebras, applied in the proofs of the preliminary results and the main theorems. In Section 4 we present and prove several results applied in the proof of the first main result of the paper. Sections 5 and 6 are devoted to the proofs of Theorem 1.1 and Theorem 1.2, respectively. The aim of the final Section 7 is to present examples illustrating the main results of the paper.
For basic background on the representation theory of algebras we refer to the books [6,[41][42][43], for more information on simply connected algebras we refer to the survey article [2], and for more details on algebras with separating families of Auslander-Reiten components and their representation theory to the survey article [35].

2.1
Let A be an algebra and A ∼ = kQ A /I be a presentation of A as a bound quiver algebra. Then the algebra A = kQ A /I can equivalently be considered as a k-linear category, of which the object class A 0 is the set of points of Q A , and the set of morphisms A(x, y) from x to y is the quotient of the k-vector space kQ A (x, y) of all formal linear combinations of paths in Q A from x to y by the subspace I (x, y) = kQ A (x, y) ∩ I (see [11]). A full subcategory B of A is called convex (in A) if any path in A with source and target in B lies entirely in B. For each vertex v of Q A we denote by S v the corresponding simple A-module, and by P v (respectively, I v ) the projective cover (respectively, the injective envelope) of S v .

One-point Extensions and Coextensions Frequently an algebra
One defines dually the one-point coextension [M]B of B by M (see [41]).  [17] the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras and the second class is formed by the wild algebras whose representation theory comprises the representation theories of all finite dimensional algebras over k.

Tameness and Wildness Let
Hence, a classification of the finite dimensional modules is only feasible for tame algebras. It has been shown by Crawley-Boevey [16] that, if A is a tame algebra, then, for any positive integer d ≥ 1, all but finitely many isomorphism classes of indecomposable A-modules of dimension d are invariant on the action of τ A , and hence, by a result due to Hoshino [23], lie in stable tubes of rank one in A .

Hochschild Cohomology of Algebras Let
A be an algebra. Denote by C • A the Hochschild complex C • = (C i , d i ) i∈Z defined as follows: for f ∈ C i and a 1 , [14,Chapter IX]). Recall that the first Hochschild cohomology space

Concealed Canonical Algebras
An important role in our considerations will be played by certain tilts of canonical algebras introduced by Ringel [41]. Let p 1 , p 2 , . . . , p t be a sequence of positive integers with t ≥ 2, 1 ≤ p 1 ≤ p 2 ≤ . . . ≤ p t , and p 1 ≥ 2 if t ≥ 3. Denote by (p 1 , . . . , p t ) the quiver of the form normalized such that λ 1 = ∞, λ 2 = 0, λ 3 = 1, and the admissible ideal I (λ 1 , λ 2 , . . . , λ t ) in the path algebra k (p 1 , . . . , p t ) of (p 1 , . . . , p t ) generated by the elements Then the bound quiver algebra (p, λ) = k (p 1 , . . . , p t )/I (λ 1 , λ 2 , . . . , λ t ) is said to be the canonical algebra of type p = (p 1 , . . . , p t ). Moreover, for t = 2, the path algebra (p) = k (p 1 , p 2 ) is said to be the canonical algebra of type p = (p 1 , p 2 ). It has been proved in [41,Theorem 3.7] that if is a canonical algebra of type (p 1 , . . . , p t ) then = P ∪ T ∪ Q for a P 1 (k)-family T of stable tubes of tubular type (p 1 , . . . , p t ), separating P from Q . Following [27], a connected algebra C is called a concealed canonical algebra of type (p 1 , . . . , p t ) if C is the endomorphism algebra End (T ), for some canonical algebra of type (p 1 , . . . , p t ) and a tilting -module T whose indecomposable direct summands belong to P . Then the images of modules from T via the functor Hom (T , −) form a separating family T C of stable tubes of C , and in particular we have a decomposition C = P C ∪ T C ∪ Q C . It has been proved by Lenzing and de la Peña [28, Theorem 1.1] that the class of (connected) concealed canonical algebras coincides with the class of all connected algebras with a separating family of stable tubes. It is also known that the class of concealed canonical algebras of type (p 1 , p 2 ) coincides with the class of hereditary algebras of Euclidean types A m , m ≥ 1 (see [22]). Recall also that the canonical algebras of types (2, 2, 2, 2), (3,3,3), (2,4,4) and (2,3,6) are called the tubular canonical algebras, and an algebra which is tilting-cotilting equivalent to a tubular canonical algebra is called a tubular algebra (see [18,21,41]).

Simple Connectedness Let (Q, I ) be a connected bound quiver. A relation
is minimal if m ≥ 2 and, for any nonempty proper subset J ⊂ {1, . . . , m}, we have j ∈J λ j w j / ∈ I (x, y). We denote by α −1 the formal inverse of an arrow α ∈ Q 1 . A walk in Q from x to y is a formal composition α ε 1 1 α ε 2 2 . . . α ε t t (where α i ∈ Q 1 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 homotopy relation on (Q, I ), that is, the smallest equivalence relation on the set of all walks in Q such that: (a) If α : x → y is an arrow, then α −1 α ∼ e y and αα −1 ∼ e x .
then wuw ∼ wvw whenever these compositions make sense. Let x ∈ Q 0 be arbitrary. The set π 1 (Q, I, x) of equivalence classes u of closed walks u starting and ending at u has a group structure defined by the operation u · v = uv. Since Q is connected, π 1 (Q, I, x) does not depend on the choice of x. We denote it by π 1 (Q, I ) and call it the fundamental group of (Q, I ).
Let A ∼ = kQ A /I be a presentation of a triangular algebra A as a bound quiver algebra. The fundamental group π 1 (Q A , I ) depends essentially on I , so is not an invariant of A. A triangular algebra A is called simply connected if, for any presentation A ∼ = kQ A /I of A as a bound quiver algebra, the fundamental group π 1 (Q A , I ) of (Q A , I ) is trivial [7]. Example 2.7 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 3 γ y y t t t t t t t t t t and I the ideal in the path algebra kQ of Q over k generated by the elements γβ, δα − aδβ, αλ, where a ∈ k \{0}. Then π 1 (Q, I ) is trivial. Moreover, the triangular algebra A is simply connected. Indeed, any choice of a basis of rad A /rad 2 A will lead to at least one minimal relation with target 1 and source i ∈ {3, 4} or with target 5 and source 2.

Generalized Multicoil Algebras
It has been proved in [32, Theorem A] that a connected component of an Auslander-Reiten quiver A of an algebra A is almost cyclic and coherent if and only if is a generalized multicoil, that is, can be obtained, as a translation quiver, from a finite family of stable tubes by a sequence of operations called admissible. We recall briefly the generalized multicoil enlargements of algebras from [33,Section 3].
Given a generalized standard component of A , and an indecomposable module X in , the support S(X) of the functor Hom A (X, −)| is the k-linear category defined as follows [9]. Let H X denote the full subcategory of consisting of the indecomposable modules M in such that Hom A (X, M) = 0, and I X denote the ideal of H X consisting of the morphisms f : M → N (with M, N in H X ) such that Hom A (X, f ) = 0. We define S(X) to be the quotient category H X /I X . Following the above convention, we usually identify the k-linear category S(X) with its quiver.
Recall that a module X in mod A is called a brick if End A (X) ∼ = k. Let A be an algebra and be a family of generalized standard infinite components of A . For an indecomposable brick X in , called the pivot, five admissible operations are defined, depending on the shape of the support S(X) of the functor Hom A (X, −)| . These admissible operations yield in each case a modified algebra A such that the modified translation quiver is a family of generalized standard infinite components in the Auslander-Reiten quiver A of A (see [32,Section 2] or [35,Section 4] for the figures illustrating the modified translation quiver ).
(ad 1) Assume S(X) consists of an infinite sectional path starting at X: Let t ≥ 1 be a positive integer, D be the full t ×t lower triangular matrix algebra, and Y 1 , . . ., In this case, is obtained by inserting in the rectangle consisting of the modules Z ij = k, X i ⊕ Y j , 1 1 for i ≥ 0, 1 ≤ j ≤ t, and X i = (k, X i , 1) for i ≥ 0. If t = 0 we set A = A[X] and the rectangle reduces to the sectional path consisting of the modules X i , i ≥ 0.
(ad 2) Suppose that S(X) admits two sectional paths starting at X, one infinite and the other finite with at least one arrow: where t ≥ 1. In particular, X is necessarily injective. We set A = A [X]. In this case, is obtained by inserting in the rectangle consisting of the modules (ad 3) Assume S(X) is the mesh-category of two parallel sectional paths: with the upper sectional path finite and t ≥ 2. In particular, X t−1 is necessarily injective. Moreover, we consider the translation quiver of obtained by deleting the arrows Y i → τ −1 A Y i−1 . We assume that the union of connected components of containing the modules is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver * , containing the pivot X. We set A = A [X]. In this case, is obtained from * by inserting the rectangle consisting of the modules Z ij = k, X i ⊕ Y j , 1 1 for i ≥ 1, 1 ≤ j ≤ i, and X i = (k, X i , 1) for i ≥ 0. (ad 4) Suppose that S(X) consists of an infinite sectional path, starting at X Let r be a positive integer. Moreover, we consider the translation quiver of obtained by deleting the arrows We assume that the union of connected components of containing the vertices τ −1 is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver * , containing the pivot X.
In this case, is obtained from * by inserting the rectangle consisting of the modules For r ≥ 1, let G be the full r × r lower triangular matrix algebra, U 1,t+1 , U 2,t+1 , . . ., U r,t+1 denote the indecomposable projective G-modules, U r,t+1 , U r,t+2 , . . ., U r,t+r denote the indecomposable injective G-modules, with U r,t+1 the unique indecomposable projective-injective G-module. We define the matrix algebra with r + 2 columns and rows. In this case, is obtained from * by inserting the following modules for i ≥ 0 and 1 ≤ j ≤ t + r, and X i = (k, X i , 1) for i ≥ 0. In the above formulas U sl is treated as a module over the algebra A s = A s−1 [U s−1,1 ], where A 0 = A and U 01 = Y (in other words A s is an algebra consisting of matrices obtained from the matrices belonging to A by choosing the first s +1 rows and columns). We note that the quiver Q A of A is obtained from the quiver of the double one-point by adding a path of length r + 1 with source at the extension vertex of
(ad 5) We define the modified algebra A of A to be the iteration of the extensions described in the definitions of the admissible operations (ad 1), (ad 2), (ad 3), (ad 4), and their finite versions corresponding to the operations (fad 1), (fad 2), (fad 3) and (fad 4). In this case, is obtained in the following three steps: first we are doing on one of the operations (fad 1), (fad 2) or (fad 3), next a finite number (possibly zero) of the operation (fad 4) and finally the operation (ad 4), and in such a way that the sectional paths starting from all the new projective modules have a common cofinite (infinite) sectional subpath. By an (ad 5)-pivot we mean an indecomposable brick X from the last (ad 4) operation used in the whole process of creating (ad 5).
Finally, let C be a (not necessarily connected) concealed canonical algebra and T C a separating family of stable tubes of C . Following [33] we say that an algebra A is a generalized multicoil enlargement of C using modules from T C if there exists a sequence of algebras C = A 0 , A 1 , . . . , A n = A such that A i+1 is obtained from A i by an admissible operation of one of the types (ad 1)-(ad 5), (ad 1 * )-(ad 5 * ) performed either on stable tubes of T A i , or on generalized multicoils obtained from stable tubes of T A i by means of operations done so far. The sequence C = A 0 , A 1 , . . . , A n = A is then called an admissible sequence for A. Observe that this definition extends the concept of a coil enlargement of a concealed canonical algebra introduced in [10]. We note that a generalized multicoil enlargement A of C invoking only admissible operations of type (ad 1) (respectively, of type (ad 1 * )) is a tubular extension (respectively, tubular coextension) of C in the sense of [41]. An algebra A is said to be a generalized multicoil algebra if A is a connected generalized multicoil enlargement of a product C of connected concealed canonical algebras.

Proposition 3.2 [33, Proposition 3.7]
Let C be a concealed canonical algebra, T C a separating family of stable tubes of C , and A a generalized multicoil enlargement of C using modules from T C . Then A admits a generalized standard family C A of generalized multicoils obtained from the family T C of stable tubes by a sequence of admissible operations corresponding to the admissible operations leading from C to A.
The following theorem, proved in [33, Theorem A], will be crucial for our further considerations.

Theorem 3.3 Let A be an algebra. The following statements are equivalent: (i) A admits a separating family of almost cyclic coherent components. (ii) A is a generalized multicoil enlargement of a concealed canonical algebra C.
Remark 3.4 The concealed canonical algebra C is called the core of A and the number m of connected summands of C is a numerical invariant of A. We note that m can be arbitrary large, even if A is connected. Let us also note that the class of algebras with generalized standard almost cyclic coherent Auslander-Reiten components is large (see [34,Proposition 2.9] and the following comments).
We note that the class of tubular extensions (respectively, tubular coextensions) of concealed canonical algebras coincides with the class of algebras having a separating family of ray tubes (respectively, coray tubes) in their Auslander-Reiten quiver (see [27,29]). Moreover, these algebras are quasitilted algebras of canonical type.
We recall also the following theorem on the structure of the module category of an algebra with a separating family of almost cyclic coherent Auslander-Reiten components proved in [33, Theorems C and F].

Theorem 3.5 Let A be an algebra with a separating family C A of almost cyclic coherent components in
Then the following statements hold. (l) and A (r) are tame.
In the above notation, the algebras A (l) and A (r) are called the left and right quasitilted algebras of A. Moreover, the algebras A (l) and A (r) are tame if and only if A (l) and A (r) are products of tilted algebras of Euclidean type or tubular algebras.
Recall that an algebra A is strongly simply connected if every convex subcategory of A is simply connected (see [44]). Clearly, if A is strongly simply connected then A is simply connected. We need the following result proved in [31, Theorem 1.1].

Theorem 3.6 Let
A be an algebra with a separating family of almost cyclic coherent components in A without exceptional configurations of modules. Then there are quotient algebras

Branch Extensions and Coextensions Let
The concept of branch coextension is defined dually.

Lemma 4.2 Let A be a generalized multicoil enlargement of a concealed canonical algebra
. . , A n = A be an admissible sequence for A, j ≥ p, X ∈ ind A j be an (ad 2) or (ad 3)-pivot, and A j +1 be the modified algebra of A j . If v is the corresponding extension point then there is a unique vertex u ∈ A (l) \ A (r) that satisfies: . If X is an (ad 2)-pivot (respectively, (ad 3)-pivot), then in the sequence of earlier admissible operations, there is an operation of type (ad 1 * ) or (ad 5 * ) which contains an operation (fad 1 * ) which gives rise to the pivot X of (ad 2) (respectively, to the pivot X of (ad 3) and to the modules Y 1 , . . . , Y t in the support of Hom A (X, −) restricted to the generalized multicoil containing X -see definition of (ad 3)). The operations done after must not affect the support of Hom A (X, −) restricted to the generalized multicoil containing X. Note that in general, in the sequence of earlier admissible operations, there can be an operation of type (ad 5) which contains an operation (fad 4) which gives rise to the pivot X of (ad 2) (respectively, to the pivot X of (ad 3)) but from Lemma [33, Lemma 3.10] this case can be reduced to (ad 5 * ) which contains an operation (fad 1 * ).
Let X be an (ad 2)-pivot, A j +1 = A j [X], and u, u 1 , . . . , u t (where X = I u , Y i = I u i for i ∈ {1, . . . , t} -see definition of (ad 2)) be the points in the quiver Q A j of A j corresponding to the new indecomposable injective A j -modules obtained after performing the above admissible operation (ad 1 * ) or the operation (fad 1 * ). Then u, u 1 , . . . , u t ∈ A (l) . Since X = rad P v , there must be a nonzero path from v to each vertex w which is a predecessor of u. Hence, each α ∈ v → is the starting arrow of a nonzero path from v to u, and there are at least two arrows in v → , namely: one from v to u t and one from v to a point in Supp X 1 , where X 1 is the immediate successor of X on the infinite sectional path in S(X) (see definition of (ad 2)). Moreover, since P v (u) = X(u) = k, all paths from v to u are congruent modulo I j +1 . The bound quiver Let now X be an (ad 3)-pivot, A j +1 = A j [X], and assume that we had r consecutive admissible operations of types (ad 1 * ) or (fad 1 * ), the first of which had X t as a pivot, and these admissible operations built up a branch K in A j with points u, u 1 , . . . , u t in Q A j , so that X t−1 and Y t are the indecomposable injective A j -modules corresponding respectively to u and u 1 , and both Y 1 and τ −1 A j Y 1 are coray modules in the generalized multicoil containing the (ad 3)-pivot X (where X, X t−1 , X t , Y 1 and Y t are as in the definition of (ad 3)). Then u, u 1 ∈ A (l) and X is the indecomposable A j -module given by: in any other case. Since X = rad P v , there must be a nonzero path from v to each vertex w which is a predecessor of u, but those which are predecessors of u 1 . Hence, each α ∈ v → is the starting arrow of a nonzero path from v to u, and there are at least two arrows in v → , namely: one from v to u 1 and one from v to a point in Supp X t , where X t is the immediate successor of X t−1 on the infinite sectional path in S(X) (see definition of (ad 3)). Moreover, since Again, from the proofs of [33, Theorems A and C], we have u ∈ A (l) \ A (r) , v ∈ A (r) \ A (l) , u 1 and the vertices of the branch K belong to A (l) ∩ A (r) .

Lemma 4.3 Let
A be a generalized multicoil enlargement of a concealed canonical algebra C = C 1 × · · · × C m . Moreover, let C = A 0 , . . . , A p = A (l) , A p+1 , . . . , A n = A be an admissible sequence for A, j ≥ p, X ∈ ind A j be an (ad 1)-pivot, A j +1 be the modified algebra of A j , and v be the corresponding extension point. Then the following statements hold.
(i) If there is a vertex u ∈ A (l) \ A (r) such that each α ∈ v → is the starting point of a nonzero path ω α ∈ A(v, u), then: (a) The vertex u is unique.
Proof Since X is an (ad 1)-pivot, the support S(X) consists of an infinite sectional path X = X 0 → X 1 → X 2 → · · · starting at X. Let t ≥ 1 be a positive integer, D be the full t × t lower triangular matrix algebra, and Y 1 , . . ., Y t be the indecomposable injective Dmodules with Y 1 the unique indecomposable projective-injective D-module (see definition of (ad 1)).
(i) Again, we know from [33, Section 4] that A (l) is a unique maximal convex branch coextension of C = C 1 × · · · × C m inside A, that is, i is a unique maximal convex branch coextension of C i inside A, i ∈ {1, . . . , m}. More precisely, B (l) where v 1 , . . . , v t are the points in the quiver Q A j +1 of A j +1 corresponding to the new indecomposable projective A j +1 -modules. Then A j +1 is the extension of B (l) s at X by the extension branch K consisting of the points v, v 1 , . . . , v t , that is, we have A j +1 = A j [X, K]. Since u does not belong to A (r) and for any α ∈ v → it is the starting point of a nonzero path ω α ∈ A(v, u), we get that u is the coextension point of the admissible operation (ad 2 * ) or (ad 3 * ). By [10, Lemma 3.1] the admissible operations (ad 2 * ) and (ad 3 * ) commute with (ad 1), so we can apply (ad 2 * ) after (ad 1) (respectively, (ad 3 * ) after (ad 1)). Using now [10, Lemma 3.3] (respectively, [10, Lemma 3.4]), we are able to replace (ad 1) followed by (ad 2 * ) (respectively, (ad 1) followed by (ad 3 * )) by an operation of type (ad 1 * ) followed by an operation of type (ad 2) (respectively, (ad 1 * ) followed by an operation of type (ad 3)). Therefore, the statements (a), (b) and (c) follow from Lemma 4.2.
(ii) A case by case inspection (which admissible operation gives rise to the (ad 1)-pivot X) shows that X is either simple module or the support of X is a linearly ordered quiver of type A t .

Lemma 4.4 Let
A be a generalized multicoil enlargement of a concealed canonical algebra C = C 1 × · · · × C m . Moreover, let C = A 0 , . . . , A p = A (l) , A p+1 , . . . , A n = A be an admissible sequence for A, j ≥ p, X ∈ ind A j be an (ad 4) or (ad 5)-pivot, A j +1 be the modified algebra of A j , and v be the corresponding extension point. If there is a vertex u ∈ A (l) \ A (r) such that for pairwise different arrows α 1 , . . . , α q ∈ v → , q ≥ 2 there are paths ω α 1 , . . . , ω α q ∈ A(v, u), then for arbitrary f, g ∈ {1, . . . , q}, f = g, one of the following cases holds: (i) At least one of ω α f , ω α g is zero path.
Proof It follows from [33, Section 4] that A (l) is a unique maximal convex branch coex- i is a unique maximal convex branch coextension of C i inside A, i ∈ {1, . . . , m}. More precisely, Assume that there is a vertex u ∈ A (l) \ A (r) such that for pairwise different arrows α 1 , . . . , α q ∈ v → , q ≥ 2, there are paths ω α 1 , . . . , ω α q ∈ A(v, u). Then there exists s ∈ {1, . . . , m} such that u ∈ B (l) s . Let X be an (ad 4)-pivot and Y 1 → Y 2 → · · · → Y t with t ≥ 1, be a finite sectional path in A j (as in the definition of (ad 4)). Note that this finite sectional path is the linearly oriented quiver of type A t and its support algebra (given by the vertices corresponding to the simple composition factors of the modules Y 1 , Y 2 , . . . , Y t ) is a tilted algebra of the path algebra D of the linearly oriented quiver of type A t . From [41, (4.4)(2)] we know that is a bound quiver algebra given by a branch in x, where x corresponds to the unique projectiveinjective D-module. Let be a generalized multicoil of A j +1 obtained by applying the admissible operation (ad 4), where X is the pivot contained in the generalized multicoil 1 , and Y 1 is the starting vertex of a finite sectional path contained in the generalized multicoil 1 or 2 . So, is obtained from 1 or from the disjoint union of two generalized multicoils 1 , 2 by the corresponding translation quiver admissible operations. In general, 1 and 2 are components of the same connected algebra or two connected algebras. Hence, we get two cases. In the first case X, Y 1 ∈ 1 or X ∈ 1 , Y 1 ∈ 2 and 1 , 2 are two components of the same connected algebra. In the second case X ∈ 1 , Y 1 ∈ 2 and 1 , 2 are two components of two connected algebras. Therefore, the bound quiver Q A j +1 of A j +1 in the first case is of the form for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A j [X], w is the extension point of A j [Y 1 ], w 1 , . . . , w d belong to the branch in w generated by the support of Y 1 ⊕ · · · ⊕ Y t , and αβ 1 . . . β h = 0 for some h ∈ {1, . . . , d + 1}. In the second case the bound quiver Q A j +1 of A j +1 is of the form for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A j [X], w is the extension point of A j [Y 1 ], w 1 , . . . , w d belong to the branch in w generated by the support of Y 1 ⊕ · · · ⊕ Y t , αβ 1 . . . β h = 0 for some h ∈ {1, . . . , d + 1}, and y is the coextension point of A j such that y ∈ A (l) \ A (r) . More precisely, y ∈ B (l) s , where s ∈ {1, . . . , m} and s = s. Moreover in both cases, we have P v (u) = X(u) = k or P v (u) = X(u) = 0, and hence all nonzero paths from v to u are congruent modulo I j +1 . So, A j +1 (v, u) is at most one-dimensional. We note that in the first case, the definition of (ad 4) (see the shape of the bound quiver Q A j +1 of A j +1 ) implies that if the paths ω α f , ω α g ∈ A j +1 (v, u) are nonzero and ω α f − ω α g ∈ I , then there is also a zero path ω α h ∈ A j +1 (v, u) for some h ∈ {1, . . . , q}, h = f = g.
Let X be an (ad 5)-pivot and be a generalized multicoil of A j +1 obtained by applying this admissible operation with pivot X. Then is obtained from the disjoint union of the finite family of generalized multicoils 1 , 2 , . . . , e by the corresponding translation quiver admissible operations, 1 ≤ e ≤ l, where l is the number of stable tubes of C used in the whole process of creating . Since in the definition of admissible operation (ad 5) we use the finite versions (fad 1)-(fad 4) of the admissible operations (ad 1)-(ad 4) and the admissible operation (ad 4), we conclude that the required statement follows from the above considerations.

Remark 4.5 Let
A be a generalized multicoil enlargement of a concealed canonical algebra C. We know from Theorems 3.3 and 3.5 that A can be obtained from A (l) by a sequence of admissible operations of types (ad 1)-(ad 5) or A can be obtained from A (r) by a sequence of admissible operations of types (ad 1 * )-(ad 5 * ). We note that all presented above lemmas can be formulated and proved for dual operations (ad 1 * )-(ad 5 * ) in a similar way.

The Separating Vertex
Let A be a triangular algebra. Recall that a vertex v of Q A is called separating if the radical of P v is a direct sum of pairwise nonisomorphic indecomposable modules whose supports are contained in different connected components of the subquiver Q(v) of Q A obtained by deleting all those vertices u of Q A being the source of a path with target v (including the trivial path from v to v).
We have the following lemma which follows from the proof of [44, Proposition 2.3] (see also [2, Lemma 2.3]).

Lemma 4.7 Let A be a triangular algebra and assume that A = B[X], where v is the extension vertex and X = rad A P v . If B is simply connected and v is separating, then A is simply connected.
Let D be the same as in the definition of (ad 1), that is, the full t × t lower triangular matrix algebra. Denote by Y 1 , . . ., Y t the indecomposable injective D-modules with Y = Y 1 the unique indecomposable projective-injective D-module.

Lemma 4.8 Let A be a triangular algebra and assume that
where v is the extension vertex and X ⊕ Y = rad A P v . If B is simply connected and v is separating, then A is simply connected.
Proof Since the module P v is a sink in the full subcategory of ind A consisting of projectives, the vertex v is a source in Q A . Moreover, A = (B × D)[X ⊕ Y ], where X is the indecomposable direct summand of rad A P v that belongs to mod B and Y is a directing module (that is, an indecomposable module which does not lie on a cycle in ind A) such that rad A P v = X ⊕ Y . Therefore, the proof follows from the proof of [44, Proposition 2.3] (see also the proof of Lemma 2.3 in [2]).

The Pointed Bound Quiver
In order to carry out the construction of the free product of two fundamental groups of bound quivers, and in analogy with algebraic topology where pointed spaces are considered, one can define a pointed bound quiver (Q, I, x), that is, a bound quiver (Q, I ) together with a distinguished vertex x (see [13,Section 3]). Given two pointed bound quivers Q = (Q , I , x ) and Q = (Q , I , x ), we can assume, without loss of generality, that Q 0 ∩ Q 0 = Q 1 ∩ Q 1 = ∅. We define the quiver Q = Q Q as follows: Q 0 is Q 0 ∪ Q 0 in which we identify x and x to a single new vertex x, and Q 1 = Q 1 ∪Q 1 . Then, Q and Q are identified to two full convex subquivers of Q, so walks on Q or Q can be considered as walks on Q. Thus, I and I generate two-sided ideals of kQ which we denote again by I and I . We define I to be the ideal I +I of kQ. It follows from this definition that the minimal relations of I are precisely the minimal relations of I together with the minimal relations of I give the minimal relations needed to determine the homotopy relation on (Q, I ). Moreover, we can consider an element w ∈ π 1 (Q , I , x ) as an element w ∈ π 1 (Q, I, x) (we denote by w the homotopy class of a walk w). Conversely, any (reduced) walk w in Q has a decomposition w = w 1 w 1 w 2 w 2 . . . w n w n , where w i and w i are walks in Q and Q for i ∈ {1, . . . , n}, respectively. Moreover, this decomposition is unique, up to reduced walk, and compatible with the homotopy relations involved. This leads us to the following proposition.

Proof of Theorem 1.1
The aim of this section is to prove Theorem 1.1 and recall the relevant facts.
We know from Theorem 3.3 that the Auslander-Reiten quiver A of A admits a separating family of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a concealed canonical algebra C. Let C = C 1 ×C 2 ×· · ·×C l ×C l+1 ×· · ·× C m be a decomposition of C into product of connected algebras such that C 1 , C 2 , . . . , C l are of type (p 1 , p 2 ) and C l+1 , C l+2 , . . . , C m are of type (p 1 , . . . , p t ) with t ≥ 3. Following [36], by h i we denote the number of all stable tubes of rank one from C i with 1 ≤ i ≤ l, used in the whole process of creating A from C, and h i = 0, if l + 1 ≤ i ≤ m. Moreover, let for i ∈ {1, . . . , m}. We define also f C i = max{e i − h i , 0}, for i ∈ {1, . . . , m} and set Note that we can apply the operations (ad 4), (fad 4), (ad 4 * ), (fad 4 * ) in two ways. The first way is when the sectional paths occurring in the definitions of these operations come from a component or two components of the same connected algebra. The second one is, when these sectional paths come from two components of two connected algebras. By d A we denote the number of all operations (ad 4), (fad 4), (ad 4 * ) or (fad 4 * ) which are of the first type, used in the whole process of creating A from C.
The Hochschild cohomology of a connected generalized multicoil algebra A has been described in [36, Theorem 1.1] using the numerical invariants of A (f A , d A and the others), depending on the types of admissible operations (ad 1)-(ad 5) and their duals, leading from a product C of concealed canonical algebras to A. Here, we will only need information about the first Hochschild cohomology of A, namely from [36, Theorem 1.1(iii)] we have: We are now able to complete the proof of Theorem 1.1. Since A is tame, we may restrict to the generalized multicoil enlargements of tame concealed algebras. Namely, we have the following consequence of Theorem 3.3 and [33, Theorem F]: A is tame and A admits a separating family of almost cyclic coherent components if and only if A is a tame generalized multicoil enlargement of a finite family C 1 , . . . , C m of tame concealed algebras (concealed canonical algebras of Euclidean type).
We first show the necessity. Suppose that A is simply connected. We must show that the first Hochschild cohomology H 1 (A) of A vanishes. Assume to the contrary that H 1 (A) = 0. Then by Theorem 5.1, d A + f A = 0. If d A = 0, then it follows from the proof of Lemma 4.4 (and its dual version) that A is not simply connected, a contradiction. Therefore, we may assume that d A = 0 and f A = 0. Since f A = l i=1 max{e i − h i , 0} = 0, we get that max{e j − h j , 0} = 0 for some j ∈ {1, . . . , l}. Note that, from Lemmas 4.2, 4.3, 4.4 and their proofs (and also from their dual versions -see Remark 4.5), we know how the bound quiver algebra changes after applying a given admissible operation. We have three cases to consider: (1) Assume that the algebra C j is of type (p 1 , p 2 ) with p 1 , p 2 ≥ 2. Then e j = 1 and h j = 0. The bound quiver algebra A = kQ/I is given by the quiver Q which can be visualized as follows: | | y y y y y y y y y where I the ideal in the path algebra kQ of Q over k generated by the elements ε 1 α 1 , α 2 γ 1 , ε 1 γ 1 − ε 2 γ 2 , β 2 ξ , α p 1 −1 ω, δα p 1 , σ 1 β p 2 −1 , σ 2 σ 3 ϕ, elements from parts A, B, D of Q, and elements from C i . Therefore, π 1 (Q, I ) is not trivial and so A is not simply connected. More precisely, it follows from Proposition 4.10 that π 1 (Q, I ) = Z π 1 (A) π 1 (B) π 1 (D) π 1 (C i ).
(2) Assume that the algebra C j is of type (p 1 , p 2 ) with p 1 = 1, p 2 ≥ 2. Then e j = 2, h j = 0 or h j = 1 and we have two subcases to consider. If e j = 2 and h j = 0, then the bound quiver algebra A = kQ/I is given by the quiver Q which can be visualized as follows: where I the ideal in the path algebra kQ of Q over k generated by the elements γβ p 2 , β p 2 −1 ω, σ 1 β 1 , σ 2 σ 3 ϕ, elements from parts A, B of Q, and elements from C i . Therefore, π 1 (Q, I ) is not trivial and so A is not simply connected. More precisely, it follows from Proposition 4.10 that π 1 (Q, I ) = Z π 1 (A) π 1 (B) π 1 (C i ). If e j = 2 and h j = 1, then the bound quiver algebra A = kQ/J is given by the quiver Q which can be visualized as in the previous subcase with the ideal J of kQ generated by the elements γ α 1 − aγβ p 2 . . . β 2 β 1 , β p 2 −1 ω, σ 1 β 1 , σ 2 σ 3 ϕ, elements from parts A, B of Q, and elements from C i , where a ∈ k\{0}. Note that in general, we can apply to a stable tube T of one of the following admissible operations: (ad 1), (ad 4), (ad 5) or their dual versions (with an infinite sectional path belonging to T ). Since h j = 1, we applied (in the above visualization) an admissible operation from the set S = {(ad 1), (ad 4), (ad 5)} to the algebra C j with pivot the regular C j -module corresponding to the indecomposable representation of the form lying in a stable tube of rank 1 in C j (see [42,XIII.2.4(c)]), where a ∈ k \ {0}. More precisely, if we apply (ad 1) with parameter t = 0, then we have to remove the arrow ε and the part B. Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/J , where the ideal J of kQ is generated by the elements of J \ {γ α 1 − aγβ p 2 . . . β 2 β 1 } ∪ {γ α 1 } and π 1 (Q, J ) is not trivial. Again, it follows from Proposition 4.10 that π 1 (Q, J ) = Z π 1 (A) π 1 (B) π 1 (C i ). If we apply an admissible operation from the set S * = {(ad 1 * ), (ad 4 * ), (ad 5 * )} to the algebra C j , the proof follows by dual arguments.
(3) Assume that the algebra C j is of type (p 1 , p 2 ) with p 1 = p 2 = 1. Then e j = 3, h j = 0, h j = 1 or h j = 2 and we have three subcases to consider. Note that in this case all stable tubes in C j have ranks equal to 1. Now, if e j = 3 and h j = 0, then j = l = 1 and the path algebra A = kQ is given by the Kronecker quiver Q: . Therefore, π 1 (Q) ∼ = Z and so A is not simply connected. If e j = 3 and h j = 1, then the bound quiver algebra A = kQ/J is given by the quiver Q which can be visualized as follows: with the ideal J in the path algebra kQ of Q over k generated by the element γ α − aγβ and elements from part A (the rest of Q), where a ∈ k \ {0}. Since h j = 1, we applied (in the above visualization) an admissible operation from the set S to the algebra C j with pivot the regular C j -module corresponding to the indecomposable representation of the . Moreover, if we apply an admissible operation from the set S * to the algebra C j , the proof follows by dual arguments. If e j = 3 and h j = 2, then the bound quiver algebra A = kQ/L is given by the quiver Q which can be visualized as follows: with the ideal L of kQ generated by the elements γ α − aγβ, αδ − bβδ, γ αδ and elements from parts A, B of Q, where a, b ∈ k \ {0} and a = b. Since h j = 2, we applied (in the above visualization) one admissible operation from the set S and one from the set S * to the algebra C j with pivots the regular C j -modules corresponding to the indecomposable representations of the form More precisely, if we apply (ad 1) (respectively, (ad 1 * )) with parameter t = 0, then we have to remove the arrow ε and the part B (respectively, the arrow λ and the part A). Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/L , where the ideal L of kQ is generated by the elements of L \ {γ α − aγβ, αδ − bβδ} ∪ {γ α, αδ} and π 1 (Q, L ) is not trivial. Again, it follows from Proposition 4.10 that π 1 (Q, L ) = Z π 1 (A) π 1 (B). In a similar way, one can show all the cases of applying two admissible operations from the set S ∪ S * to any two stable tubes of rank one from the Auslander-Reiten quiver of the Kronecker algebra. We now show the sufficiency. We know from Theorem 3.5 that there is a unique full convex subcategory A (l) = A (l) 1 × · · · × A (l) m of A which is a tubular coextension of the product C 1 × . . . × C m = C of a family C 1 , . . . , C m of tame concealed algebras (see remarks immediately after Theorem 5.1) such that A is obtained from A (l) by a sequence of admissible operations of types (ad 1)-(ad 5). We shall prove our claim by induction on the number of admissible operations leading from A (l) to the algebra A. Note that we can apply an admissible operation (ad 2), (ad 3), (ad 4) or (ad 5) if the number of all successors of the module Y i (which occurs in the definitions of the above admissible operations) is finite for each 1 ≤ i ≤ t. Indeed, if this is not the case, then the family of generalized multicoils obtained after applying such admissible operation is not sincere, and then it is not separating. Let C = A 0 , . . . , A p = A (l) , A p+1 , . . . , A n = A be an admissible sequence for A and assume that A p = A. In this case A is tame quasitilted algebra and our claim follows from [3, Theorem A]. Let k ≥ p, A = A k+1 and assume that A k is simply connected. Moreover, let v be the extension point of A k and X ∈ ind A k be the pivot of the admissible operation. Since H 1 (A) = 0, the vertex v is separating, by [44,Lemma 3.2]. Note that if the admissible operation leading from A k to A is of type (ad 1), (ad 2) or (ad 3), then A k is a connected algebra.
If X is an (ad 1)-pivot, then where rad A P v = X or rad A P v = X ⊕ Y respectively, D is the full t × t lower triangular matrix algebra over k for some t ≥ 1, and Y is the unique indecomposable projective-injective D-module (see definition of (ad 1)). Applying Lemma 4.7 or Lemma 4.8 respectively, we conclude that A is simply connected.
If X is an (ad 2)-pivot or (ad 3)-pivot, then A = A k [X], where rad A P v = X. Applying Lemma 4.7, we conclude that A is simply connected.
Let X be an (ad 4)-pivot and Y = Y 1 → Y 2 → · · · → Y t with t ≥ 1 be a finite sectional path in A k . Then, for r = 0, A = A k [X ⊕ Y ], and for r ≥ 1, with r +2 columns and rows (see definition of (ad 4)). We note that Y i is directing A-module for each 1 ≤ i ≤ t. Indeed, since H 1 (A) = 0, we get d A = 0, and so A k is not connected. Now, if r = 0, then A = A k [X ⊕ Y ] and rad A P v = X ⊕ Y . Then it follows from Lemma 4.7 that A is simply connected.
If r ≥ 1, then observe that the modified algebra A of A k can be obtained by applying r + 1 one-point extensions in the following way: k is separating and rad A (0) k P v 1 = U 01 , applying Lemma 4.7, we conclude that the algebra A (0) k is simply connected. Further, since the vertex v 2 of Q A (1) k is separating, k is simply connected, it follows from Lemma 4.7 that A (1) k is simply connected. Iterating a finite number of times the same arguments, we get that A (r−1) k is simply connected. Finally, since the vertex v of Q A is separating and rad A P v = X ⊕ U r1 , applying again Lemma 4.7, we get that A is simply connected.
Let X be an (ad 5)-pivot. Since in the definition of admissible operation (ad 5) we use the finite versions (fad 1)-(fad 4) of the admissible operations (ad 1)-(ad 4) and the admis-sible operation (ad 4), we conclude that the required statement follows from the above considerations.
This finishes the proof of Theorem 1.1.

Proof of Theorem 1.2
Let A be a generalized multicoil algebra. Then A is a connected generalized multicoil enlargement of a concealed canonical algebra C. Let C = C 1 ×C 2 ×· · ·×C l ×C l+1 ×· · ·× C m be a decomposition of C into product of connected algebras such that C 1 , C 2 , . . . , C l are of type (p 1 , p 2 ) and C l+1 , C l+2 , . . . , C m are of type (p 1 , . . . , p t ) with t ≥ 3. Since We now show that (i) implies (iii). Since all algebras C 1 , . . . , C m are of type (p 1 , . . . , p t ) with t ≥ 3 (l = 0), we get f A = 0. Assume to the contrary that H 1 (A) = 0. Then, by Theorem 5.1, d A + f A = 0. Therefore, d A = 0 and it follows from the proof of Lemma 4.4 (and its dual version) that A is not simply connected, a contradiction with (i).
We show that (iii) implies (iv). Assume to the contrary that there exists i ∈ {1, . . . , m} such that

Examples
We start this section with the following remark.

Remark 7.1
We can apply Theorem 1.1 to important classes of algebras. For example, to the cycle-finite algebras with separating families of almost cyclic coherent Auslander-Reiten components. Indeed, it is known (see [8]) that every cycle-finite algebra is tame. and with parameter r = 4. The modified algebra is equal to A.
Then the left quasitilted algebra A (l) of A is the convex subcategory of A being the bound quiver algebra kQ (l) /I (l) , where Q (l) is a full subquiver of Q given by the vertices 1, 2, . . . , 16 and I (l) = kQ (l) ∩ I is the ideal in kQ (l) . The right quasitilted algebra A (r) of A is the convex subcategory of A being the bound quiver algebra kQ (r) /I (r) , where Q (r) is a full subquiver of Q given by the vertices 1, 2, . . . , 7, 14, 15, . . . , 18 and I (r) = kQ (r) ∩ I is the ideal in kQ (r) . Note that A (l) and A (r) are tame.
Then the left quasitilted algebra A (l) of A is the convex subcategory of A being the product A (l) = A is the branch extension of the tame concealed algebra C 2 , Q (r) 2 is a full subquiver of Q given by the vertices 14, 15  2 are simply connected (and even strongly simply connected from [5,Corollary]). Finally, we mention that C 1 , C 2 are simply connected, A is a generalized multicoil algebra, A does not contain exceptional configurations of modules, and so this example illustrates also Theorem 1.2. and I the ideal in the path algebra kQ of Q over k generated by the elements aγβαλ − δ λ, γ ε, bπωνμ − πηξ, ζ μ, ϕψκ, where a, b ∈ k \ {0}. Then A is a generalized multicoil enlargement of a concealed canonical algebra C = C 1 × C 2 , where C 1 is the hereditary algebra of Euclidean type A 4 given by the vertices 1, 2, . . . , 5, and C 2 is the hereditary algebra of Euclidean type A 4 given by the vertices 6, 7, . . . , 10. Indeed, we apply (ad 1 * ) to C 1 with pivot the simple regular C 1 -module S 3 , and with parameter t = 2. The modified algebra B 1 is given by the quiver with the vertices 1, 2, . . . , 5, 11, 12, 13 bound by γ ε = 0. Next, we apply (ad 4) to B 1 × C 2 with pivot the simple regular C 2 -module S 7 and with the finite sectional path I 12 → S 13 consisting of the indecomposable B 1 -modules, and with parameters t = 2, r = 1. The modified algebra B 2 is given by the quiver with the vertices 1, 2, . . . , 15 bound by γ ε = 0, ζ μ = 0, ϕψκ = 0. Now, we apply (ad 1 * ) with parameter t = 0 to the algebra B 2 with pivot the regular C 1 -module corresponding to the indecomposable representation of the form 2 are tame. It follows from Theorems 3.3, 3.5(iii) and the above construction that A is tame and A admits a separating family of almost cyclic coherent components. Moreover, we have h 1 = 1, e 1 = 1, h 2 = 1, e 2 = 1, f C 1 = 0, f C 2 = 0, f A = f C 1 + f C 2 = 0, and d A = 0. Therefore, by Theorem 5.1, the first Hochschild cohomology space H 1 (A) = 0. Then, a direct application of Theorem 1.1 shows that the algebra A is simply connected. We note that, by [19,Proposition 1.6], We also mention that A (r) 2 are simply connected, by [3, Theorem A], and so A is not strongly simply connected. Moreover, by the above construction we know that A is a generalized multicoil algebra, such that A does not contain exceptional configurations of modules. Therefore, this example shows that simple connectedness assumption imposed on the considered concealed canonical algebras is essential for the validity of Theorem 1.2.
Then the left quasitilted algebra A (l) of A is the convex subcategory of A being the product A (l) = A 1 are the quasitilted algebras of wild types (4,4,13), (4,4,9), respectively. Moreover, A (l) 2 and A (r) 2 are tame. It follows from [7, Corollary 1.4] that C 1 is simply connected. Moreover, C 2 is also simply connected. By the above construction we know that A is a generalized multicoil algebra obtained from C 1 , C 2 and A does not contain exceptional configurations of modules.