Representations of nodal algebras of type A

We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness.

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Datum:	2013
Hauptverfasser:	Drozd, Yu., Zembyk, V.
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spelling	irk-123456789-1523012019-06-10T01:25:58Z Representations of nodal algebras of type A Drozd, Yu. Zembyk, V. We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness. 2013 Article Representations of nodal algebras of type A / Yu. Drozd, V. Zembyk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 2. — С. 179–200. — Бібліогр.: 13 назв. — англ. 1726-3255 2010 MSC:16G60, 16G10. http://dspace.nbuv.gov.ua/handle/123456789/152301 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness.
format	Article
author	Drozd, Yu. Zembyk, V.
spellingShingle	Drozd, Yu. Zembyk, V. Representations of nodal algebras of type A Algebra and Discrete Mathematics
author_facet	Drozd, Yu. Zembyk, V.
author_sort	Drozd, Yu.
title	Representations of nodal algebras of type A
title_short	Representations of nodal algebras of type A
title_full	Representations of nodal algebras of type A
title_fullStr	Representations of nodal algebras of type A
title_full_unstemmed	Representations of nodal algebras of type A
title_sort	representations of nodal algebras of type a
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/152301
citation_txt	Representations of nodal algebras of type A / Yu. Drozd, V. Zembyk // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 2. — С. 179–200. — Бібліогр.: 13 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT drozdyu representationsofnodalalgebrasoftypea AT zembykv representationsofnodalalgebrasoftypea
first_indexed	2025-07-13T02:46:55Z
last_indexed	2025-07-13T02:46:55Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 15 (2013). Number 2. pp. 179 – 200 c© Journal “Algebra and Discrete Mathematics” Representations of nodal algebras of type A Yuriy A. Drozd and Vasyl V. Zembyk Abstract. We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness. Introduction Nodal (infinite dimensional) algebras first appeared (without this name) in the paper [4] as pure noetherian1 algebras that are tame with respect to the classification of finite length modules. In [3] their derived categories of modules were described showing that such algebras are also derived tame. Voloshyn [13] described their structure. The definition of nodal algebras can easily be applied to finite dimensional algebras too. The simplest examples show that in finite dimensional case the above mentioned results are no more true: most nodal algebras are wild. It is not so strange, since they are obtained from hereditary algebras, most of which are also wild, in contrast to pure noetherian case, where the only hereditary algebras are those of type Ã. Moreover, even if we start from hereditary algebras of type A, we often obtain wild nodal algebras. So the natural question arise, which nodal algebras are indeed tame, at least if we start from a hereditary algebra of type A or Ã. In this paper we give an answer to this question (Theorem 5.2). The paper is organized as follows. In Section 1 we give the definition of nodal algebras and their description when the base field is algebraically 2010 MSC: 16G60, 16G10. Key words and phrases: representations of finite dimensional algebras, nodal algebras, gentle algebras, skewed-gentle algebras, inessential gluing. 1 Recall that pure noetherian means noetherian without minimal submodules. 180 Representations of nodal algebras closed. This description is alike that of [13]. Namely, a nodal algebra is obtained from a hereditary one by two operations called gluing and blowing up. Equivalently, it can be given by a quiver and a certain symmetric relation on its vertices. In Section 2 we consider a special sort of gluings which do not imply representation types. In Section 3 gentle and skewed- gentle nodal algebras are described. Section 4 is devoted to a class of nodal algebras called exceptional. We determine their representation types. At last, in Section 5 we summarize the obtained results and determine representation types of all nodal algebras of type A. 1. Nodal algebras We fix an algebraically closed field ❦ and consider algebras over ❦. Moreover, if converse is not mentioned, all algebras are supposed to be finite dimensional. For such an algebra A we denote by A-mod the category of (left) finitely generated A-modules. If an algebra A is basic (i.e. A/ rad A ≃ ❦ s), it can be given by a quiver (oriented graph) and relations (see [1] or [6]). Namely, for a quiver Γ we denote by ❦Γ the algebra of paths of Γ and by JΓ its ideal generated by all arrows. Then every basic algebra is isomorphic to ❦Γ/I, where Γ is a quiver and I is an ideal of ❦Γ such that J2 Γ ⊇ I ⊇ Jk Γ for some k. Moreover, the quiver Γ is uniquely defined; it is called the quiver of the algebra A. For a vertex i of this quiver we denote by Ar(i) the set of arrows incident to i. Under this presentation rad A = JΓ/I, so A/ rad A can be identified with the vector space generated by the “empty paths” εi, where i runs the vertices of Γ. Note that 1 = ∑ i εi is a decomposition of the unit of A into a sum of primitive orthogonal idempotents. Hence simple A-modules, as well as indecomposable projective A-modules are in one-to-one correspondence with the vertices of the quiver Γ. We denote by Āi the simple module corresponding to the vertex i and by Ai = εiA the right projective A-module corresponding to this vertex. We also write i = α+ (i = α−) if the arrow α ends (respectively starts) at the vertex i. Usually the ideal I is given by a set of generators R which is then called the relations of the algebra A. Certainly, the set of relations (even a minimal one) is far from being unique. An arbitrary algebra can be given by a quiver Γ with relations and multiplicities mi of the vertices i ∈ Γ. Namely, it is isomorphic to EndA P , where A is the basic algebra of A and P = ⊕ imiAi. (We denote by mM the direct sum of m copies of module M .) Recall also that path algebras of quivers without (oriented) cycles are just all hereditary basic algebras (up to isomorphism) [1, 6]. Yu. Drozd, V. Zembyk 181 Definition 1.1. A (finite dimensional) algebra A is said to be nodal if there is a hereditary algebra H such that 1) rad H ⊂ A ⊆ H, 2) rad A = rad H, 3) lengthA(H ⊗A U) ≤ 2 for each simple A-module U . We say that the nodal algebra A is related to the hereditary algebra H. Remark 1.2. From the description of nodal algebras it follows that the condition (3) may be replaced by the opposite one: 3′) lengthA(U ⊗A H) ≤ 2 for each simple right A-module U (see Corollary 1.10 below). Proposition 1.3. If an algebra A′ is Morita equivalent to a nodal algebra A related to a hereditary algebra H, then A′ is a nodal algebra related to a hereditary algebra H′ that is Morita equivalent to H. Proof. Denote J = rad H = rad A. Let P be a projective generator of the category mod-A of right A-modules such that A′ ≃ EndA P . Then also A ≃ EndA′ P ≃ P∨ ⊗A′ P , where P∨ = HomA′(P,A′) ≃ HomA(P,A). Let P ′ = P ⊗A H. Then P ′ is a projective generator of the category mod-H. Set H′ = EndH P ′. Note that HomH(P ′,M) ≃ HomA(P,M) for every right H-module M . In particular, EndH P ′ ≃ HomA(P, P ′). Hence, the natural map A′ → H′ is a monomorphism. Moreover, since P ′J = PJ , rad EndH P ′ = HomH(P ′, P ′J) ≃ ≃ HomA(P, P ′J) = HomA(P, PJ) = rad EndA P (see [6, Chapter III, Exercise 6]). Thus rad A′ = rad H′ ⊂ A′ ⊆ H′. Every simple A′-module is isomorphic to U ′ = P ⊗A U for some simple A-module U . Therefore H′ ⊗A′ U ′ = H′ ⊗A′ (P ⊗A U) ≃ (H′ ⊗A′ P ) ⊗A U ≃ ≃ (P ⊗A H) ⊗A U ≃ P ⊗A (H ⊗A U), since H′ ⊗A′ P ≃ HomA(P, P ⊗A H) ⊗A′ P ≃ ((P ⊗A H) ⊗A P∨) ⊗A′ P ≃ ≃ (P ⊗A H) ⊗A (P∨ ⊗A′ P ) ≃ P ⊗A (H ⊗A A) ≃ P ⊗A H. Hence lengthA′(H′ ⊗A′ U ′) = lengthA(H ⊗A U) ≤ 2, so A′ is nodal. 182 Representations of nodal algebras This proposition allows to consider only basic nodal algebras A, i.e. such that A/ rad A ≃ ❦ m for some m. We are going to present a con- struction that gives all basic nodal algebras. Definition 1.4. Let B be a basic algebra, B̄ = B/ rad B = ⊕m i=1 B̄i, where B̄i ≃ ❦ are simple B-modules. 1) Fix two indices i, j. Let Ā be the subalgebra of B̄ consisting of all m-tuples (λ1, λ2, . . . , λm) such that λi = λj , A be the preimage of Ā in B. We say that A is obtained from B by gluing the components B̄i and B̄j (or the corresponding vertices of the quiver of B). 2) Fix an index i. Let P = 2Bi ⊕ ⊕ k 6=i Bk, B′ = EndB P , B̄′ = B̄/ rad B̄ = ∏m k=1 B̄′ i, where B̄′ i ≃ Mat(2,❦) and B̄′ k ≃ ❦ for k 6= i. Let Ā′ be the subalgebra of B̄′ consisting of all m-tuples (b1, b2, . . . , bm) such that bi is a diagonal matrix, and A be the preimage of Ā in B′. We say that Ā is obtained from B by blowing up the component B̄i (or the corresponding vertex of the quiver B). This definition immediately implies the following properties. Proposition 1.5. We keep the notations of Definition 1.4. 1) If A is obtained from B by gluing components B̄i and B̄j, then it is basic and A/ rad A = Āij × ∏ k /∈{i,j} B̄k, where Āij = { (λ, λ) \| λ ∈ ❦ } ⊂ B̄i × B̄j. Moreover, rad A = rad B and B ⊗A Āij ≃ B̄i × B̄j. 2) If A is obtained from B by blowing up a component B̄i, then it is basic and A/ rad A = Āi1 × Āi2 × ∏ k 6=i B̄k, where Āis = {λess \| λ ∈ ❦ } and ess (s ∈ {1, 2}) denote the diagonal matrix units in B̄′ i ≃ Mat(2,❦). Moreover, rad A = rad B′ and B′ ⊗A Āis ≃ V , where V is the simple B̄′ i-module. We call the component Āij in the former case and the components Āis in the latter case the new components of A. We also identify all other simple components of Ā with those of B̄. Proposition 1.6. Under the notations of Definition 1.4 suppose that the algebra B is given by a quiver Γ with a set of relations R. 1) Let A be obtained from B by gluing the components corresponding to vertices i and j. Then the quiver of A is obtained from Γ by identifying the vertices i and j, while the set of relations for A is R ∪R′, where R′ is the set of all products αβ, where α starts at i (or at j) and β ends at j (respectively, at i). Yu. Drozd, V. Zembyk 183 2) Let A be obtained from B by blowing up the component corresponding to a vertex i and there are no loops at this vertex.2 Then the quiver of A and the set of relations for A are obtained as follows: • replace the vertex i by two vertices i′ and i′′; • replace every arrow α : j → i by two arrows α′ : j → i′ and α′′ : j → i′′; • replace every arrow β : i → j by two arrows β′ : i′ → j and β′′ : i′′ → j; • replace every relation r containing arrows from Ar(i) by two relations r′ and r′′, where r′ (r′′) is obtained from r by replacing each arrow α ∈ Ar(i) by α′ (respectively, by α′′); • keep all other relations; • for every pair of arrows α starting at i and β ending at i add a relation α′β′ = α′′β′′. Definition 1.7. We keep the notations of Definition 1.4 and choose pair- wise different indices i1, i2, . . . , ir+s and j1, j2, . . . , jr from { 1, 2, . . . ,m }. We construct the algebras A0,A1, . . . ,Ar+s recursively: A0 = B; for 1 ≤ k ≤ r the algebra Ak is obtained from Ak−1 by gluing the components B̄ik and B̄jk ; for r < k ≤ r+ s the algebra Ak is obtained from Ak−1 by blowing up the component B̄ik . In this case we say that the algebra A = Ar+s is obtained from B by the suitable sequence of gluings and blowings up defined by the sequence of indices (i1, i2, . . . , ir+s, j1, j2, . . . , jr). Note that the order of these gluings and blowings up does not imply the resulting algebra A. Usually such sequence of gluings and blowings up is given by a sym- metric relation ∼ (not an equivalence!) on the vertices of the quiver of B or, the same, on the set of simple B-modules B̄i: we set ik ∼ jk for 1 ≤ k ≤ r and ik ∼ ik for r < k ≤ r + s. Note that # { j \| i ∼ j } ≤ 1 for each vertex i. Theorem 1.8. A basic algebra A is nodal if and only if it is isomorphic to an algebra obtained from a basic hereditary algebra H by a suitable sequence of gluings and blowings up components. 2 One can modify the proposed procedure to include such loops, but this modification looks rather cumbersome and we do not need it. 184 Representations of nodal algebras In other words, a basic nodal algebra can be given by a quiver and a symmetric relation ∼ on the set of its vertices such that # { j \| i ∼ j } ≤ 1 for each vertex i. Proof. Proposition 1.5 implies that if an algebra A is obtained from a basic hereditary algebra H by a suitable sequence of gluings and blowings up, then it is nodal. To prove the converse, we use a lemma about semisimple algebras. Lemma 1.9. Let S̃ = ∏m i=1 S̃i be a semisimple algebra, where S̃i ≃ Mat(di,❦) are its simple components, S = ∏r k=1 Sk be its subalgebra such that Sk ≃ ❦ and lengthS(S̃ ⊗S Sk) ≤ 2 for all 1 ≤ k ≤ r. Then, for each 1 ≤ k ≤ r 1) either Sk = S̃i for some i, 2) or Sk ⊂ S̃i × S̃j for some i 6= j such that S̃i ≃ S̃j ≃ ❦ and Sk ≃ ❦ embeds into S̃i × S̃j ≃ ❦× ❦ diagonally, 3) or there is another index q 6= k such that Sk × Sq ⊂ S̃i for some i, S̃i ≃ Mat(2,❦) and this isomorphism can be so chosen that Sk × Sq embeds into S̃i as the subalgebra of diagonal matrices. Proof. Denote Lik = S̃i ⊗S Sk. Certainly Lik 6= 0 if and only if the projection of Sk onto S̃i is non-zero. Since Lk = S̃ ⊗S Sk = ⊕m i=1 Lik, there are at most two indices i such that Lik 6= 0. Therefore either Sk ⊆ S̃i for some i or Sk ⊆ S̃i × S̃j for some i 6= j and both Lik and Ljk are non-zero. Note that dim❦ Lik ≥ di and dim❦ Lk ≤ 2. So in the latter case S̃i ≃ S̃j ≃ ❦. Obviously, ❦ can embed into ❦× ❦ only diagonally. Suppose that Sk ⊆ S̃i but Sk 6= S̃i. Then di = 2, so S̃i ≃ Mat(2,❦). Then the unique simple S̃i-module is 2-dimensional. If Sk is the only simple S-module such that Sk ⊂ S̃i, then it embeds into S̃i as the subalgebra of scalar matrices, thus Lik ≃ S̃i is of dimension 4, which is impossible. Hence there is another index q 6= k such that Sq ⊂ S̃i. Then the image of Sk × Sq ≃ ❦ 2 in Mat(2,❦) is conjugate to the subalgebra of diagonal matrices [6, Chapter II, Exercise 2]. Let now A be a nodal algebra related to a hereditary algebra H̃, S̃ = H̃/ rad H̃ and Ā = A/ rad A. We denote by H the basic algebra of H̃ [6, Section III.5] and for each simple component S̃i of S̃ we denote by H̄i the corresponding simple components of H̄ = H/ rad H. We can apply Lemma 1.9 to the algebra S̃ = H̃/ rad H̃ and its subalgebra Ā = A/ rad A. Let (i1, j1), . . . , (ir, jr) be all indices such that the products S̃ik × S̃jk Yu. Drozd, V. Zembyk 185 occur as in the case (2) of the Lemma, while ir+1, . . . , ir+s be all indices such that S̃ik occur in the case (3). Then it is evident that A is obtained from H by the suitable sequence of gluings and blowings up defined by the sequence of indices (i1, i2, . . . , ir+s, j1, j2, . . . , jr). Since the construction of gluing and blowing up is left–right symmetric, we get the following corollary. Corollary 1.10. If an algebra A is nodal, so is its opposite algebra. In particular, in the Definition 1.1 one can replace the condition (3) by the condition (3′) from Remark 1.2. Thus, to define a basic nodal algebra, we have to define a quiver Γ and a sequence of its vertices (i1, i2, . . . , ir+s, j1, j2, . . . , jr). Actually, one can easily describe the resulting algebra A by its quiver and relations. Namely, we must proceed as follows: 1) For each 1 ≤ k ≤ r (a) we glue the vertices ik and jk keeping all arrows starting and ending at these vertices; (b) if an arrow α starts at the vertex ik (or jk) and an arrow β ends at the vertex jk (respectively ik), we impose the relation αβ = 0. 2) For each r < k ≤ r + s (a) we replace each vertex ik by two vertices i′k and i′′k; (b) we replace each arrow α : j → ik by two arrows α′ : j → i′k and α′′ : j → i′′k; (c) we replace each arrow β : ik → j by two arrows β′ : i′k → j and β′′ : i′′k → j; (d) if an arrow β starts at the vertex ik and an arrow α ends at this vertex, we impose the relation β′α′ = β′′α′′. We say that A is a nodal algebra of type Γ. In particular, if Γ is a Dynkin quiver of type A or a Euclidean quiver of type Ã, we say that A is a nodal algebra of type A. To define a nodal algebra which is not necessarily basic, we also have to prescribe the multiples li for each vertex i so that lik = ljk for 1 ≤ k ≤ r. In what follows we often present a basic nodal algebra by the quiver Γ, just marking the vertices i1, i2, . . . , ir, j1, j2, . . . , jr by bullets, with the 186 Representations of nodal algebras indices 1, 2, . . . , r, and marking the vertices ir+1, . . . , ir+s by circles. For instance: · α1 // •1 α2 // · α3 // α5 �� •1 •2 α4oo ◦3 α6 �� •2 In this example the resulting nodal algebra A is given by the quiver with relations ◦3′ α′ 6 �� · α1 // •1 α2 // · α3 oo α′ 5 OO α′′ 5 �� •2α4dd α2α3 = α2α4 = 0 α4α ′ 6 = α4α ′′ 6 = 0 α′ 6α ′ 5 = α′′ 6α ′′ 5 ◦3′′ α′′ 6 OO 2. Inessential gluings In this section we study one type of gluing which never implies the representation type. Definition 2.1. Let a basic algebra B be given by a quiver Γ with relations and an algebra A be obtained from B by gluing the components corresponding to the vertices i and j such that there are no arrows ending at i and no arrows starting at j. Then we say that this gluing is inessential. It turns out that the categories A-mod and B-mod are “almost the same.” Theorem 2.2. Under the conditions of Definition 2.1, there is an equiva- lence of the categories B-mod/〈 B̄i, B̄j 〉 and A-mod/〈 Āij 〉, where C/〈M 〉 denotes the quotient category of C modulo the ideal of morphisms that factor through direct sums of objects from the set M. Proof. We identify B-modules and A-modules with the representations of the corresponding quivers with relations. Recall that the quiver of A Yu. Drozd, V. Zembyk 187 is obtained from that of B by gluing the vertices i and j into one vertex (ij). For a B-module M denote by FM the A-module such that FM(k) = M(k) for any vertex k 6= (ij), FM(ij) = M(i) ⊕M(j), FM(γ) = M(γ) if γ /∈ Ar(ij), FM(α) = ( M(α) 0 ) if α ∈ Ar(i) \ Ar(j), FM(β) = ( 0 M(β) ) if β ∈ Ar(j) \ Ar(i) FM(α) = ( 0 0 M(α) 0 ) if α : i → j. (2.1) If f : M → M ′ is a homomorphism of B-modules, we define the homo- morphism Ff : FM → FM ′ setting Ff(k) = f(k) if k 6= (ij), Ff(ij) = ( f(i) 0 0 f(j) ) Thus we obtain a functor F : B-mod → A-mod. Obviously, FB̄i = FB̄j = Āij , so F induces a functor f : B-mod/〈 B̄i, B̄j 〉 → A-mod/〈 Āij 〉. Let now N be an A-module. We define the B-module GN as follows: GN(k) = N(k) if k /∈ {i, j}, GN(i) = N(ij)/N0(ij), where N0(ij) = ⋂ α−=(ij) KerN(α), GN(j) = ∑ β+=(ij) ImN(β), GN(β) = N(β) if β /∈ Ar(i), GN(α) is the induced map GN(i) → GN(k) if α : i → k. Note that if β+ = j, then ImN(β) ⊆ GN(j). If g : N → N ′ is a homomor- phism of A-modules, then g(ij)(GN(j)) ⊆ GN ′(j) and g(ij)(N0(ij)) ⊆ N ′ 0(ij). So we define the homomorphism Gg : GN → GN ′ setting Gg(k) = g(k) if k 6= i, Gg(i) is the map GN(i) → GN ′(i) induced by g(ij), Gg(j) is the resriction of g(ij) onto GN(j). 188 Representations of nodal algebras Thus we obtain a functor G : A-mod → B-mod. Since GĀij = 0, it induces a functor g : A-mod/〈 Āij 〉 → B-mod/〈 B̄i, B̄j 〉. Suppose that Gg = 0. It means that g(k) = 0 for k 6= (ij), Im g(ij) ⊆ ⋂ α−=(ij) KerN ′(α) and Ker g(ij) ⊇ ∑ β+=(ij) ImN(β). So g(ij) induces the map ḡ : N(j)/ ∑ β+=j ImN(β) → N ′(ij) with Im ḡ ⊆ ⋂ α−=(ij) KerN ′(α). So g = g′′g′, where g′ :N → N and g′′ : N → N ′, N(k) = 0 if k 6= (ij), N(ij) = N(j)/ ∑ β+=j ImN(β), g′(k) = g′′(k) = 0 if k 6= (ij), g′(ij) is the natural surjection N(ij) → N(ij), g′′(ij) = ḡ. Obviously, N ≃ mĀij for some m, so Ker G is just the ideal 〈 Āij 〉. By the construction, GFM(i) = M(i)/ ⋂ α−=i Kerα, GFM(j) = ∑ β+=j Im β, FGN(ij) = N(ij)/ ⋂ α−=i Kerα⊕ ∑ β+=(ij) ImN(β). So we fix for every B-module M a retraction ρM : M(j) → ∑ β+=j Im β, for every A-module N a retraction ρN : N(ij) → ∑ β+=(ij) Im β and define the morphisms of functors φ : IdB-mod → GF such that φM (k) = IdM(k) if k /∈ {i, j}, φM (j) = ρM , φM (i) is the natural surjection M(i) → GFM(i), Yu. Drozd, V. Zembyk 189 and ψ : IdA-mod → FG such that ψN (k) = IdN(k) if k 6= (ij), ψN (ij) = ρN . Evidently, if M has no direct summands B̄i and B̄j , then φM is an isomorphism. Also if N has no direct summands Āij , then ψN is an isomorphism. Therefore, g and f are mutually quasi-inverse, defining an equivalence of the categories B-mod/〈 B̄i, B̄j 〉 and A-mod/〈 Āij 〉. 3. Gentle and skewed-gentle case In what follows we only consider non-hereditary nodal algebras, since the representation types of hereditary algebras are well-known. Evidently, blowing up a vertex i such that there are no arrows starting at i or no arrows ending at i, applied to a hereditary algebra, gives a hereditary algebra. The same happens if we glue vertices i and j such that there are no arrows starting at these vertices or no arrows ending at them. Recall that a basic (finite dimensional) algebra A is said to be gentle if it is given by a quiver Γ with relations R such that 1) for every vertex i ∈ Γ, there are at most two arrows starting at i and at most two arrows ending at i; 2) all relations in R are of the form αβ for some arrows α, β; 3) if there are two arrows α1, α2 starting at i, then, for each arrow β ending at i, either α1β ∈ R or α2β ∈ R, but not both; 4) if there are two arrows β1, β2 ending at i, then, for each arrow α starting at i, either αβ1 ∈ R or αβ2 ∈ R, but not both. A basic algebra A is said to be skewed-gentle if it can be obtained from a gentle algebra B by blowing up some vertices i such that there is at most one arrow α starting at i, at most one arrow β ending at i and if both exist then αβ /∈ R.3 It is well-known that gentle and skewed-gentle algebras are tame, and even derived tame (i.e. their derived categories of finite dimensional mod- ules are also tame). Skowronski and Waschbüsch [12] proved a criterion 3 The original definition of skew-gentle algebras in [7] as well as that in [2] differ from ours, but one can easily see that all of them are equivalent. 190 Representations of nodal algebras of representation finiteness for biserial algebras, the class containing, in particular, all gentle algebras. We give a complete description of nodal algebras which are gentle or skewed-gentle. Theorem 3.1. A nodal algebra A is skewed-gentle if and only if it is obtained from a direct product of hereditary algebras of type A or Ã by a suitable sequence of gluings and blowings up defined by a sequence of vertices such that, for each of them, there is at most one arrow starting and at most one arrow ending at this vertex. It is gentle if and only if, moreover, it is obtained using only gluings. Proof. If A is related to a hereditary algebra H such that its quiver is not a disjoint union of quivers of type A or Ã, there is a vertex i in the quiver of H such that Ar(i) has at least 3 arrows. The same is then true for the quiver of A. Moreover, there are no relations containing more that one of these arrows, which is impossible in a gentle or skewed-gentle algebra. So we can suppose that the quiver of H is a disjoint union of quivers of type A or Ã. Let A is obtained from H by a suitable se- quence of gluings and blowings up defined by a sequence of vertices i1, i2, . . . , ir+s, j1, j2, . . . , jr. Suppose that there is an index 1 ≤ k ≤ r + s such that there are two arrows α1, α2 ending at ik (the case of two starting arrows is analogous). If k ≤ r and there is an arrow ending at jk, there are 3 vertices ending at the vertex (ij) in the quiver of A, neither two of them occurring in the same relation, which is impossible in gentle or skewed-gentle case. If there is an arrow β starting at jk, it occurs in two zero relations βα1 = βα2 = 0, which is also impossible. Finally, if we apply blowing up, we obtain three arrows incident to a vertex without zero relations between them which is impossible in a gentle algebra. Thus the conditions of the theorem are necessary. On the contrary, let H be a hereditary algebra and its quiver be a disjoint union of quivers of type A or Ã, i1 6= i2 be vertices of this quiver such that there is a unique arrow αk starting at ik as well as a unique arrow βk ending at ik (k = 1, 2). Then gluing of vertices i1, i2 gives a vertex i = (i1i2) in the quiver of the obtained algebra, two arrows αk starting at i and two arrows βk ending at i (i = 1, 2) satisfying relations α1β2 = α2β1 = 0. Therefore, gluing such vertices give a gentle algebra. Afterwards, blowing up vertices j such that there is one arrow α starting at j and one arrow β ending at it gives a skewed-gentle algebra since αβ 6= 0 in H. Yu. Drozd, V. Zembyk 191 4. Exceptional algebras We consider some more algebras obtained from hereditary algebras of type A. Definition 4.1. Let H be a basic hereditary algebra with a quiver Γ. 1) We call a pair of vertices (i, j) of the quiver Γ exceptional if they are contained in a full subquiver of the shape · β // i · · α1oo α2 . . . · αn−1 j · αnoo · γoo (4.1) or · i · βoo α1 // · α2 . . . · αn−1 αn // j · γ // · (4.2) where the orientation of the arrows α2, . . . , αn−1 is arbitrary. Possi- bly n = 1, then α1 = αn : j → i in case (4.1) and α1 = αn : i → j in case (4.2). 2) We call gluing of an exceptional pair of vertices exceptional gluing. 3) A nodal algebra is said to be exceptional if it is obtained from a hereditary algebra of type A by a suitable sequence of gluings consisting of one exceptional gluing and, maybe, several inessential gluings. Recall that inessential gluing does not imply the representations type of an algebra. So we need not take them into account only considering exceptional algebras obtained by a unique exceptional gluing. Note that such gluing results in the algebra A given by the quiver with relations · α1 �� · αnα1 = αnβ = 0 · βm . . . · β1 β // (ij) · αn AA · γoo γ1 . . . · γl (4.3) in case (4.1) or · · αn �� α1αn = βαn = 0 · βm . . . · β1 (ij) · βoo α1 ]] γ // · γ1 . . . · γl (4.4) 192 Representations of nodal algebras in case (4.2). The dotted line consists of the arrows α2, . . . , αn−1; if n = 1, we get a loop α at the vertex (ij) with the relations α2 = 0 and, respectively, αβ = 0 or βα = 0. We say that A is an (n,m, l)-exceptional algebra. We determine representation types of exceptional algebras. Theorem 4.2. An (n,m, l)-exceptional algebra is 1) representation finite in cases: (a) m = l = 0, (b) l = 0, m = 1, n ≤ 3, (c) l = 0, 2 ≤ m ≤ 3, n = 1, (d) m = 0, l = 1, n ≤ 2. 2) tame in cases: (a) l = 0, m = 1, n = 4, (b) l = 0, m = 2, n = 2, (c) l = 0, m = 4, n = 1, (d) m = 0, l = 1, n = 3. 3) wild in all other cases. Proof. We consider an algebra A given by the quiver with relations (4.4). Denote by C the algebra given by the quiver with relations · · αnxx α1αn = 0 •α1 ff (the bullet shows the vertex (ij)). It is obtained from the path algebra of the quiver Γn = 0 · α1 // 1 · α2 . . . n−1 · αn−1 αn // n · by gluing vertices 0 and n. This gluing is inessential, so we can use Theorem 2.2. We are interested in the indecomposable representations of C that are non-zero at the vertex • . We denote by L the set of such representations. They arise from the representations of the quiver Γn that are non-zero at the vertex 0 or n. Such representations are L̃i and L̃′ i (0 ≤ i ≤ n), where L̃i(k) { ❦ if k ≤ i, 0 if k > i; Yu. Drozd, V. Zembyk 193 L̃′ i(k) = { ❦ if k ≥ i, 0 if k < i. We denote by Li and L′ i respectively the representations FL̃i and FL̃′ i (see page 187, formulae (2.1)). Obviously Ln = L′ 0, L0 = L′ n = C• and dim❦ Li(•) = 1 for i 6= n as well as dim❦ L ′ i(•) = 1 for i 6= 0, while dim❦ Ln(•) = 2. We denote by ei (0 ≤ i < n) and e′ j (0 < i < n) basic vectors respectively of Li and L′ j , and by en, e ′ n basic vectors of Ln = L′ 0 such that e′ n ∈ ImLn(αn); then Ln(α1)(e′ n) = 0, while Ln(α1)(en) 6= 0. We consider the set E = { e0, e1, . . . , en } and the relation ≺ on E, where u ≺ v means that there is a homomorphism f such that f(u) = v. From the well-known (and elementary) description of representations of the quiver Γ and Theorem 2.2 it follows that ≺ is a linear order and ei ≺ e0 ≺ e′ j for all i, j. Let u0, u1, . . . , u2n be such a numeration of the elements of E that ui ≺ uj if and only if i ≤ j (then un = e0), and en = uk, e ′ n = ul (k < n < l). Note also that if f ∈ EndLn, the matrix f(•) in the basis en, e ′ n is of the shape ( λ 0 µ λ ) . Let M be an A-module, M̄ be its restriction onto C. Then M̄ ≃⊕ imiLi ⊕ ⊕ j m ′ jL ′ j . Respectively M(•) = ⊕2n+1 i=1 Ui, where Ui is gener- ated by the images of the vectors ui. Note that, for i > n, ui = e′ j for some j, so M(β)Ui = 0. Therefore, the matrices M(β) and M(γ) shall be considered as block matrices M(β) = ( B0 B1 . . . Bn 0 . . . 0 ) , M(γ) = ( C0 C1 . . . Cn Cn+1 . . . C2n ) , (4.5) where the matrices Bi, Ci correspond to the summands Ui. If f ∈ HomA(M,N), then f(•) is a block lower triangular matrix (fij), where fij : Uj → Ui and fij = 0 if i < j. Moreover, the non-zero blocks can be ar- bitrary with the only condition that fkk = fll. Hence given any matrix f(•) with this condition and invertible diagonal blocks fii, we can construct a module N isomorphic to M just setting N(β) = M(β)f(•), N(γ) = M(γ)f(•). Then one can easily transform the matrix M(γ) so that there is at most one non-zero element (equal 1) in every row and in every column, if i /∈ {k, l}, the non-zero rows of Ci have the form ( I 0 ) and 194 Representations of nodal algebras the non-zero rows of the matrix (Ck \|Cl) are of the form   0 0 0 0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 0 0   We subdivide the columns of the matrices Bi respectively to this subdivi- sion of Ci. It gives 2(n+ 2) new blocks B̃s(1 ≤ s ≤ 2(n+ 2)). Namely, the blocks B̃s with s odd correspond to the non-zero blocks of the matrices Ci and those with s even correspond to zero columns of Ci. Two extra blocks come from the subdivision of Ck into 4 vertical stripes. We also subdivide the blocks fij of the matrix f(•) in the analogous way. From now on we only consider such representations that the matrix M(γ) is of the form reduced in this way. One can easily check that it imposes the restrictions on the matrix f(•) so that the new blocks f̃st obtained from fij with 0 ≤ i, j ≤ n can only be non-zero (and then arbitrary) if and only if s > t and, moreover, either t is odd or s is even. It means that these new blocks can be considered as a representation of the poset (partially ordered set) Sn+2: 1 2 3 4 ... ... 2n+ 3 2(n+ 2) (in the sense of [11]). It is well-known [11] that Sn+2 has finitely many indecomposable representations. It implies that A is representation finite if m = l = 0. If l = 1, let γ : j → j1, γ1 : j2 → j1 (the case γ1 : j1 → j2 is analogous). We can suppose that the matrix M(γ1) is of the form ( I 0 0 0 ) . Yu. Drozd, V. Zembyk 195 Then the rows of all matrices Ci shall be subdivided respectively to this division of M(γ1): Ci = ( Ci1 Ci0 ) . Moreover, if f is a homomorphism of such representations, then f(j1) = ( c1 c2 0 c3 ) . Quite analogously to the previous considerations one can see that, reducing M(γ) to a canonical form, we subdivide the columns of Bi so that the resulting problem becomes that of representations of the poset C ′ n+3: · · · · · ... · ... · ... · · (n + 3 points in each column). The results of [8, 10] imply that this problem is finite if n ≤ 2, tame if n = 3 and wild if n > 3. Therefore, so is the algebra A if m = 0 and l = 1. If l > 1 then after a reduction of the matrices M(γ2),M(γ1) and M(γ) we obtain for M(β) the problem of the representations of the poset S′′ n+4 analogous to S and S ′ but with 4 columns and n + 4 point in every column. This problem is wild [10], hence the algebra A is also wild. If both l > 0 and m > 0, analogous consideration shows that if we reduce the matrix M(β1) to the form ( I 0 0 0 ) , the rows of the matrix M(β) will also be subdivided, so that we obtain the problem on representations of a pair of posets [8], one of them being S ′ n+3 and the other being linear ordered with 2 elements. It is known from [9] that this problem is wild, so the algebra A is wild as well. 196 Representations of nodal algebras Let now l = 0 and fix the subdivision of columns of M(β) described by the poset Sn+2 as above. If we reduce the matrices M(βi), which form representations of the quiver of type Am, the rows of M(β) will be subdivided so that as a result we obtain representations of the pair of posets, one of them being Sn+2 and the other being linear ordered with m + 1 elements. The results of [8, 9] imply that this problem is representation finite if either m = 1, n ≤ 3 or 2 ≤ m ≤ 3, n = 1, tame if either m = 1, n = 4, or m = 2, n = 2, or m = 4, n = 1. In all other cases it is wild. Therefore, the same is true for the algebra A, which accomplishes the proof. We use one more class of algebras. Definition 4.3. A nodal algebra is said to be super-exceptional if it is obtained from an algebra of the form (4.3) or (4.4) with n = 3 by gluing the ends of the arrow α2 in the case when such gluing is not inessential, and, maybe, several inessential gluings. Obviously, we only have to consider super-exceptional algebras ob- tained without inessential gluings. Using [9, Theorem 2.3], one easily gets the following result. Proposition 4.4. A super-exceptional algebra is 1) representation finite if m = l = 0, 2) tame if m+ l = 1, 3) wild if m+ l > 1. 5. Final result Now we can completely describe representation types of nodal algebras of type A. Definition 5.1. 1) We call an algebra A quasi-gentle if it can be ob- tained from a gentle or skewed-gentle algebra by a suitable sequence of inessential gluings. 2) We call an algebra good exceptional (good super-exceptional) if it is exceptional (respectively, super-exceptional) and not wild. Theorem 4.2 and Proposition 4.4 give a description of good exceptional and super-exceptional algebras. Yu. Drozd, V. Zembyk 197 Theorem 5.2. A non-hereditary nodal algebra of type A is representation finite or tame if and only if it is either quasi-gentle, or good exceptional, or good super-exceptional. In other cases it is wild. Before proving this theorem, we show that gluing or blowing up cannot “improve” representation type. Proposition 5.3. Let an algebra A be obtained from B by gluing or blowing up. Then there is an exact functor F : B-mod → A-mod such that FM ≃ FM ′ if and only if M ≃ M ′ or, in case of gluing vertices i and j, M and M ′ only differ by trivial direct summands at these vertices. Proof. Let A is obtained by blowing up a vertex i. We suppose that there are no loops at this vertex. The case when there are such loops can be treated analogously but the formulae become more cumbersome. Note that in the further consideration we do not need this case. For a B-module M set FM(k) = M(k) if k 6= i, FM(i′) = FM(i′′) = M(i), FM(α) = M(α) if α /∈ Ar(i) and FM(α′) = FM(α′′) = M(α) if α ∈ Ar(i). If f : M → M ′, set Ff(k) = f(k) if k 6= i and Ff(i′) = Ff(i′′) = f(i). It gives an exact functor F : B-mod → A-mod. Conversely, if N is an A-module, set GN(k) = N(k) if k 6= i and GN(i) = N(i′). It gives a functor G : A-mod → B-mod. Obviously GFM ≃ M , hence FM ≃ FM ′ implies that M ≃ M ′. Let now A be obtained from B by gluing vertices i and j. As above, we suppose that there are no loops at these vertices. For a B-module M set FM(k) = M(k) if k 6= (ij), FM(ij) = M(i)⊕M(j), FM(α) = M(α)) if α /∈ Ar(i) ∪ Ar(j), FM(α) = ( M(α) 0 ) ( or ( 0 M(α) )) if α− = i (respectively α− = j) and FM(β) = ( M(β) 0 ) ( or M(β) = ( 0 M(β) )) if β+ = i (respectively β+ = j). If f : M → M ′, set Ff(k) = f(k) if k 6= (ij) and f(ij) = f(i) ⊕ f(j). It gives an exact functor F : B-mod → A-mod. Suppose that φ : FM ∼ → FM ′, φ(ij) = ( φ11 φ12 φ21 φ22 ) , φ−1(ij) = ( ψ11 ψ12 ψ21 ψ22 ) . Then φ11M(β) = M ′(β)φ(k) and φ21M(β) = 0 if β : k → i, 198 Representations of nodal algebras φ22M(β) = M ′(β)φ(k) and φ12M(β) = 0 if β : k → j, M ′(α)φ11 = φ(k)M(α) and M ′(α)φ12 = 0 if α : i → k, M ′(α)φ22 = φ(k)M(α) and M ′(α)φ21 = 0 if α : j → k. and analogous relations hold for the components of φ−1(ij) with inter- change of M and M ′. We suppose that M has no direct summands B̄i and B̄j . It immediately implies that ⋂ α−=i KerM(α) ⊆ ∑ β+=i ImM(β) and ⋂ α−=j KerM(α) ⊆ ∑ β+=j ImM(β). If M ′ also contains no direct summands B̄i and B̄j , it satisfies the same conditions. Therefore Imψ21 ⊆ ⋂ α−=j KerM(α) ⊆ ∑ β+=j ImM(β), whence φ12ψ21 = 0 and φ11ψ11 = 1. Quite analogously, φ22ψ22 = 1 and the same holds if we interchange φ and ψ. Therefore we obtain an isomorphism φ̃ : M ∼ → M ′ setting φ̃(i) = φ11, φ̃(j) = φ22 and φ̃(k) = φ(k) if k /∈ {i, j}. Corollary 5.4. If an algebra A is obtained from B by gluing or blowing up and B is representation infinite or wild, then so is also A. Proof of Theorem 5.2. We have already proved the “if” part of the the- orem. So we only have to show that all other nodal algebras are wild. Moreover, we can suppose that there were no inessential gluings during the construction of a nodal algebra A. As A is neither gentle nor quasi-gentle, there must be at least one exceptional gluing. Hence A is obtained from an algebra B of the form (4.3) or (4.4) by some additional gluings (not inessential) or blowings up. One easily sees that any blowing up of B gives a wild algebra. Indeed, the crucial case is when n = 1, m = l = 0 and we blow up the end of the arrow β. Then, after reducing α1 and γ, just as in the proof of Theorem 4.2, we obtain for the non-zero parts of the two arrows obtained from β the problem of the pair of posets (1, 1) and S1 (see page 194), which is wild by [9, Theorem 2.3]. The other cases are even easier. Thus no blowing up has been used. Suppose that we glue the ends of β (or some βk) and γ (or some γk). Then, even for n = 1, m = l = 0, we obtain the algebra ·α ## β ** γ 44 · α2 = βα = 0 Yu. Drozd, V. Zembyk 199 (or its dual). Reducing α, we obtain two matrices of the forms β = ( 0 B2 B3 ) and γ = ( G1 G2 G3 ) . Given another pair (β′, γ′) of the same kind, its defines an isomorphic representation if and only if there are invertible matrices X and Y such that Xβ = β′Y and Xγ = γ′Y , and T is of the form Y =   Y1 Y3 Y4 0 Y2 Y5 0 0 Y1   , where the subdivision of Y corresponds to that of β, γ. The Tits form of this matrix problem (see [5]) is Q = x2 + 2y2 1 + y2 2 + 2y1y2 − 3xy1 − 2xy2. As Q(2, 1, 1) = −1, this matrix problem is wild. Hence the algebra A is also wild. Analogously, one can see that if we glue ends of some of βi or γi, we get a wild algebra (whenever this gluing is not inessential). Gluing of an end of some αi with an end of β or γ gives a wild quiver algebra as a subalgebra (again if it is not inessential). Just the same is in the case when we glue ends of some αi so that this gluing is not inessential and n > 3. It accomplishes the proof. References [1] M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1995. [2] V. Bekkert, E. N. Marcos, H. Merklen, Indecomposables in derived categories of skewed-gentle algebras, Commun. Algebra, 31, (2003), pp. 2615–2654. [3] I. Burban, Y. Drozd., Derived categories of nodal algebras, J. Algebra, 272, (2004), pp. 46–94. [4] Y. A. Drozd, Finite modules over pure Noetherian algebras, Trudy Mat. Inst. Steklov Acad. Sci. USSR, 183, (1990), pp. 56–68. [5] Y. Drozd, Reduction algorithm and representations of boxes and algebras, Comtes Rendue Math. Acad. Sci. Canada, 23 (2001), pp. 97–125. [6] Y. A. Drozd, V. V. Kirichenko. Finite Dimensional Algebras, Vyshcha Shkola, Kiev, 1980. (English translation: Springer–Verlag, 1994.) [7] C. Geiß, J. A. de la Peña, Auslander–Reiten components for clans, Bol. Soc. Mat. Mex., III. Ser., 5, N. 2, (1999), pp. 307–326. [8] M. M. Kleiner, Partially ordered sets of finite type, Zapiski Nauch. Semin. LOMI, 28, (1972), pp. 32–41. [9] M. M. Kleiner, Pairs of partially ordered sets of tame representation type, Linear Algebra Appl. 104, (1988), pp. 103–115. 200 Representations of nodal algebras [10] L. A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR, Ser. Mat., 39, (1975), pp. 963–991. [11] L. A. Nazarova, A. V. Roiter, Representations of partially ordered sets, Zapiski Nauch. Semin. LOMI, 28, (1972), pp. 5–31. [12] A. Skowronski and J. Waschbusch, Representation-finite biserial algebras, J. Reine Angew. Math., 345, (1983), pp. 172–181. [13] D. E. Voloshyn, Structure of nodal algebras, Ukr. Mat. Zh., 63, N. 7, (2011), pp. 880–888. Contact information Yu. Drozd, V. Zembyk Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3, 01601 Kyiv, Ukraine E-Mail: y.a.drozd@gmail.com, drozd@imath.kiev.ua, vaszem@rambler.ru URL: www.imath.kiev.ua/∼drozd Received by the editors: 18.02.2013 and in final form 18.02.2013.

Representations of nodal algebras of type A

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