Representations of nodal algebras of type E

We define representation types of nodal algebras of type E.

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Дата:	2017
Автори:	Drozd, Y.A., Golovashchuk, N.S., Zembyk, V.V.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2017
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Representations of nodal algebras of type E / Y.A. Drozd, N.S. Golovashchuk, V.V. Zembyk // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 16-34. — Бібліогр.: 12 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1559282019-06-18T01:29:20Z Representations of nodal algebras of type E Drozd, Y.A. Golovashchuk, N.S. Zembyk, V.V. We define representation types of nodal algebras of type E. 2017 Article Representations of nodal algebras of type E / Y.A. Drozd, N.S. Golovashchuk, V.V. Zembyk // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 16-34. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC:16G60, 16G10, 16G20. http://dspace.nbuv.gov.ua/handle/123456789/155928 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 representation types of nodal algebras of type E.
format	Article
author	Drozd, Y.A. Golovashchuk, N.S. Zembyk, V.V.
spellingShingle	Drozd, Y.A. Golovashchuk, N.S. Zembyk, V.V. Representations of nodal algebras of type E Algebra and Discrete Mathematics
author_facet	Drozd, Y.A. Golovashchuk, N.S. Zembyk, V.V.
author_sort	Drozd, Y.A.
title	Representations of nodal algebras of type E
title_short	Representations of nodal algebras of type E
title_full	Representations of nodal algebras of type E
title_fullStr	Representations of nodal algebras of type E
title_full_unstemmed	Representations of nodal algebras of type E
title_sort	representations of nodal algebras of type e
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/155928
citation_txt	Representations of nodal algebras of type E / Y.A. Drozd, N.S. Golovashchuk, V.V. Zembyk // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 16-34. — Бібліогр.: 12 назв. — англ.
series	Algebra and Discrete Mathematics
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 23 (2017). Number 1, pp. 16–34 c© Journal “Algebra and Discrete Mathematics” Representations of nodal algebras of type E Yuriy A. Drozd, Natalia S. Golovashchuk and Vasyl V. Zembyk Abstract. We define representation types of nodal algebras of type E. Introduction Finite dimesional nodal algebras were introduced in [7] as finite dime- sional analogues of nodal orders considered in [2, 4]. In this paper and in papers [8, 12] their represention types (finite, tame or wild) were studied. Unfortunately, as the second author noticed, some results of [12] were incorrect. In this paper we improve them using the technique of coverings from the papers [3, 6] and a lemma on representations of quivers with relations which seems to be of independent interest. Namely, in each considered case we construct a Galois covering Ã of the algebra A (in the sence of [3, 6]) with Galois group Z. Then, according to [6], A and Ã are of the same representation type. In most cases Lemma 2 reduces the description of representations of Ã to those of some quiver, which gives the answer. Moreover, this Lemma implies that the supports of indecomposable representations of Ã are bounded. Hence, according to [3], in finite or tame case all representations of A are just the natural images of those of Ã. Therefore, in these cases we also obtain a description of all representations of A. 2010 MSC: 16G60, 16G10, 16G20. Key words and phrases: nodal algebras, representation type, quivers. Yu. Drozd, N. Golovashchuk, V. Zembyk 17 1. Definitions and results We consider finite dimensional algebras over an algebraically closed field k. Recall the definition of nodal algebras [7]. Definition 1. An algebra A is called nodal if there is a hereditary algebra H such that 1) H ⊃ A ⊃ rad H = rad A 2) lengthA(H ⊗A U) 6 2 for any simple left A-module U . We say that the nodal algebra A is related to the hereditary algebra H. As any finite dimensional algebra is Morita equivalent to its basic alge- bra [1,5] and, moreover, an algebra is nodal if and only if its basic algebra is nodal, we only consider basic algebras A, i.e. such that A/radA ≃ k m for some m. We present such algebras by quivers with relations as in [1,5]. A basic nodal algebra can be obtained from We say that a nodal algebra A is of type E if it is related to a hereditary algebra which is Morita equivalent to the path algebra H of a Dynkin quiver of type E or of a Euclidean quiver of type Ẽ. If A is basic, it can be obtained from H by a sequence of glueing and blowing up vertices [7, 11]. Recall that glueing vertices i and j is said to be inessential if one of these vertices is a sink, while the other is a source. It is known that inessential glueing does not imply representation type [7]. We use the following numeration of the vertices and arrows of quivers of types E and Ẽ (independently of their orientations): E6 : 4 1 α 2 β 3 γ 5 δ ε 6 E7 : 4 1 α 2 β 3 γ 5 δ ε 6 ξ 7 E8 : 4 1 α 2 β 3 γ 5 δ ε 6 ξ 7 η 8 18 Nodal algebras Ẽ6 : 6′ 4 θ 1 α 2 β 3 γ 5 δ ε 6 Ẽ7 : 4 7′ σ 1 α 2 β 3 γ 5 δ ε 6 ξ 7 Ẽ8 : 4 1 α 2 β 3 γ 5 δ ε 6 ξ 7 η 8 µ 8′ The following theorems describe representations types of nodal al- gebras of type E. If the orientaions of arrows are not prescribed on the pictures, they are arbitrary and do not imply the reprsentations type. Theorem 1.1. Let a nodal algebgra A be isomorphic or anti-isomorphic to an algebra obtained from a quiver of type E with one of the following operations and some inessential glueings: 1) glueing verices 1 and 3 in the quiver E6 of the form 4 1 α // 2 β // 3 γ OO 5 δoo ε 6 or 4 γ �� 1 α // 2 β // 3 δ // 5 ε 6 2) glueing vertices 1 and 5 in the quiver E6 of the form 4 1 α // 2 β 3 γ δ // 5 ε // 6 Yu. Drozd, N. Golovashchuk, V. Zembyk 19 3) glueing vertices 2 and 6 in the quiver E6 of the form 4 1 α // 2 β // 3 γ 5 δ ε // 6 4) glueing vertices 2 and 7 in the quiver E7 of the form 4 1 α // 2 β // 3 γ 5 δ ε 6 ξ // 7 5) glueing vertices 3 and 7 in the quiver E7 of the form 4 γ �� 1 α 2 3 βoo δ // 5 ε 6 ξ // 7 Then A is reresentation finite. Theorem 1.2. Let a nodal algebgra A be isomorphic or anti-isomorphic to an algebra obtained from a quiver of type E with one of the following operations and some inessential glueings: 1) glueing vertices 1 and 3 in the quiver E7 of the form 4 1 α // 2 β // 3 γ OO 5 δoo ε 6 ξ 7 2) glueing vertices 1 and 6 in the quiver E7 of the form 4 1 α // 2 β 3 γ δ 5 ε // 6 ξ // 7 3) glueing vertices 2 and 4 in the quiver E6 of the form 4 1 α // 2 β // 3 γ OO 5 δ ε 6 20 Nodal algebras 4) glueing vertices 2 and 5 in the quiver E6 of the form 4 1 α // 2 β // 3 γ δ // 5 ε // 6 5) glueing vertices 2 and 8 in the quiver E8 of the form 4 1 α // 2 β // 3 γ 5 δ ε 6 ξ 7 η // 8 6) glueing vertices 3 and 7 in the quiver E7 of the form 4 1 α 2 β // 3 γ OO δ // 5 ε 6 ξ // 7 7) glueing vertices 3 and 8 in the quiver E8 of the form 4 γ �� 1 α 2 3 βoo δ // 5 ε 6 ξ 7 η // 8 Then A is tame (of infinite representation type). Theorem 1.3. If a nodal algebra A of type E is neither isomorphic nor anti-isomorphic to an algebra occuring in Theorems 1.1 or 1.2, it is wild. 2. Main lemma The proof of the theorems from Section 1 is substantially based upon the following result. Lemma 2. Let the quiver Γ of an algebra A be a union of three parts Γ+, Γ− and L such that: 1) Every intersection Γ+ ∩ Γ−, Γ+ ∩ L and Γ− ∩ L consists of a unique vertex o which is a sink in the quiver Γ+ and the source in the quiver Γ−. Yu. Drozd, N. Golovashchuk, V. Zembyk 21 2) L is a chain (a quiver of type Ak) and has at most one arrow starting at o and at most one arrow ending at o (for instance, o is a source or a sink in L). 3) If α is an arrow of Γ+ ending at o and β is an arrow of Γ− starting at o, then βα = 0 in A. 4) If λ+ is an arrow of L ending at o and λ− is an arrow of L starting at o, then λ−λ+ = 0 in A. If M be an indecomposable representation of A, then either M(α) = 0 for every arrow α from Γ+ ending at o or M(β) = 0 for every arrow β from Γ− starting at o. Proof. Note that L is of the shape a · · · o+ λ+ // o λ − // o− · · · b Let M(o) = M1 ⊕ M0, where M1 = ∑ α Im α, where α runs through all arrows of Γ+ ending at o. Then β(M1) = 0 for all arrows β of Γ− starting at o. With respect to this decomposition λ+ = ( λ1 + λ0 + ) and λ− = ( λ1 − λ0 − ) so that λ1 − λ1 + + λ0 − λ0 + = 0. If we decompose the restrictions of M onto the parts of L between a and o+ and between o− and b, the columns of λ+ and the rows of λ− split onto several blocks: λ+ = λ− = The number of columns in λ+ and the number of rows in λ− equal the lengths of the corresponding parts of L. The horizontal subdivision of λ+ and the vertical subdivision of λ− correspond, as above, to the decomposition M(o) = M1 ⊕ M0. As one can replace any vector v from a basis of M0 by v + u, where u ∈ M1, one can replace any row (say, i-th) from the upper part of λ+ by a sum of this row and a multiple of some row (say, j-th) from the lower part. Doing it, one also has to subtract the same multiple of the i-th column of the matrix λ− from its j-th column. Moreover, one can arrange the columns of λ+ (and the rows of λ−) so that one can add a multiple of any column from a vertical block to any column of the block which is on the right (respectively, add a multiple of any row from a horizontal block to any row from a block below). Using these transformations and the condition λ−λ+ = 0, one can decompose 22 Nodal algebras M1 = M11 ⊕ M10 and M0 = M01 ⊕ M11 so that the matrices λ+ and λ− become of the form: λ+ =   λ11 + 0 λ01 + 0   λ− = ( 0 λ10 − \| 0 λ00 − ) , where the subdivision of the rows of λ+ is the same as the subdivision of the columns of λ− and the first two rows (columns) are from λ1 + (respectively, from λ1 − ), while the other two are from λ0 + (respectively, from λ0 − ). Evidently, if M is indecomposable, at most one of the spaces Mij can be non-zero. It implies the claim of the lemma. 3. Proofs We prove Theorems 1.1–1.3 simultaneously, considering all sorts of glueing vertices in quivers of types E and Ẽ. One can easily see that any blowing up vertices in these quivers gives a wild algebra. Recall that a Galois covering [3, 6] of an algebra A given by a quiver with relations (Q, R) consists of an algebra Ã given by a quiver with relations (Q̃, R̃), a homomorphism of algebras Ã → A given by a homo- morphism of quivers φ : Q̃ → Q preserving relations and a free action of a group G (the Galois group of this covering) on the quiver Q̃ such that the preimages of vertices and arrows of the quiver Q under the map φ coincide with the orbits of this action. For any representation M of the algebra Ã one can consider the induced representation φ∗M = A ⊗Ã M of the algebra the A. The Galois group G also acts on the category of representations of Ã and φ∗M ≃ φ∗N if and only if N ≃ Mg for some g ∈ G. Note that usually the algebra Ã is infinite dimensional and the quiver Q̃ is infinite. They say that Ã is representation support bounded if there is a number C such that #{v ∈ Ver Q̃ \| M(v) 6= 0} 6 C for every finite dimensional representation M of the algebra Ã. Recall the main results about representation types of Galois coverings from [3,6]. Theorem 3.1. Let Ã is a Galois covering of an algebra A with Galois group G 1) If Ã is representation finite, so is A and vice versa. Moreover, in this case every indecomposable representation of A is isomorphic to φ∗M for some indecomposable representation M of Ã. Yu. Drozd, N. Golovashchuk, V. Zembyk 23 2) Let the group G be torsion free. If Ã is tame, so is A and vice versa. Moreover, if Ã is tame and representation support bounded, every indecomposable finite dimensional representation of A is isomorphic to φ∗M for some indecomposable representation M of Ã. We only consider several typical cases of glueings including one which seems the most complicated. All other cases are similar (as a rule, simpler). Case 1. Glueing vertices 1 and 3. We can suppose that the arrow α starts at the vertex 1. Depending on the orientation, we have several possibilities of essential glueings which do not give an algebra that is evidently wild. They are (for E6): 1.1. 4 1 α // 2 β // 3 γ OO 5 δoo ε 6 Then A is given by the quiver with relations 4 2 β 66 3 α vv γ OO 5 δoo ε 6 αβ = 0, αδ = 0. There is a Galois covering Ã of A with Galois group Z given by the quiver 4 2 β // 3 α �� γ OO 5 δoo ε 6 4 2 β // 3 α �� γ OO 5 δoo ε 6 4 2 β // 3 γ OO 5 δoo ε 6 24 Nodal algebras with the same relations αβ = 0, αδ = 0 for all α, β, δ. Here and later on we denote vertices and arrows of Q̃ by the same letters as their images in Q and suppose that the described quiver repeats infinitely up and down. Using Lemma 2 in two subsequent vertices 3, we can make zero the arrows β, δ, then α. We obtain the quiver of type E7 which is representation finite. Hence, Ã is representation finite and so is A. The same observations for the quiver E7 leads to a quiver of type Ẽ7 which is tame. Moreover, they show that Ã is representation support bounded. In case of the quiver Ẽ6, Q̃ has a wild subquiver without relations • • • • • • • • Therefore, in this case Ã and A are wild. The same is the case of other Dynkin and Euclidean quivers. 1.2. 4 γ �� 1 α // 2 β // 3 δ // 5 ε 6 Then A is given by the quiver with relations 4 γ �� 2 β 66 3 α vv δ // 5 ε 6 αβ = 0, αγ = 0. Yu. Drozd, N. Golovashchuk, V. Zembyk 25 There is a Galois covering Ã of A with Galois group Z given by the quiver 4 γ �� 2 β // 3 δ // α �� 5 ε 6 4 γ �� 2 β // 3 δ // α �� 5 ε 6 4 γ �� 2 β // 3 δ // 5 ε 6 with relations αβ = 0, αγ = 0. Using Lemma 2 in two subsequent vertices 3, we can make zero the arrows β, γ, then α. We obtain the quiver of type E8 which is representation finite. Hence, Ã is representation finite and so is A. In case of the quiver E7 we obtain, as a subquiber of Q̃ the wild quiver without relations. • • • • • • • • • • Therefore, in this case Ã and A are wild. The same is the case of other Dynkin and Euclidean quivers. 1.3. 4 γ �� 1 α // 2 3 βoo δ // 5 ε 6 Then A is given by the quiver with relations 4 γ �� 2 3 β hh α vv δ // 5 ε 6 αγ = 0, αδ = 0. 26 Nodal algebras There is a Galois covering Ã of A with Galois group Z given by the quiver 4 γ �� 2 3 βoo α �� 5 δoo ε 6 4 γ �� 2 3 βoo α �� 5 δoo ε 6 4 γ �� 2 3 α �� βoo 5 δoo ε 6 2 with relations αγ = 0, αδ = 0. It contiains a wild subquiver without relations • • • • • · · · • • with arbitrary long branch to the right. 1.4. 4 1 α // 2 β // 3 γ OO δ // 5 ε 6 Then A is given by the quiver with relations 4 2 β 66 3 α vv γ OO δ // 5 ε 6 αβ = 0. Yu. Drozd, N. Golovashchuk, V. Zembyk 27 One esaily sees that Q̃ also has a wild subquiver without relations, so Ã and A are wild. All the more so they are in other Dynkin and Euclidean cases. Thus we see that all cases of glueing vertices 1 and 3 which are representation finite or tame are listed in Theorems 1.1 and 1.2, while all other cases give wild algebras, as claimed in Theorem 1.3. One can also easily verify that any additional essential glueing leads to a wild algebra. Case 2. Glueing vertices 2 and 5. If both arrows α and β end at the vertex 2 and the glueing is essential, it gives an algebra which is evidently wild. So we can suppose that they have the orientation 1 α // 2 β // 3 . Then there are two possibilities of essential glueings which do not give an algebra that is evidently wild. They are (for E6): 2.1. (Perhaps, the most complicated case.) 4 1 α // 2 β // 3 γ δ // 5 ε // 6 Then A is given by the quiver with relations 6 4 1 α // 2 ε OO β (( 3 γ δ hh εα = 0, βδ = 0. 28 Nodal algebras There is a Galois covering Ã of A with Galois group Z given by the quiver 6 4 1 α // 2 ε OO β // 3 γ 6 4 1 α // 2 ε OO β // 3 δ WW γ 6 4 1 α // 2 ε OO β // 3 δ WW γ with relations εα = 0, βδ = 0. Using Lemma 2 in two subsequent vertices 2, we can make zero the arrow β, then δ. It gives the quiver • • • • ε __ • βoo γ δ // • ε ?? • α ?? • α __ with relations εα = 0. If we reduce the matrices corresponding to α, ε and γ, the rows of the matrices corresponding to β and δ subdivide into 3 parts and their columns subdivide into 2 parts. Moreover, one can make elementary transformations in every horizontal stripe (independently) and in every vertical stripe simultaneously in β and δ. Moreover, one can also add columns and rows from one part tp those of another part according to the following picture: β : δ : // �� Yu. Drozd, N. Golovashchuk, V. Zembyk 29 It is the problem of representations of a pair of posets (2) and (3, 3) in the sense of [9, 10], which is tame. Therefore, Ã and A are tame and Ã is representation support bounded. If we start with the quiver Ẽ6, we obtain in the same way representation of the pair of posets (3) and (3, 3), which is a wild problem. Hence Ã and A are wild. Analogously, in all other Dynkin and Euclidean cases we obtain wild algebras. 2.2. 4 1 α // 2 β // 3 γ δ // 5 6 εoo Then A is given by the quiver with relations 6 ε �� 4 1 α // 2 β (( 3 γ δ hh βε = 0, βδ = 0. There is a Galois covering Ã of A with Galois group Z given by the quiver 6 ε �� 4 1 α // 2 β �� 3 δoo γ 6 ε �� 4 1 α // 2 β �� 3 δoo γ 6 ε �� 4 1 α // 2 3 δoo γ 30 Nodal algebras with relations βε = 0, βδ = 0. It contains a wild subquiver without relations • • • • • • • Hence Ã and A are wild. All the more, it is so for all other Dynkin and Euclidean quivers. Thus, the only case when glueing 2 and 5 is not wild is case 4 from Theorem 1.2 and all other cases are wild. One can also easily verify that any additional essential glueing leads to a wild algebra. Case 3. Glueing vertices 3 and 7. We consider the quiver E7 and suppose that the arrow ξ ends at 7. There are the following variants of essential glueing which do not give an algebra that is evidently wild: 3.1. 4 γ �� 1 α 2 3 βoo δ // 5 6 ε ξ // 7 Then A is given by the quiver with relations 4 γ �� 1 α 2 3 βoo δ // 5 6 ε ξ hh βξ = 0, δξ = 0. Yu. Drozd, N. Golovashchuk, V. Zembyk 31 There is a Galois covering Ã of A with Galois group Z given by the quiver 4 γ �� 1 α 2 3 βoo δ // 5 6 ε 4 γ �� 1 α 2 3 βoo δ // 5 6 ε ξ ^^ 4 γ �� 1 α 2 3 βoo δ // 5 6 ε ξ ^^ with relations βξ = 0, δξ = 0. Using Lemma 2 in two subsequent vertices 3, we can make zero the arrows β, δ, then ξ. We obtain the quiver of type E8 which is representation finite. Hence, Ã is representation finite and so is A. One easily sees that for all other Dynkin and Euclidean quivers the same consideration gives a wild quiver. 3.2. 4 1 α 2 β // 3 γ OO δ // 5 6 ε ξ // 7 Then A is given by the quiver with relations 4 1 α 2 β // 3 γ OO δ // 5 6 ε ξ hh γξ = 0, δξ = 0. 32 Nodal algebras There is a Galois covering Ã of A with Galois group Z given by the quiver 4 1 α 2 β // 3 γ OO δ // 5 6 ε 4 1 α 2 β // 3 γ OO δ // 5 6 ε ξ ^^ 4 1 α 2 β // 3 γ OO δ // 5 6 ε ξ ^^ with relations γξ = 0, δξ = 0. Using Lemma 2 in two subsequent vertices 3, we can make zero the arrows γ, δ, then ξ. We obtain the quiver of type Ẽ8 which is tame. Hence, Ã is tame and so is A. Moreover, Ã is representation support bounded. One easily sees that for all other Dynkin and Euclidean quivers the same consideration gives a wild quiver. 3.3. 4 1 α 2 3 βoo γ OO 5 δoo 6 ε ξ // 7 Then A is given by the quiver with relations 4 1 α 2 3 βoo γ OO 5 δoo 6 ε ξ hh βξ = 0, γξ = 0. Yu. Drozd, N. Golovashchuk, V. Zembyk 33 There is a Galois covering Ã of A with Galois group Z given by the quiver 4 1 α 2 3 βoo γ OO 5 δoo 6 ε 4 1 α 2 3 βoo γ OO 5 δoo 6 ε ξ ^^ 4 1 α 2 3 βoo γ OO 5 δoo 6 ε ξ ^^ with relations βξ = 0, γξ = 0. It contiains a wild subquiver without relations • • • • • · · · • • with arbitrary long branch to the right. Hence the algebras Ã and A are wild. The same is if we start from another Dynkin or Euclidean quiver. Thus we see that all cases of glueing vertices 3 and 7 which are representation finite or tame are listed in Theorems 1.1 and 1.2, while all other cases give wild algebras, as claimed in Theorem 1.3. One can also easily verify that any additional essential glueing leads to a wild algebra. Just in the same way we check all other cases of gluing, which accom- plishes the proof of Theorems 1.1, 1.2 and 1.3. References [1] M. Auslander, I. Reiten, S. Smalø. Representation Theory of Artin Algebras. Cambridge University Press, 1995, xiv+423 pp. [2] I. Burban, Yu. Drozd, Derived categories of nodal algebras, J. Algebra, 272, 2004, pp 46–94. [3] P. Dowbor, A. Skowronski, On Galois coverings of tame algebras, Arch. Math., 44, 1985, pp. 522–529. 34 Nodal algebras [4] Yu. Drozd, Finite modules over pure Noetherian algebras, Tr. Mat. Inst. Steklova, 183, 1990, pp. 86-96. [5] Yu. Drozd, V. Kirichenko, Finite Dimensional Algebras, Vyshcha Shkola, Kyiv, 1980, 192 pp. [6] Yu. Drozd, S. Ovsienko Coverings of tame boxes and algebras, Preprint MPI 00-26, Max-Plank-Institut für Mathematik, Bonn, 2000, 44 pp. [7] Yu. Drozd, V. Zembyk, Representations of nodal algebras of type A, Algebra Discrete Math. 15, N.1, 2013, pp. 179–200. [8] Yu. Drozd, V. Zembyk, Representation type of nodal algebras of type D, Ukr. Mat. Zh., 66, N.7, 2014, pp. 930–938. [9] M. Kleiner. Partially ordered sets of finite type, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, 28, 1972, pp. 32-41. [10] Kleiner M. M Pairs of partially ordered sets of tame representation type, Linear Algebra Appl., 104, 1988, pp 103–115. [11] V. Zembyk, Structure of finite dimensional nodal algebras, Ukr. Mat. Zh., 66, N.3, 2014, pp. 415–419. [12] V. Zembyk, Representation type of nodal algebras of type E, Zb. Pr. Inst. Mat. NAN Ukr., 12, N.1, 2015, pp. 119-140. Contact information Yu. Drozd, V. Zembyk Institute of Mathematics of the Na- tional Academy of Sciences of Ukraine, Tereschenkivska 3, 01601 Kyiv, Ukraine E-Mail(s): y.a.drozd@gmail.com, drozd@imath.kiev.ua, vaszem@rambler.ru Web-page(s): www.imath.kiev.ua/∼drozd N. Golovashchuk Department of Mathematics, Kyiv National Taras Shevchenko University, Prosp. Acad. Glushkov 4e, Kyiv 03127, Ukraine E-Mail(s): golova@univ.kiev.ua Received by the editors: 12.03.2017 and in final form 17.03.2017.

Representations of nodal algebras of type E

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