On wildness of idempotent generated algebras associated with extended Dynkin diagrams

On wildness of idempotent generated algebras associated with extended Dynkin diagrams

Let Λ denote an extended Dynkin diagram with vertex set Λ0 = {0, 1,... ,n}. For a vertex i, denote by S(i) the set of vertices j such that there is an edge joining i and j; one assumes the diagram has a unique vertex p, say p = 0, with |S(p)| = 3. Further, denote by Λ \ 0 the full subgraph of Λ...

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Datum:2004
1. Verfasser: Bondarenko, V.M.
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Veröffentlicht: Інститут прикладної математики і механіки НАН України 2004
Schriftenreihe:Algebra and Discrete Mathematics
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Zitieren:On wildness of idempotent generated algebras associated with extended Dynkin diagrams / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 1–11. — Бібліогр.: 4 назв. — англ.

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spelling irk-123456789-1564572019-06-19T01:27:48Z On wildness of idempotent generated algebras associated with extended Dynkin diagrams Bondarenko, V.M. Let Λ denote an extended Dynkin diagram with vertex set Λ0 = {0, 1,... ,n}. For a vertex i, denote by S(i) the set of vertices j such that there is an edge joining i and j; one assumes the diagram has a unique vertex p, say p = 0, with |S(p)| = 3. Further, denote by Λ \ 0 the full subgraph of Λ with vertex set Λ0 \ {0}. Let ∆ = (δi |i ∈ Λ0) ∈ Z |Λ0| be an imaginary root of Λ, and let k be a field of arbitrary characteristic (with unit element 1). We prove that if Λ is an extended Dynkin diagram of type D₄, E₆ or E₇, then the k-algebra Qk(Λ, ∆) with generators ei , i ∈ Λ0 \ {0}, and relations e 2 i = ei , eiej = 0 if i and j 6= i belong to the same connected component of Λ \ 0, and Pn i=1 δi ei = δ01 has wild representation time. 2004 Article On wildness of idempotent generated algebras associated with extended Dynkin diagrams / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 1–11. — Бібліогр.: 4 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16G60; 15A21, 46K10, 46L05. http://dspace.nbuv.gov.ua/handle/123456789/156457 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description Let Λ denote an extended Dynkin diagram with vertex set Λ0 = {0, 1,... ,n}. For a vertex i, denote by S(i) the set of vertices j such that there is an edge joining i and j; one assumes the diagram has a unique vertex p, say p = 0, with |S(p)| = 3. Further, denote by Λ \ 0 the full subgraph of Λ with vertex set Λ0 \ {0}. Let ∆ = (δi |i ∈ Λ0) ∈ Z |Λ0| be an imaginary root of Λ, and let k be a field of arbitrary characteristic (with unit element 1). We prove that if Λ is an extended Dynkin diagram of type D₄, E₆ or E₇, then the k-algebra Qk(Λ, ∆) with generators ei , i ∈ Λ0 \ {0}, and relations e 2 i = ei , eiej = 0 if i and j 6= i belong to the same connected component of Λ \ 0, and Pn i=1 δi ei = δ01 has wild representation time.
format Article
author Bondarenko, V.M.
spellingShingle Bondarenko, V.M.
On wildness of idempotent generated algebras associated with extended Dynkin diagrams
Algebra and Discrete Mathematics
author_facet Bondarenko, V.M.
author_sort Bondarenko, V.M.
title On wildness of idempotent generated algebras associated with extended Dynkin diagrams
title_short On wildness of idempotent generated algebras associated with extended Dynkin diagrams
title_full On wildness of idempotent generated algebras associated with extended Dynkin diagrams
title_fullStr On wildness of idempotent generated algebras associated with extended Dynkin diagrams
title_full_unstemmed On wildness of idempotent generated algebras associated with extended Dynkin diagrams
title_sort on wildness of idempotent generated algebras associated with extended dynkin diagrams
publisher Інститут прикладної математики і механіки НАН України
publishDate 2004
url http://dspace.nbuv.gov.ua/handle/123456789/156457
citation_txt On wildness of idempotent generated algebras associated with extended Dynkin diagrams / V.M. Bondarenko // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 1–11. — Бібліогр.: 4 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT bondarenkovm onwildnessofidempotentgeneratedalgebrasassociatedwithextendeddynkindiagrams
first_indexed 2025-07-14T08:49:46Z
last_indexed 2025-07-14T08:49:46Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2004). pp. 1 – 11 c© Journal “Algebra and Discrete Mathematics” On wildness of idempotent generated algebras associated with extended Dynkin diagrams Vitalij M. Bondarenko Communicated by V. Mazorchuk Abstract. Let Λ denote an extended Dynkin diagram with vertex set Λ0 = {0, 1, . . . , n}. For a vertex i, denote by S(i) the set of vertices j such that there is an edge joining i and j; one assumes the diagram has a unique vertex p, say p = 0, with |S(p)| = 3. Further, denote by Λ \ 0 the full subgraph of Λ with vertex set Λ0 \ {0}. Let ∆ = (δi | i ∈ Λ0) ∈ Z |Λ0| be an imaginary root of Λ, and let k be a field of arbitrary characteristic (with unit element 1). We prove that if Λ is an extended Dynkin diagram of type D̃4, Ẽ6 or Ẽ7, then the k-algebra Qk(Λ,∆) with generators ei, i ∈ Λ0 \ {0}, and relations e2i = ei, eiej = 0 if i and j 6= i belong to the same connected component of Λ \ 0, and ∑n i=1 δi ei = δ01 has wild representation type. 1. Formulation of the main result Throughout the paper, we keep the right-side notation. By k we will denote a fixed field of arbitrary characteristic; for a natural number n and 1 ∈ k, we identify n1 with n. Let Λ be an nonoriented graph without loops and multiple edges, and let i be a vertex of Λ. Denote by S(i) the set of vertices j such that there is an edge joining i and j. The vertex i is said to be outer if |S(i)| ≤ 1, inner if |S(i)| > 1, weakly inner if |S(i)| = 2 and strongly inner if |S(i)| > 2. 2000 Mathematics Subject Classification: 16G60; 15A21, 46K10, 46L05. Key words and phrases: idempotent, extended Dynkin diagram, representation, wild type. Jo u rn al A lg eb ra D is cr et e M at h .2 Idempotent generated algebras Now let Λ be a finite connected tree with vertex set Λ0 = {0, 1, . . . , n}. We assume that 0 is the unique strongly inner vertex, and denote by Λ \ 0 the full subgraph of Λ with vertex set Λ0 \ {0}. Given a vector P = (p0, p1 . . . , pn) ∈ Z 1+n, we denote by Qk(Λ, P ) the k-algebra with generators ei, 1 ≤ i ≤ n, and relations 1) e2i = ei (1 ≤ i ≤ n); 2) eiej = 0 if i and j 6= i belong to the same connected component of Λ \ 0; 3) ∑n i=1 pi ei = p0. In this paper we study finite-dimensional representations of the al- gebra Qk(Λ, P ) with Λ being an extended Dynkin diagram. What we consider here is concerned with Yu. S. Samoilenko’s investigations [1]. Before we formulate the main results of this paper, we recall some definitions. Let Λ and Γ be algebras over a field k. A matrix representation of Λ over Γ is a homomorphism ϕ : Λ → Γs×s of algebras, where s is a natural number and Γs×s the set of all s× s-matrices with entries in Γ) s is called degree of ϕ and is denoted by degϕ. Two representations ϕ and ψ of Λ over Γ are called equivalent if degϕ = degψ and there exists an invertible matrix α, with entries in Γ, such that ϕ(λ)α = αψ(λ) for every λ ∈ Λ. The indecomposability and direct sum of representations are defined in a natural way. Let Λ be a k-algebra, and Σ = k〈x, y〉 be the free associative k-algebra in two noncommuting variables x and y. A representation γ of Λ over Σ is said to be strict if it satisfies the following conditions: 1) the representation γ ⊗ ϕ of Λ over k is indecomposable if a repre- sentation ϕ of Σ over k is indecomposable; 2) the representations γ⊗ϕ and γ⊗ϕ′ of Λ over k are nonequivalent if representations ϕ and ϕ′ of Σ over k are nonequivalent. Following [2] a k-algebra Λ is called wild (or of wild representation type) if it has a strict representation over Σ. Note that the matrix (γ ⊗ϕ)(λ) is obtained from the matrix γ(λ) by change x and y, respectively, on the matrices ϕ(x) and ϕ(y) (and a ∈ k on the scalar matrix aEs, where Es is the identity matrix of dimension s = degϕ). We now formulate the main result of the paper. Theorem. Let Λ be an extended Dynkin diagram of type D̃4, Ẽ6 or Ẽ7 and ∆ ∈ Z |Λ0| an imaginary root of Λ. Then the algebra Qk(Λ,∆) is wild. Jo u rn al A lg eb ra D is cr et e M at h .V. M. Bondarenko 3 In proving the theorem we can obviously take ∆ to be minimal posi- tive, which we denote by ∆0. 2. Proof of the theorem for Λ = D̃4 In this case the diagram Λ and vector ∆0 are 2 1 1 1 1 By the convention indicated above 0 denotes the strongly inner vertex, and 1, 2, 3 and 4 the outer vertices. Then the algebra Qk(Λ,∆0), with generators e1, e2, e3, e4, has the relations 1′) e2i = ei (1 ≤ i ≤ 4); 2′) e1 + e2 + e3 + e4 = 2. Consider the following representation γ of Qk(Λ,∆0) over Σ = k < x, y > : γ(e1) =           1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0           , γ(e2) =           0 0 −1 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1           , Jo u rn al A lg eb ra D is cr et e M at h .4 Idempotent generated algebras γ(e3) =           1 0 x y x2 − x+ y xy − y 1 0 1 1 0 x− 1 y 0 0 0 0 0 −x+ 1 −y 0 0 0 0 0 −1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0           , γ(e4) =           0 0 −x+ 1 −y −x2 + x− y −xy + y −1 0 0 −1 1 −x+ 1 −y 0 0 0 1 0 x y 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1           . In [3] the author has proved that this representation is strict. 3. Proof of the theorem for Λ = Ẽ6 In this case the diagram Λ and vector ∆0 are 3 2 1 2 21 1 We assume that the vertices 1, 3, 5 are outer, the vertices 2, 4, 6 are weakly inner (the vertex 0 is strongly inner), and the edges join the vertices 1 and 2, 3 and 4, 5 and 6, and consequently 0 with 2, 4, 6. Then the algebra Qk(Λ,∆0), with generators e1, e2, . . . , e6, has the relations 1′) e2i = ei (1 ≤ i ≤ 6); 2′) e1e2 = e2e1 = 0, e3e4 = e4e3 = 0, e5e6 = e6e5 = 0; 3′) e1 + e3 + e5 + 2(e2 + e4 + e6) = 3. Consider the following representation γ of Qk(Λ,∆0) over Σ = k < x, y > : Jo u rn al A lg eb ra D is cr et e M at h .V. M. Bondarenko 5 γ(e1) =           1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 1           , γ(e2) =           0 0 −2 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           , γ(e3) =           0 0 0 0 0 0 0 0 1 0 −x −y 2 0 0 0 1 −1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           , γ(e4) =           0 0 0 −1 1 1 −1 0 0 0 x y −x −y 0 0 0 1 0 −1 0 0 0 0 1 0 −1 0 0 0 0 0 1 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0           , γ(e5) =           0 0 0 0 2 0 0 0 0 0 −x −y 0 0 0 0 0 −1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           , Jo u rn al A lg eb ra D is cr et e M at h .6 Idempotent generated algebras γ(e6) =           1 0 1 1 −2 −1 1 0 0 0 0 0 x− 1 y 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1           . We will prove that the representation γ is strict. Let ϕ and ϕ′ be representations of Σ over k having the same de- gree: degϕ = degϕ′ = d. The representation γ ⊗ ϕ (respectively, γ ⊗ ϕ′) is uniquely defined by the matrices As = (γ ⊗ ϕ)(es) (respec- tively, A′ s = (γ ⊗ ϕ′)(es)), where s = 1, 2, . . . , 6. It is natural to consider these matrices as block matrices with blocks (As)ij and (A′ s)ij of degree d (i, j = 1, 2, . . . , 7). Then Hom(γ ⊗ ϕ, γ ⊗ ϕ′) = {T ∈ k7d×7d | AsT = TA′ s for each s = 1, 2, . . . , 6}. Lemma 1. Let T = (Tij), i, j = 1, 2, . . . , 7, be a block matrix (over k) with blocks Tij of degree d, belonging to Hom(γ⊗ϕ, γ⊗ϕ′). Then Tij = 0 if i 6= j and (i, j) 6= (1, 6), (1, 7), and T11 = T22 = . . . = T77. Proof. Denote by I, II, . . . ,VI the matrix equalities A1T = TA′ 1, A2T = TA′ 2, . . . , A6T = TA′ 6, respectively. The (matrix) equality (AsT )ij = (TA′ s)ij , i, j ∈ {1, 2, . . . , 7}, induced by an equality AsT = TA′ s, is de- noted by I(i, j) for s = 1, II(i, j) for s = 2, . . ., VI(i, j) for s = 6. It is easy to see that I(2, 1) implies T21 = 0; I(3, 1) implies T31 = 0; I(6, 4) implies T64 = 0; I(6, 5) implies T65 = 0; I(7, 4) implies T74 = 0; I(7, 5) implies T75 = 0; II(2, 4) implies T24 = 0; II(2, 5) implies T25 = 0; II(2, 6) implies T26 = 0; II(2, 7) implies T27 = 0; II(3, 4) implies T34 = 0; II(3, 5) implies T35 = 0; II(3, 6) implies T36 = 0; II(3, 7) implies T37 = 0; III(1, 2) implies T12 = 0; III(1, 3) implies T13 = 0; III(4, 2) implies T42 = 0; III(4, 3) implies T43 = 0; III(5, 2) implies T52 = 0; III(5, 3) implies T53 = 0; III(6, 2) implies T62 = 0; III(6, 3) implies T63 = 0; III(7, 2) implies T72 = 0; III(7, 3) implies T73 = 0; V(4, 6) implies T46 = 0; V(4, 7) implies T47 = 0; V(5, 6) implies T56 = 0; V(5, 7) implies T57 = 0; I(1, 4) and T34 = 0 imply T14 = 0; I(1, 5) and T35 = 0 imply T15 = 0; IV(6, 4), T62 = 0, T63 = 0 and T64 = 0 imply T61 = 0; IV(7, 4), T72 = 0, T73 = 0 and T74 = 0 imply T71 = 0; IV(4, 1) and T61 = 0 imply T41 = 0; IV(5, 1) and T71 = 0 imply T51 = 0; VI(1, 2), T12 = 0, T42 = 0, T52 = 0, T62 = 0 and T72 = 0 imply T32 = 0; V(3, 5), T31 = 0, T32 = 0 and T35 = 0 imply T45 = 0; IV(3, 7), T31 = 0, T32 = 0, T35 = 0 and T47 = 0 imply T67 = 0; IV(1, 4), VI(1, 4), T12 = 0, T13 = 0, T14 = 0, T34 = 0, T64 = 0 and T74 = 0 imply T54 = 0; IV(5, 6), T51 = 0, T52 = 0, T53 = 0, T54 = 0 Jo u rn al A lg eb ra D is cr et e M at h .V. M. Bondarenko 7 and T56 = 0 imply T76 = 0; IV(2, 5),IV(5, 7), VI(2, 7), T21 = 0, T25 = 0, T27 = 0, T45 = 0, T51 = 0, T52 = 0, T57 = 0, T65 = 0, T67 = 0 and T75 = 0 imply T23 = 0. So Tij = 0 when i 6= j and (i, j) 6= (1, 6), (1, 7). Then it follows from IV(1, 4),IV(1, 5), IV(1, 6), IV(1, 7), III(3, 4), III(2, 4) and VI(2, 6) that T11 = T22 = . . . = T77. It follows from the lemma that a matrix T = (Tij) belonging to Hom(γ ⊗ ϕ, γ ⊗ ϕ′) satisfies the following conditions: a) T is invertible if and only if T0 = T11 = T22 = . . . = T77 is invertible; b) ϕ(x)T0 = T0ϕ ′(x) and ϕ(y)T0 = T0ϕ ′(y). (In fact it follows from the lemma that the equalities I-VI are equiv- alent to the equalities b)). Therefore the representation γ satisfies the condition 2) (of the defi- nition of a strict representation). It remains to prove that γ satisfies the condition 1) or, in other words, ϕ is decomposable if so is γ ⊗ ϕ. We will denote by 0s and Es the s× s zero and identity matrices, respectively. Denote by Hom(ϕ,ϕ) the algebra of endomorphisms of ϕ, i.e. Hom(ϕ,ϕ) = {S ∈ kd×d | ϕ(x)S = Sϕ(x), ϕ(y)S = Sϕ(y)}. Decomposability of γ⊗ϕ implies that the k-algebra Hom(γ⊗ϕ, γ⊗ϕ) (of endomorphisms of γ ⊗ ϕ) contains an idempotent T 6= 07d, E7d (see, for example, [4, ch.V]). Then, by the lemma, the matrix T0 = T11 = T22 = . . . = T77 is an idempotent; moreover, T0 6= 0d, Ed, because otherwise it would follow from the equality T 2 = T that T = T0 ⊕ T0 ⊕ . . . ⊕ T0, where T0 occurs 7 times, or in other words T = 07d or T = E7d, respec- tively. Since T0 belong to the algebra Hom(ϕ,ϕ) = {S ∈ kd×d | ϕ(x)S = Sϕ(x), ϕ(y)S = Sϕ(y)} (see the condition b)), the representation ϕ is decomposable (see again [4, ch.V]). 4. Proof of the theorem for Λ = Ẽ7 In this case the diagram Λ and vector ∆0 are 4 2 3 32 21 1 We assume that the vertices 1, 4, 7 are outer, the vertices 2, 3, 5, 6 are weakly inner (the vertex 0 is strongly inner), and the edges join the Jo u rn al A lg eb ra D is cr et e M at h .8 Idempotent generated algebras vertices 1 and 2, 2 and 3, 4 and 5, 5 and 6, and consequently 0 with 3, 6, 7. Then the algebra Qk(Λ,∆0), with generators e1, e2, . . . , e7, has the relations 1′) e2i = ei (1 ≤ i ≤ 7); 2′) e1e2 = e2e1 = 0, e2e3 = e3e2 = 0, e1e3 = e3e1 = 0, e4e5 = e5e4 = 0, e5e6 = e6e5 = 0, e4e6 = e6e4 = 0; 3′) e1 + e4 + 2(e2 + e5 + e7) + 3(e3 + e6) = 4. Consider the following representation γ of Qk(Λ,∆0) over Σ = k < x, y > : γ(e1) =               0 0 1 0 −3 0 0 3 0 0 1 0 −3 0 0 0 x y 0 0 1 0 −3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               , γ(e2) =               0 0 0 0 0 0 0 0 0 0 0 0 3 0 −3 0 0 0 0 0 0 0 3 0 −3 0 0 0 0 0 1 0 −1 0 0 0 0 0 0 0 1 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               , γ(e3) =               0 0 0 0 0 1 0 1 0 0 0 0 0 0 3 0 3 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               , Jo u rn al A lg eb ra D is cr et e M at h .V. M. Bondarenko 9 γ(e4) =               0 0 0 0 0 3 3 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 −3 0 0 0 0 0 0 0 −1 0 −3 0 0 0 0 0 1 0 3 0 0 0 0 0 0 0 1 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               , γ(e5) =               1 0 1 0 0 −3 −3 −12 −9 0 0 0 0 0 0 0 x y 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 0 0 0 0 0 0 0 0 0 −3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1               , γ(e6) =               0 0 −1 0 1 0 1 3 3 0 1 0 −1 0 −1 0 −x− 3 −y 0 0 1 0 −1 0 −1 −3 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               , γ(e7) =               1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1               . Jo u rn al A lg eb ra D is cr et e M at h .10 Idempotent generated algebras We will prove that the representation γ is strict. Let ϕ and ϕ′ be representations of Σ over k having the same de- gree: degϕ = degϕ′ = d. The representation γ ⊗ ϕ (respectively, γ ⊗ ϕ′) is uniquely defined by the matrices As = (γ ⊗ ϕ)(es) (respec- tively, A′ s = (γ ⊗ ϕ′)(es)), where s = 1, 2, . . . , 7. It is natural to consider these matrices as block matrices with blocks (As)ij and (A′ s)ij of degree d (i, j = 1, 2, . . . , 9). Then Hom(γ ⊗ ϕ, γ ⊗ ϕ′) = {T ∈ k9d×9d | AsT = TA′ s for each s = 1, 2, . . . , 7}. Lemma 2. Let T = (Tij), i, j = 1, 2, . . . , 9, be a block matrix (over k) with blocks Tij of degree d, belonging to Hom(γ⊗ϕ, γ⊗ϕ′). Then Tij = 0 if i 6= j and (i, j) 6= (1, 8), (1, 9), and T11 = T22 = . . . = T99. Proof. Denote by I, II, . . . ,VII the matrix equalities A1T = TA′ 1, A2T = TA′ 2, . . . , A7T = TA′ 7, respectively. The (matrix) equality (AsT )ij = (TA′ s)ij , i, j ∈ {1, 2, . . . , 9}, induced by an equality AsT = TA′ s, is de- noted by I(i, j) for s = 1, II(i, j) for s = 2, . . ., VII(i, j) for s = 7. It is easy to see that VII(i, j) implies Tij = 0 for each (i, j) ∈ {1, 4, 5, 8, 9} × {2, 3, 6, 7} and each (i, j) ∈ {2, 3, 6, 7} × {1, 4, 5, 8, 9}; II(1, 4) and T12 = 0 imply T14 = 0; II(1, 5) and T13 = 0 imply T15 = 0; II(4, 1) and T61 = 0 imply T41 = 0; II(4, 8) and T68 = 0 imply T48 = 0; II(4, 9) and T69 = 0 imply T49 = 0; II(5, 1) and T71 = 0 imply T51 = 0; II(5, 8) and T78 = 0 imply T58 = 0; II(5, 9) and T79 = 0 imply T59 = 0; II(8, 4) and T82 = 0 imply T84 = 0; II(8, 5) and T83 = 0 imply T85 = 0; II(9, 4) and T92 = 0 imply T94 = 0; II(9, 5) and T93 = 0 imply T95 = 0; III(6, 2) and T82 = 0 imply T62 = 0; III(6, 3) and T83 = 0 imply T63 = 0; III(7, 2) and T92 = 0 imply T72 = 0; III(7, 3) and T93 = 0 imply T73 = 0; III(6, 1) and T61 = 0 imply T81 = 0; III(1, 9), T13 = 0, T15 = 0, T17 = 0 and T69 = 0 imply T89 = 0; III(6, 9), T63 = 0, T65 = 0, T69 = 0 and T89 = 0 imply T67 = 0; III(7, 1) and T71 = 0 imply T91 = 0; IV(2, 6), T21 = 0 and T24 = 0 imply T26 = 0; IV(2, 7), T21 = 0 and T25 = 0 imply T27 = 0; IV(3, 6), T31 = 0 and T34 = 0 imply T36 = 0; IV(3, 7), T31 = 0 and T35 = 0 imply T37 = 0; VI(1, 2), T12 = 0, T52 = 0, T72 = 0, T82 = 0 and T92 = 0 imply T32 = 0; VI(2, 7), T21 = 0, T27 = 0, T47 = 0, T67 = 0, T87 = 0 and T97 = 0 imply T23 = 0; VI(2, 5), T21 = 0, T23 = 0, T25 = 0, T65 = 0, T85 = 0 and T95 = 0 imply T45 = 0; VI(1, 4), T12 = 0, T34 = 0, T74 = 0, T84 = 0 and T94 = 0 imply T54 = 0; II(5, 6), T52 = 0, T54 = 0 and T56 = 0 imply T76 = 0; III(5, 8), T51 = 0, T52 = 0, T54 = 0, T56 = 0 and T78 = 0 imply T98 = 0. So Tij = 0 when i 6= j and (i, j) 6= (1, 8), (1, 9). Then it follows from III(1, 6), III(1, 8), III(5, 7), III(5, 9), VI(1, 3), VI(1, 5) VI(2, 4) and VI(2, 6). that T11 = T22 = . . . = T99. Jo u rn al A lg eb ra D is cr et e M at h .V. M. Bondarenko 11 The final part of the proof is analogous to that in the case Λ = Ẽ6 (see Section 3). References [1] Ostrovskyi V., Samoilenko Yu. Introduction to the Theory of Representations of Finitely Presented ∗-Algebras.I. Representations by bounded operators. The Gordon and Breach Publishing Group. 1999. [2] Drozd, Yu. A., Tame and wild matrix problems, Lecture Notes in Math. 831, vol. 2, Springer, Berlin, 1980, 242-258. [3] Bondarenko, V. M., On certain wild algebras generated by idempotents, Methods Funct. Anal. Topology 5, no. 3 (1999), 1-3. [4] Pierce R.S. Associative Algebras. Springer-Verlag. 1982. Contact information V. M. Bondarenko Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine E-Mail: vit-bond@imath.kiev.ua Received by the editors: 26.04.2004 and final form in 05.10.2004.

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