On schurity of one-sided bimodule problems

We consider a class of normal bimodule problems satisfying some structure, triangularity and finiteness conditions (one-sided bimodule problems). We study the structure of non-schurian bimodule problems from our class and describe explicitly the minimal non-schurian one-sided bimodule problems.

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Veröffentlicht in:	Algebra and Discrete Mathematics
Datum:	2019
Hauptverfasser:	Babych, V., Golovashchuk, N.
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spelling	Babych, V. Golovashchuk, N. 2023-03-02T19:08:14Z 2023-03-02T19:08:14Z 2019 On schurity of one-sided bimodule problems / V. Babych, N. Golovashchuk // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 157–170. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC: 16D90, 16P60, 15A04, 16G20, 15A63 https://nasplib.isofts.kiev.ua/handle/123456789/188485 We consider a class of normal bimodule problems satisfying some structure, triangularity and finiteness conditions (one-sided bimodule problems). We study the structure of non-schurian bimodule problems from our class and describe explicitly the minimal non-schurian one-sided bimodule problems. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On schurity of one-sided bimodule problems Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	On schurity of one-sided bimodule problems
spellingShingle	On schurity of one-sided bimodule problems Babych, V. Golovashchuk, N.
title_short	On schurity of one-sided bimodule problems
title_full	On schurity of one-sided bimodule problems
title_fullStr	On schurity of one-sided bimodule problems
title_full_unstemmed	On schurity of one-sided bimodule problems
title_sort	on schurity of one-sided bimodule problems
author	Babych, V. Golovashchuk, N.
author_facet	Babych, V. Golovashchuk, N.
publishDate	2019
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	We consider a class of normal bimodule problems satisfying some structure, triangularity and finiteness conditions (one-sided bimodule problems). We study the structure of non-schurian bimodule problems from our class and describe explicitly the minimal non-schurian one-sided bimodule problems.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/188485
citation_txt	On schurity of one-sided bimodule problems / V. Babych, N. Golovashchuk // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 157–170. — Бібліогр.: 18 назв. — англ.
work_keys_str_mv	AT babychv onschurityofonesidedbimoduleproblems AT golovashchukn onschurityofonesidedbimoduleproblems
first_indexed	2025-11-24T21:53:33Z
last_indexed	2025-11-24T21:53:33Z
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fulltext	“adm-n4” — 2020/1/24 — 13:02 — page 157 — #7 Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 28 (2019). Number 2, pp. 157–170 c© Journal “Algebra and Discrete Mathematics” On schurity of one-sided bimodule problems Vyacheslav Babych and Nataliya Golovashchuk Communicated by A. P. Petravchuk Abstract. We consider a class of normal bimodule problems satisfying some structure, triangularity and finiteness conditions (one-sided bimodule problems). We study the structure of non- schurian bimodule problems from our class and describe explicitly the minimal non-schurian one-sided bimodule problems. Introduction The notion of bimodule problem arose as a formal language for so called matrix problem solving methods, i. e. the equivalence classes repre- sentatives description problem for a set of matrices with respect to some set of transformations ([16]). Transition from matrix to bimodule problem provide us with effective algorithms for obtaining the representation type of bimodule problem and for its representation category description. Other important tools closely related to this task are quadratic and bilinear forms ([8, 14]). We develop approach to investigation of bimodule problem representa- tion category and representation type based on results of [1,9] for another class of bimodule problems. From the representation theory point of view, the simplest bimodule problems are those for which the dimensions of indecomposable representations are in bijection with the roots of corre- sponding Tits quadratic form. This observation leads to the notion of a 2010 MSC: 16D90, 16P60, 15A04, 16G20, 15A63. Key words and phrases: bimodule problem, quasi multiplicative basis, Tits quadratic form, representation, schurity. “adm-n4” — 2020/1/24 — 13:02 — page 158 — #8 158 On schurity of one-sided bimodule problems schurian bimodule problem for which any indecomposable representation has only scalar endomorphisms. In particular, quivers and posets with weakly positive Tits form are schurian ([4, 13]). We consider the class of bimodule problems called one-sided bimodule problems which generalizes well known classes of quivers and posets. We impose two important restrictions on considered bimodule problems. Firstly we assume that the quadratic form of bimodule problem is weakly positive since the problem has an infinite representation type in the opposite case. The second assumption limits the number of objects in the category to at least three, otherwise the problems are considered directly. Bimodule problem from this class has a quasi multiplicative basis introduced in [2] which is a generalization of the notion of a multiplicative basis and allows to distinguish the minimal non-schurian subproblems effectively. Theorem 1 asserts that the minimal non-schurian admitted bimodule problem having more than two objects is standard. This theorem is a special case of the main result of [4] on DGC, but we give a new proof using more invariant language. We use the definitions, notations and statements from [1–3,16]. The considered class ∁ of bimodule problems and the notion of quasi multi- plicative basis are defined in [3]. The notions and facts from the theory of quadratic forms can be found in [6, 8, 14,17,18]. 1. Preliminaries Basic roots and singular vertices. Denote by WP the set of all weakly positive locally finite quadratic forms and by ℜ+ q a set of all positive roots of a form q. Let q(x1, . . . , xn) ∈ WP be an unit form, n > 2. A sincere x ∈ ℜ+ q is called a basic root, if there exist i1, i2 ∈ I = {1, . . . , n} such that 2(ei1 , x) = 2(ei2 , x) = 1, xi1 = xi2 = 1 and 2(ei, x) = 0, i ∈ I\{i1, i2}. We call i1, i2 the singular vertices of x. Lemma 1. Let q(x1, . . . , xn) ∈ WP be an unit form, n > 3, let x ∈ ℜ+ q be sincere non-basic, and let i1, i2 ∈ I, i1 6= i2. Then there exists non-sincere y ∈ ℜ+ q such that y < x, and i1, i2 ∈ supp y. Proof. Since q ∈ WP, 2(ei, x) ∈ {−1, 0, 1} for any i ∈ I, so the equality 2 = 2q(x) = 2(x, x) = ∑n i=1 2(ei, x)xi implies 2(ej , x) = 1 for some j ∈ I. Then z = x− ej ∈ ℜ+ q . If z is sincere, it is non-basic since 2(ej , z) = −1. Therefore, there exists a minimal sincere non-basic z ∈ ℜ+ q such that z 6 x. Let J ⊂ I be the set of all j ∈ I such that 2(ej , z) = 1. Then for any j ∈ J , the root y = z − ej < z is non-sincere by the minimality of z, “adm-n4” — 2020/1/24 — 13:02 — page 159 — #9 V. Babych, N. Golovashchuk 159 and hence zj = 1. Since 2 = ∑n i=1 2(ei, x)xi, \|J \| > 2. If \|J \| = 2, then z is basic. Therefore, \|J \| > 3, and there is j ∈ J\{i1, i2}. Representation category. For a bimodule problem A = (K,V), a rep- resentation M of A is a pair M = (MK,MV) of MK ∈ ObaddK and MV ∈ addV(MK,MK). If M ,N are representations of A, then a morphism f :M → N is a morphism f ∈ addK(MK, NK) such thatNV·f−f ·MV = 0. The unit morphisms and composition of morphisms in the representation category repA and in addK coincide. All indecomposable representations form the subcategory in repA which we denote by indA. With a locally finite dimensional bimodule problem A = (K,V) we associate the Z-lattice dimA = ⊕ ObK Z of elements x = (xA)A∈ObK with finite support suppx = {A ∈ ObK \| xA 6= 0}. The lattice dimA has the standard basis {eA, A ∈ ObK} such that (eA)A = 1, and (eA)B = 0 for B ∈ ObK\{A}. Besides, dimA is endowed with the partial product order: for a vector x ∈ dimA, we write x > 0 if and only if xA > 0 for all A ∈ ObK. For a representation M ∈ repA such that MK ≃ ⊕ A∈ObK AxA where almost all xA = 0, a dimension vector of M is defined by equality dimM = dimAM = (xA)A∈ObK ∈ dimA. By definition, a support suppM of the representation M is supp dimAM and is always finite. There exists the identical on morphisms forgetful functor F(= FA) : repA−→ addK such that F(M) = MK. A morphism f : M −→N in repA is an isomorphism if and only if F(f) :MK−→NK is an isomorphism in addK. We will denote an isomorphism by ≃, and for a family of representations S from repA, we denote by S/≃ the set the isoclasses of S. The direct sum in repA is induced by the direct sum in addK. It turns repA into a fully additive category, and the Krull-Schmidt theorem holds in repA (see [5, 8]). The following result is clear. Lemma 2. Let p = (p0, p1) : A−→Ared = (K/AnnK V,V) be a natu- ral bimodule problem morphism, where p0 : K−→Kred is the canonical projection and p1 = 1V. Then the functor rep p : repA−→ repAred be- tween the representation categories induced by p is an epimorphism on the morphisms, preserves isomorphisms and rep p(indA) = indAred. The category repA is fully additive, and it is called of finite represen- tation type provided repA has finitely many isoclasses of indecomposable objects, and of infinite representation type in the opposite case. A representation M ∈ repA is called sincere provided (dimM)A 6= 0 for anyA ∈ ObK. In this case ObK is obviously finite. A bimodule problem “adm-n4” — 2020/1/24 — 13:02 — page 160 — #10 160 On schurity of one-sided bimodule problems A is called sincere if there exists a sincere indecomposable representation M ∈ repA. A representation M ∈ indA is called schurian provided it has only scalar endomorphisms. A bimodule problem A is called schurian if every M ∈ indA is schurian ([10, 14]). Lemma 3 ([11, 14]). Let A be a finite dimensional schurian bimodule problem. Then A is representation finite, its Tits form qA is unit integral and WP, the map dimA : indA/≃ → ℜ+ qA is a bijection, where indA/≃ denote the set of all isoclasses of indecomposable representations. Some results on representations of partially ordered sets. Let A = (K,V) be a faithful admitted bimodule problem with ObK−= {O}. If dimk V(O,A) = 1 for every A ∈ ObK+, then we say that A describes representations of a poset Q = Q(A) defined in the following way ([12], 4.1). The elements of Q are Σ+ 0 , and for A,B ∈ Q, A > B if and only if K(A,B) 6= 0 (in this case dimk K(A,B) = 1), and the composition of two non-zero morphisms is again non-zero. The category repA is isomorphic to the representation category of the poset Q. Lemma 4 ([12]). Let A describes the representation of the poset Q. 1) If qA is WP, then A is schurian. 2) A is of finite type if and only if qA is WP. 3) A is of finite type if and only if Q does not contain the subposets 1̂⊔ 1̂⊔ 1̂⊔ 1̂, 2̂⊔ 2̂⊔ 2̂, 1̂⊔ 3̂⊔ 3̂, 1̂⊔ 2̂⊔ 5̂, 4̂⊔ 2̂ տ 2̂. The quadratic forms of these posets are critical (critical posets are indicated in [13]). 4) If Q contains a chain of the length 3, x ∈ dimA is sincere positive and xO 6 2, then every M ∈ repA with dimAM = x is decomposable. 2. Schurity Problem Sincere schurian bimodule problem. The following lemma states that each sincere schurian bimodule problem is faithful. Lemma 5. If bimodule problem A = (K,V) is non-faithful, then every sincere M ∈ indA is non-schurian. Proof. Let AnnK V(A,B) 6= 0, A,B ∈ ObK. Then the subspace generated by AnnK V(A,B) in addK(MK,MK) consists of nilpotent endomorphisms of M . Since M is sincere, M is non-schurian. “adm-n4” — 2020/1/24 — 13:02 — page 161 — #11 V. Babych, N. Golovashchuk 161 Minimal non-schurian bimodule problem. Let A = (K,V) be a sincere non-schurian bimodule problem such that qA ∈ WP, and for every proper subset S ⊂ ObK, the restriction AS is schurian. Then A is called a minimal non-schurian bimodule problem. Let B, C be bimodule problems, defined by their basic bigraphs (ΣB)0 = {X}, (ΣB)1 = ∅, and (ΣC)0 = {X,Y }, (ΣC)1 = {a : X → Y }. For a bimodule problem A having basis with at most 2 vertices, A is sincere schurian if and only if either A = B or A = C. The following lemma is similar to the result given in [4] for box problems (see comments on Theorem 1). Lemma 6. Let A = (K,V) be a minimal non-schurian admitted bimodule problem with a basis Σ and \|Σ0\| > 3. If Ared is a sincere schurian, then there exist two uniquely defined vertices A,B ∈ Σ+ 0 such that: 1) for any sincere M ∈ indA, the vector dimAM is a basic root of Tits quadratic form qAred with the singular vertices A,B; 2) if AnnK V(A1, B1) 6= 0, then the sets {A1, B1}, {A,B} coincide. Proof. Since Ared is sincere, the set ObK is finite. By Lemma 3, the map dimA : indAred/≃ −→ ℜ+ qAred is a bijection. Let us consider a pair A,B ∈ Σ+ 0 such that AnnK V(A,B) 6= 0. Since \|Σ0\| 6= 1 and A is minimal non-schurian, A 6= B. If there exists sincere non-basic y ∈ ℜ+ qAred , then by Lemma 1 there exists non-sincere z ∈ ℜ+ qAred such that A,B ∈ supp z and subproblem Asupp z is not schurian by Lemma 5. If x is a basic root of qAred with singular vertices A1, B1, then x− eA1 and x− eB1 are the roots, non-sincere in A1 and B1 correspondingly. If {A,B} 6= {A1, B1}, then by Lemma 5 at least one of the subproblems AΣ0\{A} or AΣ0\{B} is non-shurian. A minimal non-schurian bimodule problem A satisfying the conditions of Lemma 6 is called standard minimal non-schurian bimodule problem with singular vertices A, B. Small minimal non-schurian bimodule problems. The following lemma is proved by direct verification. Lemma 7. Let A be an admitted minimal non-schurian bimodule problem with a basis Σ = ΣA. Then: 1) \|Σ0\| 6= 2; 2) if \|Σ\| = 1, 3, then A is non-working contour; “adm-n4” — 2020/1/24 — 13:02 — page 162 — #12 162 On schurity of one-sided bimodule problems 3) if \|Σ0\| = 4, then Ared = D4 is given by its bigraph Ω where Ω0 = {A1, A2, E1, E2}, Ω 0 1 = {xi : E1 → Ai, yi : E2 → Ai, i = 1, 2}, Ω1 1 = {ϕ : A1 → A2} and Pϕ = {(x1, x2), (y1, y2)}. The dimension of non-schurian representation with singular vertices A1 and A2 is eA1 + eA2 + eE1 + eE2 . In this case we say that the minimal non-schurian A is of the type D4. If A is a minimal non-schurian bimodule problem, Σ is its bigraph, and Σ− 0 = ∅, then \|Σ+ 0 \| = 1, Σ0 1 = ∅, Σ1 1 6= ∅, hence Σ is a union of dotted loops. Unless otherwise indicated, we exclude this case. Moreover, we will assume that Tits quadratic form q ∈ WP, and so Σ does not contain parallel solid arrows. Lemma 8. Let A be an admitted finite dimensional minimal non-schurian bimodule problem with Tits form qAred ∈ WP. Then A ∈ ∁, and therefore, Ared possesses a quasi multiplicative basis, and the bigraph of Ared does not contain a solid cycle O4. Proof. Let the bigraph of Ared contain one of the listed in [3, Lemma 2] bigraphs or a solid cycle O4 as a subbigraph. Since qAred ∈ WP, there exists two vertices A,B ∈ Σ+ 0 such that AnnK V(A,B) 6= 0. Hence Ared contains a non-shurian proper subproblem (see Lemma 7). By [3, Theorem 1] Ared possesses a quasi multiplicative basis. Reduction of direct summand of bimodule. We will use a partial case of so called reduction functor presented in [5, 7]. We assume that: 1. A = (K,V) is a bimodule problem. 2. W, V̄ ⊂ V are subbimodules of V such that V = V̄⊕W, in particular, for every M = (MK,MV) ∈ repA, there exists a canonical decomposition MV = M V̄ ⊕ MW, M V̄ ∈ add V̄, MW ∈ addW. Let iW : addW → addV be the morphism induced by the canonical inclusion W →֒ V. 3. Let Ā = (K, V̄) be the induced bimodule problem, let L ⊂ rep Ā be a set of representations of Ā, let L = repA\|L, let iL : L →֒ rep Ā be the inclusion, let jL be the composition jL : L iL−−→ rep Ā FĀ−−−→ addK of iL with the identical on morphisms forgetful functor F(= FA) : repA−→ addK such that F(M) =MK, and let jW : WL−→ addV be the corresponding L-bimodule morphism. Without loss of generality we assume that the representations of L are pairwise non-isomorphic. We construct a bimodule problems B = (L,WL) and define the functor FL : repB−→ repA as follows: • if M = (ML,MWL ) ∈ Ob repB for MWL ∈ addWL(ML,ML), ML ∈ add L, then FL(M) is a couple (jL(ML), (ML)V̄ ⊕ jW(MWL )); “adm-n4” — 2020/1/24 — 13:02 — page 163 — #13 V. Babych, N. Golovashchuk 163 • if f :M −→N , then FL(f) = jL(f). The functor FL is called the reduction functor for A with respect to subbimodule V̄ ⊂ V. Lemma 9. FL is an equivalence on its image FL(repB). Proof. If M = (ML,MWL ), N = (NL, NWL ) ∈ repB, then repB(M,N) = {f ∈ add L(ML, NL) \|MWL f − fNWL = 0}. (1) To include repB in repA as a subcategory, we identify f ∈repB(M,N) and add iL(f) :MK → NK. Then f is a morphism in add iL(repB) if and only if M V̄ f − fN V̄ = 0 (for add V̄ this means that f ∈ rep Ā(M,N)), and (1) holds. But these conditions mean that f :MK → NK defines a morphism from M = (MK,MV̄ ⊕MW) to N = (NK, NV̄ ⊕NW). The composition in repA and in add iL(repB) coincide with the one in addK. Remark that reduction functor FL induces a homomorphism dimFL : dimB → dimA which keeps the order on dimensions. Reduction for an admitted bimodule problem. Let P ⊂ ObK−, and W = V (P ) ⊂ V be a subbimodule such that W(X,Y ) = V(X,Y ) if X 6∈ P and W(X,Y ) = 0 otherwise. Since A is admitted, then V = W⊕ V̄ with V̄ = V(ObK−\P ) be a K-subbimodule in V. Let Ā = (K, V̄) be the induced bimodule problem. There are defined the morphisms of bimodule problems pP = (p0, p1) : A−→Ā = (K, V̄) where p0 = 1K and p1 : V−→ V̄ is a projection, and iP = (i0, i1) : Ā−→A where i0 = 1K and i1 : V̄−→V is an inclusion. For M ∈ rep Ā, let M \|P = rep pP (M). For M ∈ repA, let Ob L = {M1, . . . ,Mt} where L ⊂ rep Ā is the full subcategory defined above. Let M \|P = Mk1 1 ⊕ . . . ⊕ Mkt t be the decomposition in indecomposables in rep Ā. Then the reduction morphism repB FL−→ repA FA−→ addK is called M–reduction of A (with respect to P ) and write RP,M instead of FL and AP,M = (KP,M ,VP,M ) instead of B. Lemma 10. 1) The bimodule problem AP,M is admitted. 2) The set ObKP,M is a disjoint union (ObK\P̄ )∪Ob L where P̄ ⊃ P is the set of all vertices in Σ0 incident to the arrows starting in P . 3) There exists MP ∈ repAP,M such that M ≃ repRP,M (MP ). 4) Restrictions of dimAP,M MP and dimAM on ObK\P̄ coincide. 5) If M is sincere representation, then MP is sincere. 6) Let a ∈ Σ0 1(E,A), A ∈ P̄ , E 6∈ P . Then the bigraph ΣAP,M of AP,M contains p = dimkMi(A) solid arrows ak : E → Mi such that RP,M (a1) + · · ·+ RP,M (ap) =M(a). “adm-n4” — 2020/1/24 — 13:02 — page 164 — #14 164 On schurity of one-sided bimodule problems 7) Let Eϕ = {s(a), (a, b) ∈ Pϕ}. If Eϕ ∩ P = ∅ for ϕ ∈ Σ1 1(A,B), A,B ∈ P̄ , then there are pq (p = dimkMi(A), q = dimkMj(B)) dotted arrows ϕkl :Mi→Mj in ΣAP,M such that RP,M (ϕkl) :MiK(A)→MjK(B) having only one non-zero component [RP,M (ϕkl)](ϕ), 16k6p, 16 l6q. Proof. Statements 3)-6) are the corollaries of the reduction construction. The property of a bimodule problem to be admitted depends only on its bigraph which implies 1). To prove 7), remark that KP,M (Mi,Mj) = rep Ā(Mi,Mj). The conditions on ϕ imply ϕ ∈ AnnK V̄. Without loss of generality suppose that Mi and Mj are presented as matrices with coeffi- cients in V. Then the matrix [ϕij ] with only non-zero entry [ϕkl](A,k),(B,l) = ϕ defines a morphism ϕkl :Mi−→Mj . We call the arrows ak and ϕkl from Lemma 10 the components of a and ϕ by the reduction RP,M correspondingly. Minimal pairs and reductions. Further we will carry out the proof for pairs (A,M) where M ∈ indA is a sincere. The pair (A,M) is called minimal if every N ∈ indA with dimN < dimM is schurian. Lemma 11. Let A = (K,V) ∈ ∁ be a faithful connected finite dimensional bimodule problem with basis Σ and Tits form q ∈ WP. If M ∈ indA, x = dimAM , (A,M) is a minimal pair and P ⊂ ObK−, then: 1) (AP,M ,MP ) is a minimal pair. 2) If A,B ∈ P̄ and ϕ ∈ Σ1 1(A,B) is such that Eϕ ∩ P = ∅, then: (i) There does not exist Mi∈ indA such that Mi(A) 6= 0, Mi(B) 6= 0. If Mi(A) 6= 0 (Mi(B) 6= 0), then Mi(A) ≃ A (Mi(B) ≃ B). (ii) If Mi(A) 6= 0, Mj(B) 6= 0, then VP,M (E,Mi) = ka, VP,M (E, Mj) = kb, and a‖bϕ in VP,M , where ϕ = ϕij. (iii) Let subcategory Iϕ ⊂ addK consists of all representations F such that F (ϕ) = 0. Then RP,M0(AnnKP,M (VP,M )) ⊂ Iϕ. Proof. By [3, Theorem 1], we can assume that the basis Σ is quasi multi- plicative. Statement 1) is obvious. Suppose there is i = 1, . . . , t such that Mi is sincere in both A and B. The condition on ϕ gives that ϕ ∈ AnnK V̄, and Mi is non-schurian by Lemma 5. Hence N = rep iP (Mi) ∈ indA is non-schurian and dimAN < x. If for some i, Mi(A) ≃ Ap, p > 2, then by Lemma 10, 6), \|Σ0 AP,M1(E,Mi)\| = p > 2. Since qAP,M ∈ WP, \|Σ0 AP,M1(Mi,Mi)\| > 2 andMi is non-schurian, that proves (i). By definition of reduction VP,M (E,Mi) = addV(EK,MiK) = ka, and VP,M (E,Mj) = kb. Statement a‖bϕ follows from the definition of VP,M . “adm-n4” — 2020/1/24 — 13:02 — page 165 — #15 V. Babych, N. Golovashchuk 165 To prove (iii) suppose there exists F ∈ AnnKP,M (VP,M ) such that f = RP,M0(F ) 6∈ Iϕ, or equivalently, f(ϕ) 6= 0. Obviously, F ∈ KP,M (Mi,Mj) for Mi,Mj defined in (ii). Since for every E ∈ Eϕ, F annihilated VP,M (E,Mi), then RP,M1(VP,M (E,Mi))RP,M0(F ) = 0. Using the reduction of direct summand of bimodule we obtain that the equality is equivalent to addV(EK,MiK)f = 0. For v ∈ addV(EK,MiK) with the unique non-zero component v(a), we have vf(b) 6= 0, that proves (iii). Special bimodule problems. We use in this section the following class of admitted bimodule problems. We call a faithful bimodule problem S = (L,W) with a quasi multiplicative basis Φ special provided Φ0 = {E1, E2;A1, . . . , Am}, Φ0 1 = {xi : E1−→Ai, yi : E2 → Ai, i = 1, . . . ,m} and the following equivalent conditions hold: • if 1 6= i < j 6 m, then both pairs xi, xj and yi, yj are comparable; • if 1 6 i < j 6 m, then S\|Ek,Ai,Aj , k = 1, 2 is not a non-working contour, and S\|E1,E2,Ai,Aj is not the cycle O4. Note that the bimodule problem D4 belongs to the defined class. Lemma 12. Let S be a special bimodule problem with a basis Φ. Then: 1) S is schurian; 2) the set DS ⊂ dimS of the dimensions of representations from indS is the subset of the set DΦ = {eE1 , eE2 ; eAi , eE1 +eAi , eE2 +eAi , eE1 +eE2 + eAi , i = 1, . . . ,m; eE1 + eE2 + eAi + eAj , 1 6 i 6= j 6 m}; 3) there exists M ∈ indA, dimS M = eE1 + eE2 + eAi + eAj if and only if Ai and Aj are connected in Φ by an unique joint arrow. 4) For a sincere M ∈ repS and F ∈ repS(M,M) such that F 6= 0, F (1E1 ) = 0, F (1E2 ) = 0, and once F (ϕ) 6= 0 for ϕ ∈ Φ1 1, then ϕ is joint. Then M contains a trivial direct summand UAi for some i = 1, . . . ,m which corresponds to a simple root eAi . Proof. The statements 1), 2), 3) can be verified immediately. Let us prove 4). Let M = t ⊕ k=1 Mxk k be a decomposition in indecomposables in repS. For every non-trivial Mi, i = 1, . . . , t, any N ∈ add L and G ∈ add L(MiL, N) such that G(ϕ) 6= 0 for joint ϕ ∈ Σ1 1, there holds MiLG 6= 0. For F ∈ repS(M,M), MWF − FMW = 0. The condition F (1Ei ) = 0 for i = 1, 2 implies that MWF = 0. The main result. Now we are ready to prove that every minimal non- schurian finite dimensional admitted bimodule problem is standard. “adm-n4” — 2020/1/24 — 13:02 — page 166 — #16 166 On schurity of one-sided bimodule problems Theorem 1. Let A = (K,V) be a minimal non-schurian finite dimensional connected admitted bimodule problem with basic bigraph Σ having at least 3 vertices and weakly positive Tits quadratic form qA. Then qAred ∈ WP and A is standard minimal non-schurian bimodule problem. In fact, Theorem 1 is a special case of the main Theorem of [4] on DGC problems formulated in the other terms. Note that Theorem from [4] is also given in [7] using the language of box problems which are equivalent to DGC ones. For our class of bimodule problems, we give an alternative proof using, in addition to the basic ideals of [4], the specifics of the bimodule language. Proof. According to Lemma 7, bigraph Σ does not contain dotted loops and, since q = qA is weakly positive, parallel solid arrows. Let • B = Ared = (Kred,V), Kred = K/AnnK V, be the faithful part of A; • Ω be the bigraph of B, Ω0 = Σ0, Ω 0 1 = Σ0 1, Ω 1 1 ⊂ Σ1 1; • p(x) = qB(x), p(x) = q(x)− ∑ (A,B)∈Σ0×Σ0 \|AnnK(V)(A,B)\|kxAxB. By Lemma 2, Ob repA = Ob repB. By assumption, for any proper subset S ⊂ Σ0, the problem AS is schurian. The proof of the Theorem is carried out in several steps by induction on the lexicographically ordered pairs (\|Σ− 0 \|, \|Σ + 0 \|). Step 1. \|Σ− 0 \| = 1. Let Σ− 0 = {O}. Then B is a bimodule problem corresponding to a poset Q = Q(B). To show that the Tits form p of B is WP, we formulate the following preciser statement under Theorem conditions. Lemma 13. If p 6∈ WP, M ∈ indB is sincere, then there exists y ∈ ℜ+ p such that 0 < y < x = dimM , and: 1) p\|supp y ∈ WP and there exists N ∈ indB of dimension y; 2) AnnK V\|supp y 6= 0, in particular, the indecomposable N is non- schurian in the category repA. Proof. Since q∈WP, p /∈WP, there are A,B∈Σ+ 0 with AnnKV(A,B) 6=0. If xO = 1, then Q = ⊔k i=11̂ for k > 4 since B is sincere and p 6∈ WP. Then we set y = eO + eA + eB. Thereafter we assume xO > 2. By Lemma 4, Q contains a proper subset C ⊂ Q such that restriction p\|C is critical. To construct the vector y, we need to consider the following three cases. (1) If C = 1̂ ⊔ 1̂ ⊔ 1̂ ⊔ 1̂ then we set y = eO + eA + eB. “adm-n4” — 2020/1/24 — 13:02 — page 167 — #17 V. Babych, N. Golovashchuk 167 (2) Let C = 2̂ ⊔ 2̂ ⊔ 2̂, C = 1̂ ⊔ 3̂ ⊔ 3̂, or C = 1̂ ⊔ 2̂ ⊔ 5̂. Then every pair of vertices of C, in particular (A,B), belongs to a subposet C′ ⊂ C, C′ = (A1, A2, A3 < A4} ≃ 1̂ ⊔ 1̂ ⊔ 2̂. We set y = 2eO + ∑4 i=1 eAi . (3) Assume that C = 4̂ ⊔ 2̂ տ 2̂. By Lemma 4, 4), xO > 3. Every pair of vertices (in particular A,B) belongs to C′ ⊂ C of the form {A1 < A2;B2 > B1 < C2 > C1} ≃ 2̂⊔ 2̂ տ 2̂. Then y = 3eO + eA1 + eA2 + eB1 + eB2 + eC1 + eC2 . For the constructed y, the restriction p\|supp y is WP since it is a proper subform of the critical form. Then, by Lemma 3, there exists N ∈ indB of dimension y which is non-schurian in repA by Lemma 5. In the case \|Σ− 0 \| = 1, the assumption p /∈ WP contradicts to the minimal non-schurity of A, hence p ∈ WP. Then B is schurian by Lemma 4, and we can apply Lemma 6 to A. So the induction base is proved. Step 2. General case \|Σ− 0 \| > 2. To prove the Theorem, it is enough to verify that B is a sincere schurian bimodule problem, and apply Lemma 6. Substep 2.1. p ∈ WP. Proof. Assume p 6∈ WP, then there exists S ⊂ Ω0 such that p\|S is a critical form and µ is corresponding critical vector. Since q ∈ WP, there exist A,B ∈ S such that AnnK V(A,B) 6= 0. Then there is y ∈ ℜ+ p such that A,B ∈ supp y, but supp y 6= S. By assumption, Asupp y and Bsupp y are schurian and, by Lemma 3, there exists Y ∈ indB such that dimA Y = y. By Lemma 2, Y ∈ indA is non-schurian in repA which implies contradiction by Lemma 5. By Lemma 8, B ∈ ∁ and possesses a quasi multiplicative basis. Remark 1. Let (A,M) be a minimal pair with a non-schurian M ∈ indA, let E ∈ Σ− 0 , P = Σ− 0 \{E}, RP,M : AP,M −→A be the reduction con- structed in Lemma 11, and let AP,M = (KP,M ,VP,M ) for MP ∈ repAP,M with repRP,M (MP ) ≃M . Then AP,M is the standard non-schurian, and S = (AP,M )red is a poset bimodule problem. Proof. By Lemma 11, (AP,M ,MP ) is a minimal pair. The bigraph of AP,M has a unique minus-vertex, so the fact follows from the step 1. To show that B is schurian, we prove that every nilpotent endomor- phism of any M ∈ indB is equal to zero. Then, by Fitting’s lemma ([15]), every endomorphism of any M ∈ indB is invertible, and thus scalar. “adm-n4” — 2020/1/24 — 13:02 — page 168 — #18 168 On schurity of one-sided bimodule problems Let F ∈ repB(M,M) be a nilpotent endomorphism of M , i. e. F k = 0 for some integer k > 1. Obviously, to show that F = 0 we need to prove that F (ϕ) = 0 for any ϕ ∈ Σ1 1, and F (1A) = 0 for each A ∈ Σ0. Substep 2.2. F (1E) = 0 for every E ∈ Σ− 0 . Proof. Since \|Σ− 0 \| > 2, there exists E1 ∈ Σ− 0 \{E}. Then, according to remark 1, we apply M -reduction RM,P : AM,P −→A for P = {E1}. Then MP (E) = M(E) for a representation MP ∈ repAM,P such that repRM,P (MP ) =M , and if FP :MP −→MP is such that repRM,P (FP ) = F , then FP (1E) = F (1E). Applying the induction assumption and Lemma 11 to the minimal non-schurian problem AM,P and to the sincere repre- sentation MP , we obtain FP (1E) = 0. Substep 2.3. If ϕ ∈ Σ 1 1(A,B) is a single arrow, then F (ϕ) = 0. Proof. We apply RP,M for P = ObK−\Eϕ where Eϕ = {E}. By step 1, AM,P is standard minimal non-schurian bimodule problem, and by Lemma 11, FP ∈ addAnnKP,M (VP,M ) and F (ϕ) = 0. Substep 2.4. If ϕ : A→ B is a joint arrow, then F (ϕ) = 0. Proof. Let Eϕ = {E1, E2}, P = ObK−\Eϕ. We apply RP,M . If FP ∈ addAnnKP,M (VP,M ), then, as in the substep 2.3, all is proved. In opposite case, there exist non-zero nilpotent FP ∈ indS(MP ,MP ) for S = AP,M = (Kred,W). We choose the basis Ψ = ΣS , and extend it to the basis ΣAP,M . By substep 2.3, FP (ψ) = 0 for any single ψ ∈ Ψ1 1, and so F (ψ) = 0. For any joint ψ ∈ Ψ1 1, FP (ψ) = 0. Indeed, let A1, . . . , Am ∈ Ψ0 be all vertices incident to the joint arrows from Ψ1 1. Consider a special bimodule problem ST with T = {E1, E2, A1 . . . , Am}. Then F ∈ addKredST , and F defines an endomorphism of MP . By Lemma 12, 4), we obtain that MP ∈ repST has a trivial direct summand which is the direct summand of M ∈ repS obviously. So FP ∈ addAnnKP,M (VP,M ). Substep 2.5. F = 0. Proof. It is enough to prove that F (1A) = 0 for all A ∈ Σ+ 0 . Suppose it fails for some A ∈ Σ+ 0 . We denote by aj : Ej −→A, j = 1, . . . , i, all the solid arrows incident to A. We consider the ( i ⊕ j=1 M(Ej),M(A))-cut of M and the matrix [M ]A = ( [M ](a1) . . . [M ](ai) ) . The rank of this matrix equals dimMA, otherwise M contains a trivial direct summand UA. Since F (ϕ) = 0 for all ϕ ∈ Σ0 1, by the substeps 2.3 and 2.4, the equation “adm-n4” — 2020/1/24 — 13:02 — page 169 — #19 V. Babych, N. Golovashchuk 169 [M ][F ]− [F ][M ] = 0 implies [F ]( i ⊕ j=1 1Ej )[M ]A − [M ]A[F ](1A) = 0. Since F (1Ej ) = 0 for any j, by the substep 2.2, [M ]A[F ](1A) = 0. Corollary 1. Let A be a finite dimensional connected admitted bimodule problem with at least 3 objects and Tits form qA ∈ WP. If qAred is not WP, then A can not be a minimal non-schurian bimodule problem. Conclusion The paper is devoted to the study of schurity property for a class of matrix problems called one-sided bimodule problems, and is the second step after [3] in research of the representation finiteness problem for a mentioned class. By means of the constructed quasi multiplicative basis, we prove that the minimal non-schurian admitted bimodule problem having more than two vertices is standard minimal non-schurian. References [1] Babych, V. M. and Golovashchuk, N. S. An Application of Covering Techniques. Scientific Bulletin of Uzhgorod Univ. Series Mathematics and Informatics 8 (2003) 4–14. [2] Babych, V. M., Golovashchuk, N. S. and Ovsienko, S. A. Quasi multiplicative bases for bimodule problems from some class. Scientific Bulletin of Uzhgorod Univ. Series Mathematics and Informatics 22 (2011) 12–17. [3] Babych, V. M., Golovashchuk, N. S. and Ovsienko, S. A. Generalized multiplicative bases for one-sided bimodule problems. Algebra and Discrete Mathematics 12, Num. 2 (2011) 1–24. [4] Bondarenko, V. M., Golovashuk, N. S., Ovsienko, S. A. and Roiter, A. V. Schurian matrix problems and quadratic forms. Inst. of Math., Academy of Sci. USSR, Preprint 78.25 (1978) 19–38. [in Russian] [5] Crawley-Boewey, W. W. Matrix problems and Drozd’s theorem. Banach Center publ. 26, part 1 (1990) 199–222. [6] Drozd Yu. A. Coxeter transformations and representations of partially ordered sets. Funkts. Anal. Prilozhen. 8 (1974) 34–42. [in Russian] [7] Drozd, Yu. A. Reduction algorithm and representations of boxes and algebras. Comptes Rendue Math. Acad. Sci. Canada 23 (2001) 97–125. [8] Drozd, Yu. A. Tame and wild matrix problems. In: Representations and quadratic forms, Kiev (1979) 39–74. [English translation in AMS Translations v. 128] [9] Drozd, Yu. A., Ovsienko, S. A. and Furchin, B. Yu. Categorical constructions in representations theory. In: Algebraic Structures and Their Applications, Kiev State Univ., Kiev (1988) 17–43. [10] Gabriel, P. Auslander-Reiten sequences and representation-finite algebras. LNM, Springer 831 (1980) 1–71. “adm-n4” — 2020/1/24 — 13:02 — page 170 — #20 170 On schurity of one-sided bimodule problems [11] Gabriel, P. Indecomposable representations II. Symposia Math. Ist. Naz. Alta Mat. 11 (1973) 81–104. [12] Gabriel, P. and Roiter, A. Representations of finite-dimensional algebras. Algebra VIII, Encyclopaedia of Math. Sci. 73, Springer, New York, 1992. [13] Kleiner, M. M. Partially ordered sets of finite type. Studies in the theory of repre- sentations, Zap. Nauchn. Sem. LOMI 28, Nauka, Leningrad. Otdel., Leningrad (1972) 32–41. [in Russian] [14] Kleiner, M. M. and Roiter, A. V. Representations of differential graded categories. LNM, Springer 488 (1975) 316–340. [15] Krause, H. Krull-Schmidt categories and projective covers. Expo. Math. 33, 4 (2015) 535–549. [16] Ovsienko, S. A. Bimodule and matrix problems. Progress in Mathematics 173 (1999) 323–357. [17] Peña, J. A. de la. On the dimension of the module-varieties of tame and wild algebras. Commun. Algebra 19 (1991) 1795–1807. [18] Ringel, C. M. Tame Algebras and Integral Quadratic Forms. LNM, 1099, Springer, 1984. Contact information V. Babych, N. Golovashchuk Department of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64, Volodymyrs’ka St., Kyiv, Ukraine E-Mail(s): vyacheslav.babych@univ.kiev.ua, golova@univ.kiev.ua Web-page(s): sites.google.com/view/cgtds/ Received by the editors: 06.10.2019 and in final form 19.11.2019.

On schurity of one-sided bimodule problems

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