Galois Coverings of One-Sided Bimodule Problems

Applying geometric methods of $2$-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem $\mathcal A$ is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering $\tilde{\mathcal A}$, either $\mathcal A$ is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.


Introduction
Description of bimodule problems of finite representation type is an important task of representation theory [7,12,15]. A useful tool for a finiteness problem solution is so called "covering method" ( [6,8]) which is especially effective when the basis of bimodule problem is multiplicative, i. e. the composition of two composable basic elements is either zero or a basic element too ( [17]). For a class of admitted bimodule problems, we introduced a quasi multiplicative basis ( [4]) generalizing the notion of a multiplicative one. Following [5], we introduce standard minimal non-schurian admitted bimodule problem and use the result of [2] stating that minimal admitted non-schurian bimodule problem with weakly positive Tits quadratic form is standard. A similar result was obtained by the authors for another class of bimodule problems previously in [1].
From the representation theory point of view, it is important to determine the minimal non-schurian subproblems effectively, and describe the classes of problems having a correspondence between the Tits form and the category 2 V. Babych, N. Golovashchuk of representations. In the simplest cases, the dimensions of indecomposable representations correspond to the roots of the Tits form ( [10,13]).
We apply geometrical technique for investigation of bimodule problem representation category properties for a class of bimodule problems endowed with a quasi-multiplicative basis. The proofs use geometrical methods similar to those used in the geometric group theory [14]. We associate a 2-dimensional cell complex with a faithful admitted bimodule problem in order to construct a corresponding Galois covering. This is the main tool in the geometrical investigation of bimodule problems. We use an universal covering technique to study the representation type of the initial admitted bimodule problem. A Galois covering of a bimodule problem of finite type induces a covering of corresponding representation categories with the same fundamental group ( [8,9]).
The main result (Theorem 2.9) states the existence of standard minimal non-schurian subproblem for a finite dimensional non-schurian bimodule problem with weakly positive Tits form and schurian universal covering. After [2,4], this is the next step on the way of representation type characterization for finite dimensional problems from the considered class.
We use the definitions, notations and statements from [2][3][4]15]. The considered class ∁ of bimodule problems and the notion of quasi multiplicative basis are defined in [4]. The notions and facts from the theory of quadratic forms can be found in [7,13,16].
A bigraph Σ is called locally finite if any X ∈ Σ 0 is incident to finitely many arrows, and finite provided Σ 0 and Σ 1 are finite.
The walks and the paths on Σ form the categories denoted by Walks Σ and Paths Σ respectively, both with the set of objects Σ 0 . For any X, Y ∈ Σ 0 the set Walks Σ (X, Y ) (Paths Σ (X, Y )) contains all the walks (the paths) ω in Σ such that s(ω) = X, e(ω) = Y . There exists an obvious inclusion functor ı : Paths Σ ֒→ Walks Σ . We denote by W Σ the groupoid of walks on Σ, W Σ = Walks Σ /(x • x −1 = 1 e(x) , x ∈ Σ 1 ), and by r Σ : Walks Σ −→ W Σ the canonical projection.

Reduced walks.
A walk ω is called reduced if it does not contain a walk xx −1 for some x ∈ Σ 1 , and reducible in opposite case. The operation ω 1 xx −1 ω 2 → ω 1 ω 2 we call a reducing of the walk (by the pair xx −1 ).
If ω = ω 1 ω 2 is a cyclic walk, then ω ′ = ω 2 ω 1 is cyclic as well, and we say that ω and ω ′ are cyclically equivalent. A cyclic walk ω is called cyclically reduced if every cyclically equivalent walk is reduced. A class ω of the cyclic equivalence containing the cyclic walk ω is called a cycle. We denote by Cycles Σ the set of all cycles on Σ, and by RWalks Σ (RCycles Σ ) the set of all reduced walks (cyclically reduced cycles) on Σ.
A bigraph morphism f = (f 0 , f 1 ) : Σ → Σ ′ induces in a natural way a map Cycles Σ ∋ ω a boundary map ∂ = ∂ L : L 2 −→ RCycles Σ . We say that A ∈ L 0 (x ∈ L 1 = Σ 1 ) belongs or is incident to △ i and write A ∈ △ i (x ∈ △ i ) provided A (at least one of x ±1 ) is incident or belongs to some element of ∂(△ i ).
If L ′ is a complex, then a morphism L → L ′ of complexes is a triple f = (f 0 , f 1 , f 2 ) such that (f 0 , f 1 ) is a morphism of the underlying bigraphs (denoted by the same letter f ), and f 2 : The notion of subcomplex of a complex is defined in a standard way. If S ⊂ L 0 ⊔ L 1 ⊔ L 2 , then by [S] we denote the subcomplex in L, generated by S, i. e. the minimal subcomplex in L containing S. For S ⊂ Σ 0 by Σ S and L S we denote the restriction of Σ and L to S.
Let L be a complex. We say that the complex M together with a morphism p : M → L forms a complex over L. The morphism f : . A quotient bigraph Σ/ ∼ is defined by the set L 0 / ∼ of vertices, the set L 1 / ∼ of arrows, and correctly induced e, s and deg. Let p = (p 0 , p 1 ) : Σ −→ Σ/ ∼ be the natural bigraph epimorphism. If Cycles p (∂ L (△)) is cyclically reduced for any 2-cell △ ∈ L 2 , then we set (L/ ∼ ) 2 = L 2 and ∂ L/∼ = Cycles p ∂ L that defines the quotient complex L/ ∼ and the morphism p = (p 0 , p 1 , p 2 ) : 1.6. Homotopy relation. The structure of a 2-dimensional complex L over a bigraph Σ induces the homotopy relation on Walks Σ . For any ω, ω 1 , ω 2 ∈ Walks Σ , a following transformations of walks: Two walks ω, ω ′ on Σ are said to be homotopic provided there exists a sequence E = (E 1 , . . . , E N ) of elementary homotopies such that ω = ω 0 In this case, we write ω E ❀ ω ′ or ω ∼ ω ′ and say that E is a homotopy between ω and ω ′ .
We indicate the simple properties of homotopies: For a walk ω we denote by [ω] the homotopic class of ω. Once the composition ωω ′ of walks ω, ω ′ is specified, the composition of their classes is correctly defined by the equality We denote by Hot L the quotient category of homotopic classes of walks on Σ, If L is connected, then the fundamental groups with different base vertices are isomorphic, that allows to define the fundamental group G(L) of the connected complex L. for every ω ∈ Walks L (X, Y ) and X ∈ p −1 0 (X) ( Y ∈ p −1 0 (Y )), there exist an unique Y ∈ L 0 ( X ∈ L 0 ) and a unique walk ω : X −→ Y such that Walks p ( ω) = ω (the property of the uniqueness of lifting of walks, [18]); for every △ ∈ L 2 , A ∈ L 0 such that A ∈ ∂(△) and A ∈ p −1 0 (A) there exists unique △ ∈ L 2 such that A ∈ ∂( △) and p 2 ( △) = △ (the property of homotopy lifting uniqueness, [18]). Coverings with a fixed base L form the subcategory in the category of complexes over L. An object p : L −→ L in this category is called a universal covering of L if every morphism f : p ′ → p is an isomorphism.
1.9. Construction of universal covering. We construct the universal covering L of the complex L similarly to [3,8].
1) We fix B ∈ L 0 and define the set L 0 by ⊔ X∈L 0 Hot L (B, X).
2) The arrows in L are the pairs 3) The covering morphism p :Σ −→ Σ on the underlying bigraphs is defined by p 0 ([ω]) = e(ω), p 1 (([ω], x)) = x. Obviously, the constructed p has a property of the uniqueness of lifting of walks.
). Then we define L 2 by the set {△ θ } θ∈Θ , the boundary map in L by the equality The structure maps of L commutes with the action of G, hence a quotient complex L/G = (G L 0 , G L 1 , G L 2 ) is defined, and the morphism p induces an isomorphism of the complexes L/G and L.
1.10. Tits quadratic form of bimodule problem. We refer to [4] for the notations and facts from the theory of bimodule problems. A quadratic form q = q Σ of the bigraph Σ is defined on where δ is the Kronecker delta. By definition, the Tits quadratic form q A of bimodule problem A is a quadratic form q Σ A of a basic bigraph Σ A . We denote by ℜ + q the set of all positive roots of q (i. e. x > 0 such that q(x) = 1). For a quadratic form q, by (, ) or (, ) q we denote the corresponding symmetric bilinear form, (x, y) = (q(x + y) − q(x) − q(y))/2. If q has a sincere positive root, then q is called sincere.
The form q is called weakly positive provided q(x) > 0 for x > 0. We denote by WP the set of all weakly positive locally finite forms.
If q(x 1 , . . . , x n ) ∈ WP, n 2, e i is a simple root and z ∈ ℜ + q is sincere, then 2(e i , z) ∈ {0, 1, −1} and z − 2(e i , z)e i ∈ ℜ + q . A sincere x ∈ ℜ + q is called a basic root if there exist i 1 , i 2 ∈ I = {1, . . . , n} such that 2(e i 1 , x) = 2(e i 2 , x) = 1, x i 1 = x i 2 = 1 and 2(e i , x) = 0, i ∈ I\{i 1 , i 2 }. We call i 1 , i 2 the singular vertices of x. The category rep A is a Krull-Schmidt category. Bimodule problem A is called of finite representation type provided rep A has finitely many isoclasses of indecomposable objects, and of infinite representation type in opposite case. A is called locally representation-finite provided for any object A ∈ Ob K, there are finitely many isoclasses of indecomposable representa- A representation M ∈ ind A is called schurian provided it has only scalar endomorphisms. A bimodule problem A is called schurian provided every M ∈ ind A is schurian ( [11,13]). In this case the bimodule problem A is called a base of the covering. The coverings with a fixed base form a category over A in a standard way: a morphism of the coverings p :Ã −→ A and p ′ :Ã ′ −→ A is a morphism of bimodule problems ρ :Ã −→Ã ′ such that p = p ′ ρ.

Lemma 1.12 ([10, 13]). Let A be a finite dimensional schurian bimodule problem. Then A is representation finite, its Tits form q A is unit integral and WP, the map dim
The functor p * = rep p : repÃ −→ rep A between the representation categories induced by the morphism p :Ã −→ A of bimodule problems is called the push-down functor. It allows to compare the representation types of bimodule problemsÃ and A.
1.14. Galois covering. LetÃ = (K,Ṽ) be a locally finite dimensional bimodule problem, let G be a group acting freely onK andṼ that means: there is given a group monomorphism T : G −→ Aut k (K,K) which defines a free action TK : G ×K −→K of G onK such that for any f 1 , f 2 ∈K, v ∈Ṽ and g ∈ G once the composition f 1 vf 2 is specified. For a locally finite dimensional bimodule problemÃ = (K,Ṽ) and a group G acting freely onK andṼ, we construct the bimodule problem A = (K, V) and a covering morphism p = (p 0 , p 1 ) :Ã −→ A in a following way. Let the set Ob K be the set (ObK)/G of orbits, and let the functor p 0 be the natural projection ObK → Ob K on objects. For objectsÃ,B ∈ ObK, let us identify an element ϕ ∈K(Ã,B) (a ∈Ṽ(Ã,B) respectively) with the corresponding element of sum ⊕X ∈p −1 provided ϕ (a) runsK(X,Ỹ ) (Ṽ(X,Ỹ )) for allX,Ỹ ∈ ObK such that p 0 (X) = A, p 0 (Ỹ ) = B, and g runs G. We denote the class of ϕ (a) by Gϕ ∈ K(A, B) (Ga ∈ V (A, B)). For ϕ ∈K(X,Ỹ ), ψ ∈K(gỸ ,Z), the composition of Gϕ and Gψ is defined by G(b(ga)), and the sum of Ga and Gc for c ∈K(gX, gỸ ) is G(ga + c). Now we can define the functor p 0 on morphisms by the mapK(Ã,B) ∋ ϕ → Gϕ ∈ K(p 0 (Ã), p 0 (B)). The K-bimodule structure on V and the map p 1 are defined similarly. The bimodule problem A is called a G-quotient ofÃ and is denoted bỹ A/G. The constructed morphism p G = p :Ã → A of bimodule problems mapping an objectX to GX and an arrow a :X →Ỹ to Ga : GX → GỸ is correctly defined, is a covering morphism and is called a quotient morphism of the bimodule problems.
A covering isomorphic to the defined above covering p G :Ã −→Ã/G is called a Galois covering with the fundamental group G.
2.3. Universal covering associated with a quasi multiplicative basis of schurian bimodule problem. Proof. Let ObK = L 0 , and let the spaces RadK(X,Ỹ ),Ṽ(X,Ỹ ) be freely generated over k by L 1 1 (X,Ỹ ) and L 0 1 (X,Ỹ ) correspondingly,X,Ỹ ∈ ObK. Ifã :X →Ỹ ,b :Ỹ →Z are two elements of L 1 , p 1 (ã) = a, p 1 (b) = b, then for any x ∈ L 1 such that con x (ba) = 0 and for a uniquex ∈ L 1 such that p 1 (x) = x, s(x) = s(ã), we set conx(bã) = con x (ba) inÃ, and conỹ(bã) = 0 for any otherỹ ∈ L 1 . The compositionbã is correctly defined since bãx −1 is a bound of a cell in L, and hence it is a cycle in L. Associativity of such composition is obvious.K-bimodule structure onṼ is defined similarly. The covering morphism p A :Ã −→ A is uniquely defined by the commutativity of the diagram. For a bimodule problem A ∈ ∁, let p : L → L be the constructed above universal covering of 2-dimensional complex L = L A . By Lemma 1.12, there exists the corresponding to p covering morphism p A :Ã → A of bimodule problems which we call an universal covering of bimodule problem A associated with the complex L.
A bimodule problem A is said to be (geometrically) simply connected provided 2-dimensional complex L over Σ is connected and its fundamental group G(L) is trivial. Obviously, simply connected schurian bimodule problem A is isomorphic to its universal coveringÃ.
2.6. Minimal non-schurian bimodule problem. Recall that for a sincere schurian bimodule problem A, its basis Σ A is 0-connected. If bimodule problem A is non-faithful, then every sincere M ∈ ind A is non-schurian. Therefore, each sincere schurian bimodule problem is faithful.
Let A = (K, V) be a sincere non-schurian bimodule problem such that Tits form q A ∈ WP, and for every proper subset S ⊂ Ob K the restricted bimodule problem A S is schurian. Then A is called a minimal non-schurian bimodule problem.
Let B, C be bimodule problems defined by their bigraphs: This observation excludes the non-schurian problems having at most two vertices from the consideration, and helps to describe the minimal nonschurian bimodule problems containing at least 3 vertices. A minimal non-schurian bimodule problem A satisfying the conditions of Lemma 2.7 is called a standard minimal non-schurian bimodule problem with singular vertices A and B.
2.8. Schurity and coverings: the main result. By the construction of universal covering, we assume that p i ( Σ i ) = Σ i , i = 0, 1, for the bases Σ and Σ ofÃ and A respectively. Theorem 2.9. Let A ∈ ∁ be a connected finite dimensional bimodule problem with weakly positive Tits form q A , letÃ ∈ ∁ be a schurian bimodule problem, and let p :Ã−→A be an universal covering. Then either A is schurian, or contains a dotted loop, or some restriction A S is a standard minimal non-schurian bimodule problem.
Proof. Suppose that A = (K, V) is not schurian and does not contain dotted loops. Let Σ andΣ be the bases of A andÃ correspondingly such that p(Σ) = Σ. We will mark the object (vertex, arrow, etc.) related to the coveringÃ by the sign˜, and its image in A will be denoted by the same letter without˜.
Since bimodule problemÃ is schurian and A is not, there exists a rep-resentationX ∈ indÃ such that forS = suppX, the induced morphism pS :ÃS −→ A S is not an isomorphism. Then the restriction A S is sincere minimal non-schurian. Let us chooseX with minimal possible |S|.
Remark that for any S ⊂ Σ 0 , |S| 2, the restriction A S is schurian by definition of class ∁ and our assumption. So we can assume |S| 3, and hence |S| 3. By Remark 2.5, the induced by p morphism pS : LS −→ L S of the associated complexes is not an isomorphism. Then: 1) either the induced by p map p 0 |S : ( ΣS) 0 → (Σ S ) 0 is not a bijection on the vertices; 2) or the map p 0 |S is a bijection, but there existÃ 1 ,Ã 2 ∈S such that p( Σ 1 (Ã 1 ,Ã 2 )) = Σ 1 (A 1 , A 2 ).
Step 1. The first case is impossible.
Proof. Suppose, there existÃ 1 ,Ã 2 ∈S,Ã 1 =Ã 2 , such that A 1 = A 2 ∈ S. SinceÃS is schurian, by Lemma 1.12,x = dimX is a sincere positive root of weakly positive unit quadratic form qÃS . Ifx is not basic with singular verticesÃ 1 ,Ã 2 , then, by [2, Lemma 1], there is a non-sincereỹ <x such that A 1 ,Ã 2 ∈ suppỹ which contradicts to the minimality of |S| by Lemma 1.12. Therefore,x is a basic root with the singular verticesÃ 1 ,Ã 2 . By definition of basic root, the verticesÃ 1 andÃ 2 are defined uniquely, i. e. B 1 = B 2 for any pairB 1 ,B 2 ∈S of different vertices such that SinceÃS is 0-connected, there is a vertexC ∈S connected withÃ 1 by a solid arrowã :C −→Ã 1 (up to direction ofã). IfC =Ã 2 , then the quadratic form qÃS is not WP. SoC =Ã 2 . By Lemma 1.12, for a positive root z =x − eÃ 1 , there is an indecomposable representationZ of the dimensioñ z with | suppZ| < |S|. The corresponding restriction p suppZ :Ã suppZ → A S is not an isomorphism since the arrow a : C → A 1 does not have an inverse image inΣ 1 (C,Ã 2 ), which contradicts to the minimality of |S|.
Now we can assume that the restriction p 0 |S is a bijection, and the case 2) holds.
The proof is similar.
Proof. Since bimodule problem A is admitted, either X ⊂ Σ 0 1 , or X ⊂ Σ 1 1 . Suppose that X ⊂ Σ 0 1 . Let x = dim A rep p(X). Then x A =xÃ for anỹ A ∈S. Hence that contradicts to the weak positivity of q A .
Step 4. X ⊂ Ann K S V S .
Hence, (A S ) red = (K S / Ann K S V S , V S ) is isomorphic toÃS, and therefore (A S ) red is sincere schurian. By Lemma 2.7, A S is standard minimal nonschurian bimodule problem, which completes the proof of Theorem.

Conclusion
The article is a part of research of the representation finiteness problem for a wide class of multi-vector space categories consisting of so called onesided bimodule problems. We use the construction of the universal covering of an admitted bimodule problem in order to obtain some schurity criterium for the bimodule problems from our class. We are going to study representation type of one-sided bimodule problems using developed technique.