(D,O)-species of bounded representation type
Nadiya Gubareni
Journal of Applied Mathematics and Computational Mechanics |
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(D,O)-SPECIES OF BOUNDED REPRESENTATION TYPE
Nadiya Gubareni
Institute of Mathematics, Czestochowa
University of Technology
Częstochowa, Poland
nadiya.gubareni@yahoo.com
Received: 11 April
2016; accepted: 29 May 2016
Abstract. We describe weak (D,O)-species of bounded representation type in terms of Dynkin diagrams and diagrams with weights. We reduce the problem of their description to flat mixed matrix problems over discrete valuation rings and their common skew field of fractions.
Keywords: species, (D,O)-species, flat mixed matrix problems, discrete valuation ring, Dynkin diagrams
1. Introduction
The representations of species which are first introduced by P. Gabriel [1] are closely connected with representations of finitely dimensional algebras and Artinian rings and were studied by many authors (see e.g. [1-3]). This paper is devoted to the study of (D,O)-species which can be considered as a type of a generalization of these species. They play an important role in the representation theory of some classes of rings, for example, of right hereditary SPSD-rings. We use the main definitions and results from the previous paper [4].
Let W = (F_{i}, _{i}M_{j})_{i,j }_{Î}_{ I} be a weak (D,O)-species, where all F_{i} are equal to O_{i} or D, and _{i}M_{j} is a (D, D)-bimodule that is finite dimensional both as a right and left D-vector space (see [1]). Let I = {1,2, …, n}. The diagram of W is a directed graph Q(W) whose vertices are indexed by the numbers 1, 2, ..., n, and the number of arrows from a vertex i to a vertex j is
A vertex i of the graph Q(W) is said to be marked if F_{i} = O_{i} and we say that F_{i} is a weight of this vertex. A marked vertex is denoted by . By definition of a weak (D,O)-species, each marked vertex is minimal in Q(W). The graph obtained from the graph Q(W) with marked vertices by deleting the orientation of all arrows that connect the unmarked vertices of Q(W) will be called the diagram with weights of a (D,O)-species W. Recall that a (D,O)-species is of bounded representation type if the dimensions of its indecomposable finite dimensional representations have an upper bound.
In this paper we prove the following theorem, which describes the structure of (D,O)-species of bounded representation type in terms of Dynkin diagrams and diagrams with weights. The proof of this theorem is reduced to solving some flat mixed matrix problems over discrete valuation rings and their common skew field of fractions as it was developed in [4].
Theorem 1. Let {O_{i}}_{i}_{ÎI} be a family of discrete valuation rings with a common skew field of fractions D. A weak (D,O)-species is of bounded representation type if and only if its diagram Q(W) is a finite disjoint union of Dynkin diagrams of the form A_{n}, D_{n}, E_{6}, E_{7}, E_{8} and the following diagrams with weights:
Throughout this paper all rings are assumed to be associative with identity.
2. Proof of the necessity in Theorem 1
Lemma 2.1. Let O be a discrete valuation ring with a skew field of fractions D and the Jacobson radical R = pO = Op, I = {1,2,3,4}. Then a (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(2.2) |
is of unbounded representation type.
Proof. We prove this lemma for the case F_{1 }= O, F_{2 }= F_{3 }= F_{4 }= D, _{1}M_{2 }=_{ 1}M_{3 }=_{ 1}M_{4 }= = _{O}D_{D} and _{i}M_{j} = 0 for other i,j Î I. Write J = {1,2,3}.
Let X be a right O-module, Y_{i} a right D-vector space, and let be a D-linear mapping for i Î J.
Consider the category Rep(W) whose objects are representations M = (X,_{ }Y_{i},_{ }j_{i},)_{i}_{ÎJ}, and a morphism from an object M to an object M¢ = (X¢, Y¢_{i}, j¢_{i},)_{i}_{ }_{ÎJ }is a set of homomorphisms (a, b_{i})_{ i }_{Î J}, in which a: X ® X¢ is a homomorphism of O-modules, b_{i}: Y_{i} ® Y¢_{i} is a homomorphism of D-vector spaces (iÎ J), and the following equalities hold:
(2.3) |
Let us show that the category Rep(W) is of unbounded representation type. Let X be a finitely generated free O-module with basis w_{1}, ... , w_{n}; and let Y_{i} be a finite dimensional D-space with basis (i = 1,2,3). Suppose
(iÎ J) (2.4) |
where for i Î J, s = 1, ..., n. Then the matrices , iÎ J, define the representation M uniquely up to equivalence.
Let UÎM_{n}(O) be the matrix corresponding to the isomorphism a, and let be the matrices corresponding to the isomorphisms b_{i}, i Î J. If are the matrices corresponding to a representation M¢, then equalities (2.3) have the following matrix form:
. | (2.5) |
We obtain the following matrix problem:
Given a block-rectangular matrix T = with the following admissible transformations:
1. Right O-elementary transformations of columns of T.
2. Left D-elementary transformations of rows within each block A_{i} (i = 1,2,3).
Set
, |
, , |
where p Î R = rad O, p ¹ 0. By [5, Lemma 4], the matrix T is indecomposable and therefore the species with diagram (2.2) is of unbounded representation type.
Lemma 2.6. Let O be a discrete valuation ring with a common skew field of fractions D and the Jacobson radical R = pO = Op, I = {1,2,3,4}. Then a (D,O)-spe-cies W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(2.7) |
is of unbounded representation type.
Proof. Analogously as in the proof of Lemma 2.1, we obtain the following matrix problem:
Given a block-triangular matrix T =_{ }. The following transformations are admitted:
1. Right O-elementary transformations of columns within the first vertical strip of the matrix T.
2. Right D-elementary transformations of columns within the second vertical strip of the matrix T.
3. Left D-elementary transformations of rows within each horizontal strip of the matrix T.
Reduce A_{3} to the form and obtain A_{2} of the form . It is possible to add any row of the matrix B_{2} multiplied on the left by elements of D to any column of the matrix B_{1}. Thus the matrices A_{1}, B_{1} and B_{2} form the matrix problem II from [5] and so the (D,O)-species with diagram (2.7) is of unbounded representation type.
Lemma 2.8. Let O be a discrete valuation ring with a skew field of fractions D and the Jacobson radical R = pO = Op, I = {1,2,3,4,5}. Then a (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(2.9) |
is of unbounded representation type.
Proof. Write J = {1,2,3,4}. Let X be a right O-module, let Y_{i} be a right D-vector space for iÎJ, and let , _{ }(i = 2,3), be D-linear mappings.
Let A_{i} be the matrices corresponding to homomorphisms j_{i} (i Î J) that define the representation M, and (iÎ J ) be the matrices corresponding to the represen- tation M¢. If UÎM_{n}(O) is the matrix corresponding to the isomorphism a, and are the matrices corresponding to the isomorphisms b_{i}, iÎ J, then we have the following matrix equalities:
, | (2.10) |
. | (2.11) |
We obtain the following matrix problem:
Given a block-rectangular matrix T = which partitioned into 2 horizontal and 3 vertical strips. The following transformations are admitted:
1. Right O-elementary transformations of columns within the first vertical strip of T.
2. Right D-elementary transformations of columns within the second and third vertical strips of T.
3. Left D-elementary transformations of rows within each horizontal strip of T.
Reduce A_{2} and A_{4} to the form and A_{3} to the form
In according with this reducing the matrix A_{1 }is divided into 6 horizontal strips and we obtain the following matrix problem.
Given a block-rectangular matrix B partitioned into 6 vertical strips:
B_{1} |
B_{2} |
B_{3} |
B_{4} |
B_{5} |
B_{5} |
The following transformations are admitted:
1. Left O-elementary transformations of rows of B.
2. Right D-elementary transformations of columns within each vertical strip B_{i} (i = 1,2,..,6).
3. If a_{i} £ a_{j} in the poset S:
then any column of the vertical strip B_{i} multiplied on the right by the arbitrary element of D can be added to any column of the vertical strip B_{j}.
It is easy to see that the blocks B_{2}, B_{4}, B_{5} form the matrix problem II from [5]. Therefore the species with diagram (2.9) is of unbounded representation type.
Analogously one can prove the following lemma:
Lemma 2.12. Let O be a discrete valuation ring with a skew field of fractions D and the Jacobson radical R = pO = Op, I = {1,2,3,4,5}. Then a (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(2.13) |
is of unbounded representation type for any directions of arrows.
Lemma 2.14. Let O_{i} be a discrete valuation ring with a common skew field of fractions D and the Jacobson radical R_{i} = p_{i}O_{i} = O_{i}p_{i}, for i = 1,2; and I = {1,2,3,4}. Then a (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(2.15) |
is of unbounded representation type.
Proof. Write J = {1,2,3}. Let X_{i} be a right O_{i}-module for i = 1,2; Y_{j} a right D-vector space for j = 1,2; and let , (i = 1,2), be D-linear mappings.
If M = (X_{1}, X_{2}, Y_{1}, Y_{2}, j_{i},)_{i}_{ }_{ÎJ} and M¢ = (X_{1}¢, X_{2}¢, Y_{1}¢, Y_{2}¢, j¢_{i},)_{i}_{ }_{ÎJ} are two equivalent representations of a (D,O)-species W and a_{i}: X_{i} ® X_{i}¢ are isomorphisms of O-modules, b_{i}: Y_{i} ® Y¢_{i} are isomorphisms of D-vector spaces (i = 1,2), then the following equalities hold:
(2.16) |
(2.17) |
Let A_{i} be the matrices corresponding to the homomorphisms j_{i} (i Î J) that define the representation M, and (iÎ J) be the matrices corresponding to the representation M¢. If U_{i} is the matrix with entries in O_{i} corresponding to the isomorphism a_{i}, and are the matrices corresponding to the iso- morphisms b_{i} (i = 1,2), then equalities (2.16) and (2.17) have the following matrix form:
(2.18) |
. | (2.19) |
We obtain the following matrix problem:
Given a block-rectangular matrix T =_{ }. The following transformations are admitted:
1. Left D-elementary transformations of rows of T.
2. Right O_{1}-elementary transformations of columns of A_{1}.
3. Right O_{2}-elementary transformations of columns of A_{2}.
4. Right D-elementary transformations of columns of A_{3}.
Consider two possible cases.
Case 1: Assume that O_{1} = O_{2}. Set
By [5, lemma 3], the matrix T is indecomposable. Thus the corresponding representation M of the species W is indecomposable and the species W with diagram (2.15) is of unbounded representation type.
Case 2: Assume that O_{1} ¹ O_{2}. Set A_{1} = A_{2} = I. Then T = and for the matrix A_{3} we have the matrix problem II from [6]. By [6, Lemma 4.2], this matrix problem is of unbounded representation type. Therefore, the species with diagram (2.15) is of unbounded representation type.
Lemma 2.20. Let O_{i} be a discrete valuation ring with a common skew field of fractions D and the Jacobson radical R_{i} = p_{i}O_{i} = O_{i}p_{i}, for i = 1,2; and I = {1,2,3,4}. Then a (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(2.21) |
is of unbounded representation type.
This lemma is proved as Lemma 2.14.
Note that all diagrams which are not presented in the diagrams of Theorem 1 have a subdiagram of one of the types considered in this section and hence they are of bounded representation type. Therefore, the necessity of Theorem 1 follows from Lemmas 2.1, 2.6, 2.8, 2.12, 2.14, 2.20 and [4, proposition 4.4].
3. Proof of the sufficiency in Theorem 1
Lemma 3.1. Let O be a discrete valuation ring with a skew field of fractions D and the Jacobson radical R = pO = Op, and let I = {1,2,…,n, n+1}. Then a weak (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(3.2) |
is of bounded representation type.
Proof. Write J = {1, 2, ... , n}. Let X be a right O-module, let Y_{i} be a right D-vector space for i Î J, and let ; be D-linear mapping for i = 2, ..., n.
If M = (X, Y_{i}, j_{i})_{i}_{ }_{ÎJ} and M¢ = (X¢, Y¢_{i}, j¢_{i})_{i}_{ }_{ÎJ} are two equivalent representations of a (D,O)-species W, a: X ® X¢ is an isomorphism of O-modules and b_{i}: Y_{i} ® Y¢_{i} are isomorphisms of D-vector spaces (i Î J), then the following equalities hold:
, | (3.3) |
. | (3.4) |
Let A_{i} be the matrices corresponding to the homomorphisms j_{i} (i Î J) that define the representation M, and (iÎ J) be the matrices corresponding to the representation M¢. If UÎM_{s}(O) is the matrix corresponding to the isomorphism a, and are the matrices corresponding to the isomorphisms b_{i} (iÎ J), then equalities (3.3) and (3.4) have the following matrix form:
(3.5) |
Note that the problem of reducing a set of matrices A_{i} by matrices V_{i} (i = 2,_{ }...,_{ }n) satisfying equalities (3.5) leads to the problem of classifying representations of the quiver Q with diagram A_{n}_{-1}. By the Gabriel theorem [1], this quiver has indecomposable representations. In accordance with these representations, the matrix A_{1} is partitioned into 2n-2 vertical strips, and the partial ordering relation between these strips is linear. Therefore, the matrix problem (3.5) leads to the following matrix problem.
Given a rectangular matrix T with entries in a skew field D that is partitioned into 2n-2 vertical strips:
T_{1} |
T_{2} |
… |
T_{2n-3} |
T_{2n-2} |
The transformations of the following types are admitted:
1. Right O-elementary transformations of rows of T.
2. Left D-elementary transformations of columns within each vertical strip T_{i}.
3. Addition of columns of the i-th vertical strip T_{i} multiplied on the right by elements of D to columns in the j-th vertical strip T_{j} if i £ j.
The matrix T can be reduced by these transformations to the form in which any block T_{i} has the form and all blocks over and under I are zero. Thus, for any indecomposable representation M the corresponding matrix T has a finite fixed number of nonzero elements and this number depends only on n. Therefore, the species W with diagram (3.2) is of bounded representation type.
Lemma 3.6. Let O_{i} be a discrete valuation ring with a common skew field of fractions D and the Jacobson radical R_{i} = p_{i}O_{i} = O_{i}p_{i} for i = 1,2; and let I = {1,2,..., n, n +1}. Then a weak (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(3.7) |
is of bounded representation type.
Proof. Renumber the vertices of the diagram (2.16) in such a way that the vertex 1 corresponds to F_{1} = O_{1} and the vertex 2 corresponds to F_{2} = O_{2}. We obtain the diagram
Write J = {1, 2, ..., n–1}. Let X_{i} be a right F_{i}-module (i = 1,2), let Y_{j} be a right D-vector space for j_{ }Î_{ }J, and let ; ; be D-linear mapping for i = 2, ..., n-1.
If M = (X_{1}, X_{2}, Y_{i}, j_{i}, j_{n})_{i}_{ }_{ÎJ} and M¢ = (X_{1}¢, X_{2}¢, Y¢_{i}, j¢_{i}, j_{n}¢)_{i}_{ }_{ÎJ} are two equivalent representations of a (D,O)-species W and a_{i}_{ }: X_{i} ® X_{i}¢ is an isomorphism of O-modules (i = 1,2), b_{j} : Y_{j} ® Y¢_{j} is an isomorphism of D-vector spaces (j Î J), then the following equalities hold:
(3.8) |
(3.9) |
Suppose the isomorphism a_{i} is given by the matrix U_{i }with entries in O_{i} for i = 1,2; and the isomorphism b_{j} is given by the matrix V_{j }with entries in D for j Î J. If A_{i}, (i Î I) are the matrices corresponding to the representations M, M¢ then equalities (3.8) and (3.9) are equivalent to the following matrix equalities:
(3.10) |
(3.11) |
Note that the reduction of a set of matrices A_{i}, i = 1,3,4, ..., n, by admissible transformations (3.10) and (3.11) leads to the matrix problem of classifying representations of a (D,O)-species W with diagram (3.2) described in Lemma 3.1. After reducing the matrices A_{i}, i = 1,3,4, ..., n, we get A_{2} partitioned into n vertical strips and the following matrix problem for A_{2}.
Given a block matrix T with entries in a skew field D that is partitioned into n vertical strips:
T_{1} |
T_{2} |
… |
T_{n} |
The transformations of the following types are allowed:
1. Left O_{2}-elementary transformations of rows of T.
2. Right O_{1}-elementary transformations of columns within a vertical strip T_{k}, for some fixed 1 £ k £ n.
3. Right D-elementary transformations of columns within each vertical strip T_{i}, if i ¹ k.
4. Addition of columns in the vertical strip T_{i} multiplied on the right by elements of D to columns in the vertical strip T_{j} if i £ j.
Using these transformations and taking into account [5, Lemma 4.1], we can reduce T to the form in which every block T_{i} has the form: U = or W = and all matrices over, under, on the left, and on the right of the matrix I or, respectively, p^{m}I are zero. This means that the matrix A_{2} is decomposed into a direct sum of matrices of the forms U and W.
Thus, the species W with diagram (3.7) is of bounded representation type.
Analogously we can prove the following lemma:
Lemma 3.12. Let O be a discrete valuation ring with a skew field of fractions D and the Jacobson radical R = pO = Op, and let I = {1,2,…,n, n+1}. Then a weak (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(3.13) |
is of bounded representation type.
Lemma 3.14. Let O be a discrete valuation ring with a skew field of fractions D and the Jacobson radical R = pO = Op, and I = {1,2,…,n, n+1}. Then a weak (D,O)-species W = (F_{i}, _{i}M_{j})_{i}_{,j }_{Î I} whose diagram with weights has the form
(3.15) |
is of bounded representation type.
Proof. Write J = {1,2}. Let X be a right O-module, let Y_{i} be a right D-vector space, and let be a D-linear mapping for i Î J.
If M = (X, Y_{i}, j_{i})_{i}_{ }_{ÎJ} and M¢ = (X¢, Y¢_{i}, j¢_{i})_{i}_{ }_{ÎJ} are two equivalent representations of a (D,O)-species W and a: X ® X¢ is an isomorphism of O-modules, b_{i} : Y_{i} ® Y¢_{i} are isomorphisms of D-vector spaces (i Î J), then the following equalities hold:
(3.16) |
Let U Î M_{s}(O) be the matrix corresponding to the isomorphism a, and let be the matrices corresponding to the isomorphisms b_{i}, iÎ J. If A_{i}, (i Î J) are the matrices corresponding to the representations M, M¢ then equalities (3.14) are equivalent to the following matrix equalities
(3.17) |
We obtain the following matrix problem.
Given a matrix T = with entries in D. The transformations of the follow- ing types are allowed:
1. Right D-elementary transformations of columns within any vertical strip.
2. Left O-elementary transformations of rows of T.
The matrix T can be reduced by these transformations to a direct sum of the following matrices:
, , , . |
Thus, a (D,O)-species with diagram (3.15) is of bounded representation type.
The sufficiency in Theorem 1 follows from Lemmas 3.1, 3.6, 3.12, and 3.16.
4. Conclusion
We have described all weak (D, O)-species of bounded representation type. The proof is given by reduction to mixed matrix problems over discrete valuation rings and their common skew field of fractions. We use some facts about representations of quivers and species from [1-3].
References
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[2] Dlab V., Ringel C.M., On algebras of finite representation type, Journal of Algebra 1975, 33, 306-394.
[3] Dowbor P., Ringel C.M., Simson D., Hereditary Artinian rings of finite representation type, Representation theory, II. Lecture Notes in Math. 1980, 832, 232-241.
[4] Gubareni N., Representations of (D,O)-species and flat mixed matrix problems, Journal of Applied Mathematics and Computational Mechanics 2016, 3(15), 47-56.
[5] Gubareni N., Structure of finitely generated modules over right hereditary SPSD-rings, Scientific Research of the Institute of Mathematics and Computer Science 2012, 3(11), 45-56.
[6] Gubareni N., Some mixed matrix problems over several discrete valuation rings, Journal of Applied Mathematics and Computational Mechanics 2013, 4(12), 47-58.