1. Introduction
We show that the problem of classifying pairs consisting of a bilinear form and a linear map plays the same role in the theory of systems of bilinear forms and linear maps as the problem of classifying pairs of linear maps plays in the theory of representations of finite dimensional algebras.
About fifty years ago, it was noticed that most unsolved classification problems in the theory of representations of groups and algebras “contain” the matrix pair problem, which is the problem of classifying pairs of square matrices of the same size up to similarity transformations
(A classification problem contains a classification problem if solving would solve .)
For example, Bašev [
1] classified representations of the abelian group
over a field of characteristic 2. The problems of classifying representations of the abelian groups
and
over a field of characteristic 2 and the abelian group
over a field of characteristic
are considered as hopeless since these problems contain the matrix pair problem (Brenner [
2] and Krugljak [
3]).
Donovan and Freislich [
4] call a matrix problem “wild” if it contains the matrix pair problem and “tame” otherwise, in analogy with the partition of animals into wild and tame ones. A certain characterization of tame and wild problems is given by Drozd [
5] and Crawley-Boevey [
6]; a geometric form of the tame–wild theorem is proved by Gabriel, Nazarova, Roiter, Sergeichuk, and Vossieck [
7], and by Sergeichuk [
8].
The reason for complexity of the matrix pair problem was found by Gelfand and Ponomarev [
9], who showed that the problem of classifying pairs of commuting nilpotent matrices over any field up to similarity transformations (
1) contains the problem of classifying matrix
t-tuples up to similarity transformations
The matrix pair problem also contains the problem of classifying representations of every quiver and every poset (Barot [
10] [Section 2.4], Belitskii and Sergeichuk [
11], Krause [
12] [Section 10]). Moreover, it contains the problem of classifying representations of an arbitrary finite-dimensional algebra (Barot [
13] [Proposition 9.14]).
Note that each concrete pair
of
matrices over an algebraically closed field is reduced to its canonical form
with respect to similarity transformations (
1) by Belitskii’s algorithm [
14] in such a way that
is similar to
if and only if
. However, there is no nonalgorithmic description of the set of Belitskii’s canonical pairs under similarity (i.e., the pairs that are not changed by Belitskii’s algorithm). Belitskii’s algorithm was extended by Sergeichuk [
8] to a wide class of matrix problems that includes the problems of classifying representations of quivers and finite dimensional algebras.
The problem of classifying arrays up to equivalence plays the same role in the theory of tensors as the matrix pair problem in the theory of representations of algebras: Futorny, Grochow, and Sergeichuk [
15] proved that the problem of classifying three-dimensional arrays up to equivalence transformations contains the problem of classifying every system of tensors of order at most three.
We show that the problem of classifying pairs consisting of a bilinear form and a linear map contains the problem of classifying arbitrary systems of bilinear forms and linear maps.
2. Main Results
Many classification problems of linear algebra can be formulated and studied in terms of quiver representations introduced by Gabriel [
16]. A quiver is a directed graph. Its representation is given by assigning a vector space to each vertex and a linear map of the corresponding vector spaces to each arrow. This notion plays a central role in the representation theory of finite dimensional algebras since each algebra can be given by a quiver with relations and there is a natural correspondence between their representations; see [
10,
13,
17,
18,
19].
Following [
20,
21], we consider systems of forms and linear maps over a field
as representations of a mixed graph (i.e., of a graph with undirected and directed edges; multiply edges and loops are allowed): Its vertices represent vector spaces, and its undirected and directed edges represent bilinear forms and linear maps between these spaces. Two representations are isomorphic if these are a set of linear bijections of the corresponding vector spaces that transform one representation into the other; see Definition 1.
Example 1. Consider a mixed graph Q and its representation : The representation consists of vector spaces over , bilinear forms , , , and linear maps , , . The vector is called the dimension of . Changing bases in the spaces , we can reduce the matrices of as follows:where are nonsingular matrices. Thus, the problem of classifying representations of Q of dimension is the problem of classifying all tuples of matrices of sizes , , …, up to these transformations. For a mixed graph
Q, we denote by
the set of representations of dimension
, in which all vector spaces are of the form
with
. The set
is a vector space over
; its elements are matrix tuples (see Definition 2). We say that the problem of classifying representations of a mixed graph
Q is contained in the problem of classifying representations of a mixed graph
if for each dimension
there exists an affine injection (That is,
for
and all
, in which
is a linear injection).
of a very special type given in Definition 3 such that
are isomorphic if and only if
are isomorphic.
The problem of classifying representations of Q and are equivalent if each of these problems contains the other.
The main result of this paper is the following theorem, which is proved in
Section 4 and
Section 5.
Theorem 1. - (a)
The problem of classifying representations of (i.e., of pairs consisting of a bilinear form and a linear map) contains the problem of classifying representations of each mixed graph. - (b)
Let a mixed graph Q satisfy the following condition:Then the problem of classifying representations of Q is equivalent to the problem of classifying representations of .
Let us derive a corollary of Theorem 1.
Let
Q be a connected mixed graph that does not satisfy the condition (
4). The representations of mixed graphs that are cycles are classified in [
20] [§ 3]. Suppose that
Q does not contain a cycle in which the number of undirected edges is odd; in particular, it does not contain undirected loops. Then
Q can be transformed to a quiver
with the same underlying graph (obtained by deleting the orientation of the edges) using the following procedure described in [
20]:
If
v is a vertex of
Q, then we denote by
the graph that is obtained from
Q by deleting (resp., adding) the arrows at the ends of all edges in the vertex
v that have it (resp., do not have it); we say that
is obtained from
Q by dualization at
v. For example:
(we write
instead of
v). Thus,
is replaced by
. Then we replace
w with
and obtain
. Since
Q does not contain a cycle in which the number of undirected edges is odd, we can reduce
Q to some quiver
by these replacements.
There is a natural correspondence between the representations of Q and : If is a representation of Q, then the representation of is obtained by replacing the vector space V assigned to v with the dual space of all linear forms on V. This correspondence is based on the fact that each bilinear form defines the linear map via , and each linear map defines the bilinear form via .
Therefore, the theory of representations of mixed graphs without cycles in which the number of undirected edges is odd is the theory of quiver representations. The representation types of quivers are well known; the representations of tame quivers are classified by Donovan and Freislich [
4], and Nazarova [
22]. The other quivers are wild; the problem of classifying representations of each wild quiver is equivalent to the problem of classifying representations of
(i.e., of matrix pairs up to similarity transformations (
1)); see [
10,
11,
12] and Lemma 1.
We say that a mixed graph
Q is tame if it is reduced by dualizations at vertices to the disjoint union of mixed cycles and a tame quiver (thus, the classification of representations of tame mixed graphs is known). A mixed graph
Q is wild if it is reduced by dualizations at vertices to a wild quiver. A mixed graph
Q is superwild if the problem of classifying its representations is equivalent to the problem of classifying representations of
.
Corollary 1. Each mixed graph is tame, wild, or superwild.
- (i)
A mixed graph is tame if its underlying graph is the disjoint union of some copies of the Dynkin diagrams and the extended Dynkin diagrams - (ii)
A mixed graph is wild if it is not tame and it does not contain a cycle in which the number of undirected edges is odd.
- (iii)
A mixed graph is superwild if it satisfies condition (4).
Proof. Let
Q be a mixed graph. If (
4) holds, then
Q is superwild by Theorem 1(c). Suppose that (
4) does not hold. Then
Q is the disjoint union of mixed cycles and a mixed graph
without a cycle in which the number of undirected edges is odd. The graph
is reduced to a quiver
by dualizations at vertices. By definition,
Q is tame or wild if and only if
is tame or wild, respectively. By [
4,
22],
is tame if and only if its underlying graph is the disjoint union of some copies of graphs (
5). □
3. The Category of Representations
The category of representations of a mixed graph (in general, nonadditive) is defined as follows.
Definition 1 ([
21]; see also [
18])
. Let Q be a mixed graph with vertices . Its representation over a field is given by assigning a vector space over to each vertex v, a bilinear form to each undirected edge with (this inequality is given for uniqueness), and a linear map to each directed edge . A morphism of two representations of Q is a family of linear maps …, such that for each with , and for each . In the following definition, we consider representations, in which all vector spaces are
with
. Such representations are given by their matrices, as in (
3), and so we call them “matrix representations”. The category
of matrix representations is defined as follows.
Definition 2. Let Q be a mixed graph with vertices . A matrix representation A of dimension of Q over a field is given by assigning a matrix to each undirected edge with and to each directed edge . A morphism of matrix representations of dimensions and is a family of matrices of sizes such thatfor each undirected edge () and each directed edge . The set of matrix representations of dimension is a vector space over .
Definition 3. Let Q and be mixed graphs with vertices and , and with edges and , respectively. The problem of classifying representations of Q is contained in the problem of classifying representations of , we write , if there exists a functorwith the following properties: is injective on objects; moreover, for each there exists (all and are nonnegative integers) such that maps to and this map is an affine injection of vector spaces.
is injective on morphisms; moreover, are isomorphic if and only if are isomorphic.
is produced by a parameter matrix representation of (compare with [7] [p. 338]), which means that there exists a parameter matrix representationof with parameters such that each is an exactly one entry of an exactly one matrix among , the other of their entries are elements of , andfor each matrix representation of Q. The representation (7) of is constructed on (6) as follows: - -
Rearranging the basis vectors in the vector spaces of the representation , we rearrange the rows and columns of its matrices converting to a direct sum of matrix representations of of nonzero dimension with the maximum number r. We say that two rows or columns of are lined if they (with ) are converted to rows or columns from the same summand . Thus, there are r classes of linked rows and columns; we require that each class contains a row or column with a parameter.
- -
Let be natural numbers. Denote by a matrix representation of obtained from by replacing all rows and columns that belong to the same class by strips of size such that each parameter is replaced by an arbitrary matrix of suitable size, each entry that is a nonzero element is replaced by , and each zero entry is replaced by the zero block.
Two mixed graphs Q and are equivalent if and (see Definition 3).
Example 2. each morphism : A B → A′ B′ (where and ) is given by a matrix such that and ; the corresponding morphism , whereis given by the matrix pair ; is reduced to by similarity transformations (
1)
if and only if is reduced to by transformations
4. Proof of Theorem 1(a)
Let us consider a matrix representation
in which
We reduce
by those admissible transformations that preserve
J:
Partition
conformally to
J. Since
, every
has the form
where all
are arbitrary
matrices such that
S is nonsingular. Replacing all off-diagonal blocks of
S by zeros, we obtain the block diagonal matrix
in which every
is
.
By Belitskii’s theorem (see [
23] [Section 3.4] or [
8] [Theorem 1.2]), the Jordan matrix
J is permutation similar to a nilpotent Weyr matrix such that all matrices commuting with it are upper block triangular. Since
, Belitskii’s theorem ensures that the matrix
S is permutation similar to a block triangular matrix whose main block diagonal coincides with the sequence of summands in (
10), up to permutation of these summands. Hence,
S is nonsingular if and only if all blocks
are nonsingular. Moreover, if the matrix
H in (
8) is partitioned such that the sizes of its diagonal blocks coincide with the sizes of the direct summands in (
10), then
Let us prove Theorem 1(a) for the mixed graph
Q in (
2); its proof for an arbitrary mixed graph is analogous. Let
be matrix representations of
Q of dimension
. We construct the matrix representations
H J and
J as follows:
in which the points denote zero blocks; the matrix
is obtained from
H by replacing
with
.
If
, then
(see (
10)). The summands of (
15) are written in (
14) over the vertical strips of
H, and their transposes are written to the left of the horizontal strips of
H. By (
11), each nonzero block of (
14) is multiplied by them if
is reduced by transformations (
9).
We must prove that
is reduced to
by transformations (
9) if and only if the representations (
12) are isomorphic; that is, if and only if
are reduced to
by transformations (
3).
Let
be reduced to
by transformations (
9). By (
11),
The first two equalities ensure that
and
. Therefore,
are reduced to
by transformations (
3).
Conversely, let
be reduced to
by transformations (
3). Then
is reduced to
by transformations (
9) with
S of the form (
15), in which
and
.
This completes the proof of Theorem 1(a). The following lemma is proved analogously; it is also proved in [
10] [Section 2.4], [
11], and [
12] [Section 10].
Lemma 1. The problem of classifying representations of contains the problem of classifying representations of each quiver. Proof. Let
Q be a quiver, and let
A be its matrix representation. We construct a matrix representation
H J as follows.
The matrix
J is given in (
8). The equality
implies that the main block diagonal of
S is (
10).
We take
H in which each horizontal strip and each vertical strip contains at most one nonzero block. By analogy with (
11),
H is reduced by transformations
. We construct
H such that some of its blocks are the matrices of
A and they are reduced by the same transformations as in
A, and the other blocks are zero. □