Three Representation Types for Systems of Forms and Linear Maps

: We consider systems of bilinear forms and linear maps as representations of a graph with undirected and directed edges. Its vertices represent vector spaces; its undirected and directed edges represent bilinear forms and linear maps, respectively. We prove that if the problem of classifying representations of a graph has not been solved, then it is equivalent to the problem of classifying representations of pairs of linear maps or pairs consisting of a bilinear form and a linear map. Thus, there are only two essentially different unsolved classiﬁcation problems for systems of forms and linear maps.


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 For example, Bašev [1] classified representations of the abelian group (2, 2) over a field of characteristic 2. The problems of classifying representations of the abelian groups (2,4) and (2, 2, 2) over a field of characteristic 2 and the abelian group (p, p) over a field of characteristic p > 2 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 (M 1 , . . . , M t ) → (C −1 M 1 C, . . . , C −1 M t C), C is nonsingular.
Note that each concrete pair (A, B) of n × n matrices over an algebraically closed field is reduced to its canonical form (A can , B can ) with respect to similarity transformations (1) by Belitskii's algorithm [14] in such a way that (A, B) is similar to (C, D) if and only if (A can , B can ) = (C can , D can ). 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.

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 F 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 R: The representation R consists of vector spaces U, V, W over F, bilinear forms C : The vector n = (n 1 , n 2 , n 3 ) := (dim U, dim V, dim W) is called the dimension of R. Changing bases in the spaces U, V, W, we can reduce the matrices of R as follows: where S 1 , S 2 , S 3 are nonsingular matrices. Thus, the problem of classifying representations of Q of dimension n = (n 1 , n 2 , n 3 ) is the problem of classifying all tuples (A, B, . . . , F) of matrices of sizes n 1 × n 2 , n 3 × n 1 , . . . , n 3 × n 3 up to these transformations.
For a mixed graph Q, we denote by M n (Q) the set of representations of dimension n, in which all vector spaces are of the form F k with k = 0, 1, 2, . . . . The set M n (Q) is a vector space over F; 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 Q if for each dimension n there exists an affine injection (That is, F(A) = R + ϕ(A) for R ∈ M n (Q ) and all A ∈ M n (Q), in which ϕ : M n (Q) → M n (Q ) is a linear injection). The main result of this paper is the following theorem, which is proved in Sections 4 and 5.

Theorem 1.
(a) The problem of classifying representations of r (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: Q contains a cycle in which the number of undirected edges is odd, and Q contains an edge outside of this cycle but with a vertex (or both the vertices) in this cycle.

(4)
Then the problem of classifying representations of Q is equivalent to the problem of classifying representations of r .
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 Q 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 Q v 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 Q v is obtained from Q by dualization at v. For example: Then we replace w with w * and obtain v * w * . Since Q does not contain a cycle in which the number of undirected edges is odd, we can reduce Q to some quiver Q by these replacements.
There is a natural correspondence between the representations of Q and Q v : If A is a representation of Q, then the representation A v of Q v is obtained by replacing the vector space V assigned to v with the dual space V * of all linear forms on V. This correspondence is based on the fact that each bilinear form B : 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 r (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 r .
(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 r r r r · · · r r r r r r · · · r r r r r r r r r r r r r r r r r r r r r r r r r r r r r · · · r r r r r · · · r r r r r r r r r r r r r r r r r r r r r r r r r r r r (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 Q 0 without a cycle in which the number of undirected edges is odd. The graph Q 0 is reduced to a quiver Q 0 by dualizations at vertices. By definition, Q is tame or wild if and only if Q 0 is tame or wild, respectively. By [4,22], Q 0 is tame if and only if its underlying graph is the disjoint union of some copies of graphs (5).

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 1, . . . , t. Its representation A over a field F is given by assigning a vector space A v over F to each vertex v, a bilinear form A α : A v × A u → F to each undirected edge α : u v with u v (this inequality is given for uniqueness), and a linear map A β : A u → A v to each directed edge β : u v. A morphism ϕ : A → B of two representations of Q is a family of linear maps ϕ 1 : In the following definition, we consider representations, in which all vector spaces are F k with k = 0, 1, 2, . . . . Such representations are given by their matrices, as in (3), and so we call them "matrix representations". The category M(Q) of matrix representations is defined as follows.

Definition 2.
Let Q be a mixed graph with vertices 1, . . . , t. A matrix representation A of dimension n = (n 1 , . . . , n t ) of Q over a field F is given by assigning a matrix A α ∈ F n v ×n u to each undirected edge α : u v with u v and to each directed edge α : u v. A morphism ϕ : A → B of matrix representations of dimensions n and n is a family of matrices S 1 , . . . , S t of sizes n 1 × n 1 , . . . , n t × n t such that The set M n (Q) of matrix representations of dimension n is a vector space over F. with the following properties: • F is injective on objects; moreover, for each n = (n 1 , . . . , n t ) there exists n = (n 1 , . . . , n t ) (all n i and n j are nonnegative integers) such that F maps M n (Q) to M n (Q ) and this map is an affine injection of vector spaces.
for each matrix representation A = (A α 1 , . . . , A α p ) of Q. The representation (7) of Q is constructed on (6) as follows: -Rearranging the basis vectors in the vector spaces of the representation N(x), we rearrange the rows and columns of its matrices converting N(0) to a direct sum M 1 ⊕ · · · ⊕ M r of matrix representations of Q of nonzero dimension with the maximum number r. We say that two rows or columns of N(x) are lined if they (with x = 0) are converted to rows or columns from the same summand M i . Thus, there are r classes L 1 , . . . , L r of linked rows and columns; we require that each class contains a row or column with a parameter. -Let n 1 , . . . , n r be natural numbers. Denote by N(K 1 , . . . , K t ) a matrix representation of Q obtained from N(x) by replacing all rows and columns that belong to the same class L j by strips of size n j such that each parameter x i is replaced by an arbitrary matrix K i of suitable size, each entry that is a nonzero element α ∈ F is replaced by αI, and each zero entry is replaced by the zero block.
Two mixed graphs Q and Q are equivalent if Q Q and Q Q (see Definition 3).

Example 2.
The problem of classifying representations of the quiver r is contained in the problem of classifying representations of r r (i.e., r r r ) via

Proof of Theorem 1(a)
Let us consider a matrix representation H r J, J := J k 1 (0 n 1 ) ⊕ · · · ⊕ J k τ (0 n τ ), (8) in which We reduce H r J by those admissible transformations that preserve J: (H, J) → (S T HS, S −1 JS), S is nonsingular and S −1 JS = J.
Partition S = [S ij ] τ i,j=1 conformally to J. Since JS = SJ, every S ij has the form where all C ij are arbitrary n i × n j matrices such that S is nonsingular. Replacing all offdiagonal blocks of S by zeros, we obtain the block diagonal matrix in which every S i := C ii is n i × n i . 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 JS = SJ, 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) 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 (n 1 , n 2 , n 3 ). We construct the matrix represen-tations H r J and H r J as follows: 1 · · · · · · · I · · S T 2 · · C · · · · · · · S T 2 · · · · · · · · · · S T 2 · · · · · · · · · · S T 3 · · · · · · · · I · S T 3 · · · D · · · · · · S T 3 · · · · · F · · · · S T 4 (=S −1 in which the points denote zero blocks; the matrix H is obtained from H by replacing A, B, . . . , F with A , B , . . . , F . If JS = SJ, then (see (10)). The summands of (15) are written in (14)  Let (H, J) be reduced to (H , J) by transformations (9). By (11), Proof. Let Q be a quiver, and let A be its matrix representation. We construct a matrix representation H r J as follows. The matrix J is given in (8). The equality S −1 JS = J 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 S −1 diag HS diag . 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.

Proof of Theorem 1(b)
Lemma 2. Let a connected mixed graph Q contain a cycle in which the number of undirected edges is odd, and let Q not coincide with this cycle. Then the problem of classifying its representations contains the problem of classifying representations of one of the mixed graphs Proof. Let a connected mixed graph Q contain a cycle C in which the number of undirected edges is odd, and let α be an edge outside of C with vertices v 1 ∈ C and u.
in which k 1 and each dotted line is an undirected or directed edge. Consider the matrix representation of Q, in which A, B ∈ F n×n and all vertices outside of C are assigned by the zero spaces. Let us prove that r Q with a suitable direction of the left loop in r . We need to show the following equivalence of isomorphisms: (" " means "is isomorphic to"). Let us prove "⇐=". Let R(A, B) R(A , B ) via (S 1 , . . . , S k ). Then S 1 = S −T 2 if α 1 is undirected and S 1 = S 2 if α 1 is directed. Analogously, S 2 = S −T 3 if α 2 is undirected and S 2 = S 3 if α 2 is directed, and so on. Since the number of undirected edges of C is odd, S k−1 = S k if α k is undirected and S k−1 = S −T k if α k is directed. Hence, B = S T 1 BS 1 , and so A r B A r B via S 1 , in which the left loop in r is directed as the left loop in (17). The implication "=⇒" is proved analogously.
Case 2: u = v r with r = 1. If α : v 1 v r , then we take v r as v 1 . We have that α : v 1 v r or α : v 1 v r , and so Q contains the subgraph G r that is obtained from (17) by replacing α r with α : v 1 v r . Denote by R r (A, B) the matrix representation of G r that is obtained from (18)  ⇐=. Let the right isomorphism hold. Then for nonsingular S and R.
The equality BR = SB implies that S and R have the form Substituting them to S T AS = A , we obtain Since the rows of R are linearly independent, S T 2 [S 4 S 5 ] = [0 0] implies S 2 = 0. Since S T 1 S 5 = I, S T 1 S 4 = 0 implies S 4 = 0. Equating the (2,3) blocks gives S T 5 S 7 = I. Equating the (1,3) blocks gives S 6 = 0. Equating the last horizontal strips gives Since S T 3 S 5 = 0 and S 5 is nonsingular, we have S 3 = 0. Therefore, S = S 1 ⊕ S −T 1 ⊕ S 1 , and so S T 1 XS 1 = X and S T 1 YS 1 = Y . =⇒. If the left isomorphism holds via C, then (23) holds for S = C ⊕ C −T ⊕ C and R = C ⊕ C −T .

Lemma 5.
r r r and r r r Proof. It is sufficient to prove that r r r and r r r because of Lemma 3. We take A and A as in (21) and B = [0 0 I]. We must prove (22), in which is replaced by and by . The proof is the same since the equality BS = RB implies that S as in (24).

Proof of Theorem 1(b).
Let a mixed graph Q satisfy the condition (4). By Theorem 1(a), Q r . Lemma 2 ensures that H Q, in which H is one of the mixed graphs (16). By Lemmas 3-5, r H. Hence, r Q.

Conclusions
We have proved that the problem of classifying matrix pairs with respect to transformations (A, B) → (S T AS, S −1 BS), S is nonsingular (25) contains the problem of classifying an arbitrary system of bilinear forms and linear maps. There are only two essentially different unsolved classification problems for systems of forms and linear maps: The classical unsolved problem about matrix pairs under similarity and the problem of classifying matrix pairs under transformations (25). These problems are given by the graphs r and r . There is no sense in studying representations of each of these graphs (and of any graph that is equivalent to one of them) outside the general theory of representations of quivers or mixed graphs, respectively. Likewise, Belitskii's algorithm that was constructed for matrix pairs under similarity can be applied to matrices of an arbitrary system of linear maps.