Abstract
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.
Highlights
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
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
Its representation is given by assigning a vector space to each vertex and a linear map of the corresponding vector spaces to each arrow
Summary
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. The matrix pair problem 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]) It contains the problem of classifying representations of an arbitrary finite-dimensional algebra (Barot [13] [Proposition 9.14]). 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. 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
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