On a Galois Connection Between Algebras of Linear Transformations and Lattices of Subspaces of a Vector Space

Abstract

Representation theory has contributed much to the study of linear associative algebras. The central problem of representation theory per se is the determination for each algebra of all its indecomposable representations. This turns out to be a much deeper problem than the classification of algebras, in the sense that there are algebras for which any “internal question” can be answered but for which the number and nature of representations is almost completely unknown, or if known is much more complicated than the internal theory. This can be illustrated by the example of a commutative algebra of order three for which the representation theory can be shown to be essentially the same as the problem of classifying pairs of rectangular matrices under equivalence. (This algebra has indecomposable representations of every integral degree.)