Indecomposable representations of certain commutative quivers

Abstract

This paper gives a detailed classification of the indecomposable representations of one non-trivial “commutative quiver”, gives brief details of certain generalisations, and uses methods likely to be more widely applicable.

Tame algebras (on algorithms for solving vector space problems. II)

“…An additive category of modules is called finite provided it contains only a finite number of indecompo-*) Most of the result can be adapted to the case of an arbitrary (commutative) base field. Note that in contrast to a remark in [14], the extension to skew fields will provide substantial changes, since for a skew field D, the polynomial ring D [T] in one variable may be wild. misprints and inaccura ies in a first draft of these notes.…”

“…She considers with any algebra the corresponding quadratic form, an aspect which we usually will neglect. Also, there is a recent paper by Donovan-Freislich [14], which reduces the investigation of two nontame algebras (of type (~4' 2 • 2), see 3.5) to a eorrespondomestic *) ding vectorspace problem Finally, let us confess that our interest in the work of Shkabara and Zavadskij was motivated by the fact that the report at the conference was intended to include the theory of differential graded categories due to Kleiner and Roiter. in order to get a better understanding of this method it seemed to be convenient to follow its recent applications, and in particular, to see at what point the previously known methods were not strong enough for solving the problems. The differential graded categories were introduced as generalisation of the method of partially ordered sets, and, in fact, both Shkabara and Zavadskij need a further generalisation, namely differential ~-graded categories.…”

On a Category Related to the Kalman Algebra of a Linear Control System

“…The orbits of σ consist of isomorphic representations. According to [10,11], L CS is a wild quiver (i.e., each finite-dimensional unital associative F-algebra can be realized as the endomorphism algebra of some representation of L CS ); therefore, the classification of its representations is a very difficult problem. A method for finding a family of generators for the infinite-dimensional F-linear space of semi-invariants was suggested in [12][13][14].…”