Affiliation:
1. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Abstract
We study the semigroup structure on the set [Formula: see text] of conjugacy classes of left ideals of a finite-dimensional algebra [Formula: see text] over an algebraically closed field [Formula: see text], equipped with the natural multiplication inherited from [Formula: see text], and the structure of the contracted semigroup algebra [Formula: see text]. It is shown that [Formula: see text] has a finite chain of ideals with either nilpotent or completely [Formula: see text]-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver [Formula: see text] of [Formula: see text] is proved to be a subquiver of [Formula: see text], if [Formula: see text] is finite. Moreover, in this case, the structure of [Formula: see text] determines, up to isomorphism, the structure of the algebra [Formula: see text] modulo its Jacobson radical. Combining these results we show that if the semigroup [Formula: see text] is finite, then it determines the structure of any (not necessarily basic) triangular algebra [Formula: see text] which admits a normed presentation.
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,Algebra and Number Theory