While mutual congeniality of bases has been known to guarantee that basic modules from so related bases are isomorphic, the question of what can be said about isomorphism of basic modules in general has remained open.
We show that neither of two possible extremes need hold. For some algebras it is possible for basic modules to be non-isomorphic. Also, it is possible, for some algebras, that all basic modules are isomorphic. We show that there are at least as many pairwise non-isomorphic basic modules over the F-algebra F[x] of polynomials in a single variable as there are elements in F. We show that basic modules over F[x] can be non-isomorphic when they are induced by discordant bases and also even when there is a (non-mutual) congeniality among them. In the process and as a byproduct, we introduce the notion of domains of divisibility of modules over arbitrary rings and explore some of the properties of a divisibility profile.
At the opposite end of the spectrum, we present an algebra where all basic modules are isomorphic, regardless of congeniality.
This is a report on joint work with: C. Arellano, P. Aydogdu, R. Muhammad, and M. Zailaee
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
