Representation theory of Jordanian algebra

TL;DR

This paper classifies irreducible modules over a specific algebra, studies their properties, and analyzes the complexity of their representations, revealing tame and wild behaviors depending on the dimension.

Contribution

It provides a complete classification of irreducible modules, describes the structure of their image algebras, and establishes the tame-wild dichotomy based on representation dimension.

Findings

All image algebras are basic and local complete algebras.

Image algebras are tame for dimensions ≤ 4 and wild for dimensions ≥ 5.

A stratification of the representation space related to Jordan forms is proposed.

Abstract

We describe the complete set of pairwise non-isomorphic irreducible modules S(a) over the algebra R given by the defining relation xy-yx=yy, and the rule how they could be glued to indecomposables. Namely, we show that Ext_k^1(S(a),S(b))=0, if a not equal to b. Also the set of all representations is described subject to the Jordan normal form of Y. We study then properties of the image algebras in the endomorphism ring. Among facts we prove is that they are all basic algebras. Along this line we establish an analogue of the Gerstenhaber-Taussky-Motzkin theorem on the dimension of algebras generated by two commuting matrices. All image algebras of indecomposable modules turned out to be local complete algebras. We compare them with the Ringel's classification by means of finding relations of image algebras. As a result we derive that all image algebras of n-dimensional representations with full block Y are tame for n smaller or equal then 4 and wild for n starting from 5. We suggest a stratification of representation space of R related to the partitions of n defined by the Jordan normal form of Y. We give a complete classification by parameters for some strata and present examples of tame (up to automorphisms) strata, while the generic strata is wild.