Irreducible components of the Jordan varieties

TL;DR

This paper classifies the irreducible components of the representation variety of the Jordan algebra, describing their structure, the associated modules, and properties of the image algebras, including tame-wild dichotomy.

Contribution

It provides a complete description of the irreducible components of the Jordan variety and analyzes the structure of modules and image algebras, including a tame-wild classification.

Findings

Number of irreducible components equals the number of partitions of n.

Complete set of irreducible modules over the Jordan algebra is described.

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

Abstract

We announce here a number of results concerning representation theory of the algebra $R=k<x,y>/ (xy-yx-y^2)$, known as Jordan plane (or Jordan algebra). We consider the question on 'classification' of finite-dimensional modules over the Jordan algebra. Complete description of irreducible components of the representation variety $mod (R,n)$, which we call a Jordan variety' is given for any dimension $n$. It is obtained on the basis of the stratification of this variety related to the Jordan normal form of $Y$. Any irreducible component of the representation variety contains only one stratum related to a certain partition of $n$ and is the closure of this stratum. The number of irreducible components therefore is equal to the number of partitions of $n$. As a preparation for the above result we describe the complete set of pairwise non-isomorphic irreducible modules $S_{a}$ over the Jordan algebra, and the rule how they could be glued to indecomposables. Namely, we show that ${\rm Ext}^1(S_{a},S_{b})=0$, if $a \neq b $. We study then properties of the image algebras in the endomorphism ring. Particularly, images of representations from the most important stratum, corresponding to the full Jordan block $Y$. This stratum turns out to be the only building block for the analogue of the Krull-Remark-Schmidt decomposition theorem on the level of irreducible components. Along this line we establish an analogue of the Gerstenhaber--Taussky--Motzkin theorem on the dimension of algebras generated by two commuting matrices. Another fact concerns with the tame-wild question for those image algebras. We show that all image algebras of $n$-dimensional representations are tame for $n \leq 4$ and wild for $n \geq 5$.