Irreducible components of the Jordan varieties
We announce here a number of results concerning representation theory of the algebra R=k/ (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
... . 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 Ext^1(S_a,S_b)=0, if a ≠ 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 ≤ 4 and wild for n ≥ 5.