Irreducible components of the Jordan varieties

We consider the question of ’classificiation’ of finite-dimensional modules over the Jordan algebra R = k〈x, y〉/(xy − yx− y). Complete description of irreducible components of the representation variety mod(R, n) of Jordan algebra 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 contain only one stratum related to a certain partition of n and is its closure. The number of irreducible components 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 Sa over the algebra R = k〈x, y〉/(xy − yx − y), and the rule how they could be glued to indecomposables. Namely, we show that Ext1(Sa, Sb) = 0, if a 6= b. We study then properties of the image algebras in the endomorphism ring. Especially, images of representations from the most important stratum, corresponding to trivial partition (n), which is a 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 with full block Y are tame for n 6 4 and wild for n > 5. MSC: Primary: 16G30, 16G60; 16D25; Secondary: 16A24