I am studying the representations of the following algebra: $$A=\Bbb C\langle x,y,z\rangle/ \langle zx=0,yz=0, z=xy-yx\rangle$$ Do you know how to attack the problem of classifying all its representations?
Here I recollect the facts that have been reached until now (in comments or in my attempts):
- The center of $A$ is $\Bbb C\langle yx\rangle $ (and note that $ (yx)^n =y^n x^n$ by induction)
- Every monomial is equivalent to one of the form $y^ax^by^c$ with $a \le b$, and these monomials are distinct.
- There is a surjective map $A \to R:=\Bbb C[x,y]$, so we have all the representation of the 2-variable ring. Call them rep. of commutative type. Finite dimensional commutative representations can be expressed as a sum of indecomposable representations, and finite dim indecomposable representations are of the form $x \to \lambda I + N, y \to \mu I + N$ where $N$ has 1 on the superdiagonal and 0 elsewhere. This is just Jordan decomposition (+ the observation that if two matrices commute, they have a common Jordan basis; furthermore, they must have the same jordan form to commute). EDIT: This seems to be false. Infact, if $J_n$ is a nilpotent jordan block, $J_n$ and $J_n^2$ commute but they do not have a common Jordan basis. Any hints?
- There is a surjective map $A \to B:= \Bbb C \langle x,y \rangle / \langle yx =0 \rangle $. In this algebra every monomial has the form $x^a y^b$. Using this, one can verify that the derived series for $B$ is $DB= \Bbb C \langle x^a y^b \rangle _{a,b \ge 1}$, and $D^2B = 0$. Furthermore, it holds $B/DB \simeq \Bbb C[x,y]/(xy)$ and $DB \simeq \Bbb C[x_n]/ (x_n x_m)$.
- Every irreducible (but not every indecomposable) representation factors through $R$ or $B$. Infact, if $z=0$, then it factors through $R$. If $z \neq 0$, then $yz=0$ implies that $y$ is not invertible. By Schur Lemma, $yx$ in the center implies that $yx= \lambda I$. But $yx$ is not invertible, thus $yx=0$, i.e. the representation factors through $B$.
- If $x$ or $y$ are mapped to invertible matrices, then the rep. is of commutative type ($z=0$).
Thank you!