Extensions of representations I'm again confronted with an exercise from Etingof's book "Introduction to representation theory" (page 30 of http://math.mit.edu/~etingof/replect.pdf) 
Problem 2.22.


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*Let $A=\mathbb{C}[x_{1},\ldots,x_{n}]$ and $V_{a},V_{b}$ be 1-dimensional representations in which the elements $x_{i}$ act by $a_{i},b_{i}\in\mathbb{C}$ respectively. Find $\operatorname{Ext}^{1}(V_{a},V_{b})$ and classify all 2-dimensional representations of $A$.

*Let $B$ be the algebra over $\mathbb{C}$ generated by elements $x_{1},\ldots,x_{n}$ with the defining relations $$x_{i}x_{j}=0 \ \forall \ 1\leq i,j\leq n$$Show that for $n>1$ the algebra $B$ has infinitely many non-isomorphic indecomposable representations.
This time I am really at point blank... Could someone give my a hint on how to start with this problem? Many thanks in advance
 A: For Problem 1, let me prove a more general fact. 
Let $k$ be a field. Suppose $A$ is an associative, but not necessarily commutative, $k$-algebra and let $Z$ be its center. Let $V_{1}$, $V_{2}$ be simple left $A$-modules (finite dimensional or countable dimensional if the algebra is over $\mathbb{C}$) over $A$. Then, $Z$ acts via scalars on $V_{1}$ and $V_{2}$ by Schur's Lemma. Suppose there exists some $z \in Z$ such that $z$ acts via different scalars $z_{1}, z_{2}$ on $V_{1}, V_{2}$. Then, there are no nontrivial extensions between $V_{1}$ and $V_{2}$. 
To see this, suppose $V$ is an extension of $V_{1}$ and $V_{2}$ such that $V_{1} \subseteq V $ and $V_{2}$ a quotient of $V$. Then, $z$ has distinct generalized eigenvalues on $V$ and is hence diagonalizable. This tells us that $V = V_{1} \oplus V_{2}$ is an eigenspace decomposition of $V$ with respect to distinct eigenvalues of $z$. But the $z_{i}$-eigenspace of $z$ is actually a submodule of $V$ because for any $a \in A$, $v \in V_{i}$,
$$z(a \cdot v) = a(z \cdot v) = z_{i} a\cdot v.$$
Hence, the decomposition of $V$ into $V_{1} \oplus V_{2}$ is a decomposition of $A$-modules. Hence, the extension is trivial.
