Quiver representations

How many indecomposable representations (up to isomorphism) does the quiver $1 \to 2 \leftarrow 3 \to 4$ have over $\mathbb{C}$?

For $D_4$ with the centre written second, $q(a, c, b, d) = a^2 + c^2 + b^2 + d^2 - ac - bc - dc$. Which vectors are dimension vectors of indecomposable representations? (Choose all that apply.)

For the Kronecker quiver $1 \rightrightarrows 2$ (two arrows from $1$ to $2$), compute $\dim \operatorname{Ext}^1(S(1), S(2))$ using the Euler form.

The extended Dynkin graph $\tilde D_4$ is a star with centre $c$ and four leaves. Compute its Tits form at $\delta = (1, 1, 1, 1, 2)$ (leaves first, centre last).

Two connected quivers $Q$ and $Q'$ have the same underlying graph but different orientations. Which statement is correct?

In Tits' proof that finite type forces $q(d) \ge 1$, where does the “$-1$” in $\dim(\text{orbit}) \le \sum_i d_i^2 - 1$ come from?

Which statements about the Kronecker quiver $1 \rightrightarrows 2$ over $\mathbb{C}$ are true? (Choose all that apply.)

Which Lean statement correctly formalises “the positive roots of $D_4$ are exactly the twelve vectors in D4_roots”? (titsD4 takes four integers, the centre third.)

Write positive_roots(n, edges) returning the set of positive roots (as tuples) of a Dynkin graph with vertices 0, …, n-1 and edge list edges, by closing the simple roots under simple reflections $s_i(x)_i = \sum_{j \sim i} x_j - x_i$ and keeping only non-negative, non-zero vectors.