The Lumenary · First-year graduate

Quivers and path algebras

Name: ______________________

Date: ______________________

You will learn to

Compute the path algebra $kQ$ of a small quiver, with an explicit convention for multiplying paths, and show that for a finite quiver it is finite-dimensional exactly when $Q$ has no oriented cycles
Identify $kQ$ for basic examples — $k[x]$ , free algebras, upper triangular matrices — and describe its idempotents, simple and projective modules
Prove that representations of a finite quiver $Q$ correspond to $kQ$ -modules, giving an equivalence of categories
Classify the indecomposable representations of the linearly oriented quiver $A_n$ as interval modules, and compute the multiplicities from ranks

1.
The quiver $Q$ has vertices $1, 2, 3$ and arrows $\alpha : 1 \to 2$ , $\beta : 2 \to 3$ and $\gamma : 1 \to 3$ . What is $\dim_k kQ$ ?
2.
For which of these finite quivers is the path algebra $kQ$ finite-dimensional? (Choose all that apply.)
- A. $1 \to 2 \to 3 \to 4$
- B. $1 \to 2 \leftarrow 3$
- C.The Kronecker quiver $1 \rightrightarrows 2$
- D.One vertex with one loop
- E. $1 \rightleftarrows 2$ (one arrow each way)
- F. $1 \to 2 \to 3 \to 1$
3.
In $kQ$ for $Q : 1 \xrightarrow{\alpha} 2 \xrightarrow{\beta} 3$ , with the lesson's convention ( $pq$ = first $p$ , then $q$ ), which products are non-zero? (Choose all that apply.)
- A. $\alpha\beta$
- B. $\beta\alpha$
- C. $e_1 \alpha$
- D. $\alpha e_1$
- E. $\alpha e_2$
- F. $e_3 \beta$
4.
How many indecomposable representations, up to isomorphism, does the linearly oriented quiver $1 \to 2 \to \cdots \to 6$ have?
5.
A representation $V$ of $1 \to 2 \to 3$ has dimension vector $(2, 3, 2)$ , and the composite maps have ranks $r_{12} = 2$ , $r_{23} = 2$ , $r_{13} = 1$ . What is its decomposition into interval modules?
- A. $[1,2] \oplus [2,3] \oplus [1,3]$
- B. $[1,3]^2 \oplus [2,2]$
- C. $[1,1]^2 \oplus [2,2]^3 \oplus [3,3]^2$
- D. $[1,3] \oplus [1,1] \oplus [2,3] \oplus [2,2]$
6.
In the proof that representations of $Q$ are the same as $kQ$ -modules, where is it used that $Q$ has finitely many vertices?
- A.To have an identity $1 = \sum_a e_a$ , so that every module decomposes as $M = \bigoplus_a M e_a$
- B.To make $kQ$ finite-dimensional
- C.To prove that multiplication of paths is associative
- D.To have only finitely many arrows between two vertices
7.
Which Lean statement expresses the half of Theorem 2 proved in cycle_infinite (for a vertex a of a quiver V)?
- A.∀ c : Quiver.Path a a, 0 < c.length → Infinite (Quiver.Path a a)
- B.∀ p : Quiver.Path a a, 0 < p.length
- C.∀ c : Quiver.Path a a, 0 < c.length → Finite (Quiver.Path a a)
- D.Infinite (Quiver.Path a a) → ∃ e : a ⟶ a, True
8.
For an acyclic quiver with vertices $0, \dots, n-1$ and a list of arrows (i, j) (from $i$ to $j$ , repeats allowed), write projective_dims(n, arrows, a) returning the dimension vector of the projective representation $P(a)$ : entry $b$ is the number of paths from $a$ to $b$ (including the trivial path when $b = a$ ).
Computational — Practice

Answer key — Quivers and path algebras

1.
Paths: $e_1, e_2, e_3$ (length $0$ ), $\alpha, \beta, \gamma$ (length $1$ ) and $\alpha\beta$ (length $2$ , from $1$ to $3$ ). So $\dim kQ = 7$ . Note that $\gamma$ and $\alpha\beta$ are different basis elements although both run from $1$ to $3$ .
2.
Finite-dimensional: $A_4$ linear ( $10$ ), the zigzag ( $5$ ) and the Kronecker quiver ( $4$ ) — all acyclic. Infinite-dimensional: the loop, the $2$ -cycle and the $3$ -cycle, each containing an oriented cycle $c$ with infinitely many powers $c^n$ .
3.
Non-zero: $\alpha\beta$ (the path of length $2$ ), $e_1\alpha = \alpha$ and $\alpha e_2 = \alpha$ . Zero: $\beta\alpha$ , $\alpha e_1$ , $e_3\beta$ . With the opposite (function-composition) convention the roles of left and right would be exchanged: there $\beta\alpha$ would be the non-zero product.
4.
The indecomposables are the interval modules $[a, b]$ , $1 \le a \le b \le 6$ , so there are $\binom{6}{2} + 6 = 21$ of them — the number of positive roots of the root system $A_6$ , as Gabriel's theorem predicts.
5.
$m_{13} = r_{13} = 1$ ; $m_{12} = r_{12} - r_{13} = 1$ ; $m_{23} = r_{23} - r_{13} = 1$ ; $m_{11} = r_{11} - r_{12} = 0$ ; $m_{33} = r_{33} - r_{23} = 0$ ; $m_{22} = r_{22} - r_{12} - r_{23} + r_{13} = 3 - 2 - 2 + 1 = 0$ . So $V \cong [1,2] \oplus [2,3] \oplus [1,3]$ ; check: the dimension vector is $(2, 3, 2)$ .
6.
The functor from modules to representations sets $V_a = Me_a$ . To recover $M$ as $\bigoplus_a Me_a$ we use that the $e_a$ are orthogonal idempotents summing to $1$ , which requires $Q_0$ to be finite (a module over a unital algebra is unital: $m = m \cdot 1 = \sum_a m e_a$ ). For infinite $Q_0$ the theorem holds after restricting to modules with $M = \bigoplus_a Me_a$ .
7.
cycle_infinite takes c : Quiver.Path a a with 0 < c.length and concludes Infinite (Quiver.Path a a), via the injective map n ↦ cⁿ (lengths n · c.length are distinct). Together with path_length_le it gives Theorem 2.
8.
```
def projective_dims(n, arrows, a):
    dims = [0] * n
    frontier = [a]               # endpoints of paths of the current length
    while frontier:
        for b in frontier:
            dims[b] += 1
        frontier = [j for b in frontier for (i, j) in arrows if i == b]
    return dims
```
Each round extends every path of length $\ell$ by every arrow leaving its endpoint, so frontier lists the endpoints of all paths of length $\ell$ (with multiplicity); acyclicity makes it empty after fewer than $n$ rounds. For the Kronecker quiver $P(0) = (1, 2)$ : the trivial path and the two arrows. For a sink $b$ , $P(b)$ is the simple $S(b)$ .