Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 3 / 2 / b / ii

Solution

1, \dots, r

and the sink

c = r + 1

, and write

a_{i} : i \to c

for the arrows. The only basis paths are

e_{1}, \dots, e_{r}, e_{c}, a_{1}, \dots, a_{r}

. Using standard matrix units, define

e_{i} ⟼ E_{ii}, e_{c} ⟼ E_{cc}, a_{i} ⟼ E_{c i} .

The relations

e_{c} a_{i} = a_{i} = a_{i} e_{i}

a_{i} a_{j} = 0

agree exactly with multiplication of these matrix units. The map is therefore an algebra homomorphism, and its images form a basis of the algebra

k S_{r} ≅ {(diag (λ_{1}, \dots, λ_{r}) v 0 μ) : λ_{i}, μ \in k, v \in k^{1 \times r}} .

This is an explicit isomorphism of the path algebra of the subspace quiver with a

(2 r + 1)

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