Let $K$ be a field. For $n\in \mathbb{N}$ consider the equioriented $A_n$-quiver. It is easy to see that the path algebra $K[A_n]$ is isomorphic as a $K$-algebra to the algebra of upper triangular $n\times n$-matrices with entries in $K$. Do the path algebras of the $D_n$-quivers and $E_n$-quivers (oriented in some way) have a similiarly easy description?
2026-03-25 03:01:16.1774407676
Path algebra on quivers of type ADE
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If the quiver is a tree, then there is at most one path between any two vertices, and so we can always regard the path algebra as a subalgebra of a matrix algebra, with rows/columns indexed by the vertices, and such that the unique path from $i$ to $j$, if it exists, is sent to the elementary matrix $E_{ji}$.