Path Algebras

A quiver is a directed multigraph — vertices, and arrows between them, with loops and parallel arrows allowed. Its path algebra has a basis of all directed paths, one trivial path per vertex included, and the product is concatenation:

Like the incidence algebra, most products are zero. Unlike every other family here, this one need not be finite-dimensional.

One loop and the algebra is infinite

is finite-dimensional exactly when is acyclic. A single loop gives the paths — the Jordan quiver, whose path algebra is the polynomial ring . There is no finite basis and no dimension, so Basis and AlgebraDimension have nothing to return and leave the call standing rather than enumerating forever.

#Acyclic or not

The linear quiver is acyclic and has a dimension; the Jordan quiver has a loop, so its dimension does not exist and the call stays put.

source

<notatio-cell value="QuiverIsAcyclic(LinearQuiver(4))" />
<notatio-cell value="QuiverIsAcyclic(JordanQuiver)" />
<notatio-cell value="AlgebraDimension(PathAlgebra(LinearQuiver(4)))" />
<notatio-cell value="AlgebraDimension(PathAlgebra(JordanQuiver))" />

That distinction is the whole reason this family is worth having next to the others: every previous one was finite by construction, and this is the first place where "there is no answer" is a structural fact about the input rather than a missing feature.

Aₙ's path algebra is a chain's incidence algebra

The linear quiver has exactly one path from to when , and none otherwise. So its paths are the intervals of a chain, and is the incidence algebra of Chain(n) — same dimension , same product, two libraries describing one object from different directions.

#The same algebra, twice

Both are C(7,2) = 21.

source

<notatio-cell value="AlgebraDimension(PathAlgebra(LinearQuiver(6)))" />
<notatio-cell value="AlgebraDimension(IncidenceAlgebra(Chain(6)))" />

The path count is verified a second way in the tests, against powers of the adjacency matrix: paths of length are the entries of , and an acyclic quiver on vertices has none longer than . Enumeration and linear algebra agree.

Composition, and zero

#Concatenate or annihilate

1→2 then 2→3 composes. The other order does not meet, so it is zero — not an error.

source

<notatio-cell value="QuiverCompose(LinearQuiver(4), QuiverPath(1, [0]), QuiverPath(2, [1]))" />
<notatio-cell value="QuiverCompose(LinearQuiver(4), QuiverPath(2, [1]), QuiverPath(1, [0]))" />

A path is written QuiverPath(start, [arrow indices]), so the trivial path at is QuiverPath(v, []). Those trivial paths are the local identities:

and together they sum to the identity of the algebra.

Parallel arrows are distinct

Quiver(n, [[from,to],…]) builds one explicitly, and arrows are indexed by position — so two arrows with the same endpoints are two different basis elements. The Kronecker quiver is the smallest example: two vertices, two parallel arrows, four paths.

#The Kronecker quiver

Four paths: two trivial, two arrows. The two arrows do not compose with each other, so their product is zero.

source

<notatio-cell value="AlgebraDimension(PathAlgebra(Quiver(2, [[1, 2], [1, 2]])))" />
<notatio-cell value="Basis(PathAlgebra(KroneckerQuiver))" />

Things worth knowing

Named quivers are recognised, not declared. JordanQuiver and KroneckerQuiver are bare symbols. Declaring them as nullary functions gave them the type () -> value, which then failed PathAlgebra's value parameter — an undeclared symbol types as unknown and passes. Same lesson as the named hypercomplex algebras.

Composition takes its quiver. A path does not know which quiver it belongs to, so QuiverCompose(quiver, p, q) names it rather than riding the shared ordered product.

Not built yet. Representations of a quiver — the actual subject, and where Gabriel's theorem lives: a connected quiver has finitely many indecomposable representations exactly when its underlying graph is a Dynkin diagram of type , or . Also quotients by an admissible ideal, which is how a cyclic quiver is made finite-dimensional in practice.