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TauCeti.RepresentationTheory.Quiver.Kronecker.PathAlgebra

The path algebra of the generalized Kronecker quiver #

The generalized Kronecker quiver on n arrows has n + 2 paths: the two trivial paths and the arrows themselves. Over a semiring satisfying the strong rank condition, its path algebra has finite rank n + 2; for the Kronecker quiver • ⇉ • itself this is 4. The path classification and count are developed in TauCeti.RepresentationTheory.Quiver.Kronecker.Basic. Finite-dimensionality needs nothing specific to this quiver: it is acyclic, so TauCeti.finiteDimensional_pathAlgebra_of_isAcyclic applies to it as it stands, via TauCeti.Quiver.Kronecker.isAcyclic.

Main results #

References #

Derksen--Weyman, An Introduction to Quiver Representations, and Assem--Simson--Skowroński, Elements of the Representation Theory of Associative Algebras I, Ch. II.

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theorem TauCeti.Quiver.Kronecker.finrank_pathAlgebra {A : Type v} (k : Type w) [Semiring k] [StrongRankCondition k] [Fintype A] :
Module.finrank k (pathAlgebra k (Kronecker A)) = Fintype.card A + 2

The path algebra of the generalized Kronecker quiver on n arrows has finite rank n + 2 over a semiring satisfying the strong rank condition. For • ⇉ • this is 4.

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theorem TauCeti.Quiver.Kronecker.finrank_pathAlgebra_eq_four {A : Type v} [Fintype A] (h : Fintype.card A = 2) (k : Type w) [Semiring k] [StrongRankCondition k] :
Module.finrank k (pathAlgebra k (Kronecker A)) = 4

The path algebra of the Kronecker quiver has finite rank four: two trivial paths and two arrows, over a semiring satisfying the strong rank condition.

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theorem TauCeti.Quiver.Kronecker.finrank_pathAlgebra_eq_three {A : Type v} [Unique A] (k : Type w) [Semiring k] [StrongRankCondition k] :
Module.finrank k (pathAlgebra k (Kronecker A)) = 3

The path algebra of the A₂ quiver has finite rank three: the two trivial paths and the arrow, over a semiring satisfying the strong rank condition.