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TauCeti.RepresentationTheory.Quiver.Zigzag.Dimension

Dimension of the componentwise zigzag algebra #

The public zigzag algebra of a finite simple graph is a product over connected components. A component containing an edge uses the path-algebra quotient, whose vertex--arrow--volume basis is already available, while an isolated vertex uses the dual numbers. Both cases have dimension twice the number of vertices plus twice the number of edges.

Summing the component dimensions gives the uniform formula

dim_k Z_k(G) = 2 |V(G)| + 2 |E(G)|

for every finite simple graph, including the empty graph and graphs with isolated vertices. The proof counts vertices and oriented edges componentwise; this makes explicit why replacing the coefficient ring by dual numbers on singleton components restores the missing volume class.

Main results #

References #

See Huerfano--Khovanov, A category for the adjoint representation, Section 3, and Ehrig--Tubbenhauer, Algebraic properties of zigzag algebras, Section 2.

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instance TauCeti.instFreeZigzagComponentAlgebra (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] (C : G.ConnectedComponent) :
Module.Free k ↑(zigzagComponentAlgebra k G C)

Each component factor of a finite graph's zigzag algebra is free over the coefficient ring.

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instance TauCeti.instFiniteZigzagComponentAlgebra (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] (C : G.ConnectedComponent) :
Module.Finite k ↑(zigzagComponentAlgebra k G C)

Each component factor of a finite graph's zigzag algebra is finite over the coefficient ring.

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@[simp]
theorem TauCeti.finrank_zigzagComponentAlgebra (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [Nontrivial k] (C : G.ConnectedComponent) :
Module.finrank k ↑(zigzagComponentAlgebra k G C) = 2 * Nat.card ↥C + Nat.card C.toSimpleGraph.Dart

The component factor of a zigzag algebra has dimension twice its number of vertices plus its number of darts, equivalently twice its number of edges. On a nontrivial component this is the vertex--arrow--volume basis count; on a singleton component it is the two-dimensional dual-number factor.

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instance TauCeti.instFreeZigzagAlgebra (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] :
Module.Free k ↑(zigzagAlgebra k G)

The zigzag algebra of a finite graph is free over the coefficient ring.

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instance TauCeti.instFiniteZigzagAlgebra (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] :
Module.Finite k ↑(zigzagAlgebra k G)

The zigzag algebra of a finite graph is finite over the coefficient ring.

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@[simp]
theorem TauCeti.finrank_zigzagAlgebra (k : Type w) {V : Type u} (G : SimpleGraph V) [CommRing k] [Finite V] [Nontrivial k] [Fintype V] [DecidableRel G.Adj] :
Module.finrank k ↑(zigzagAlgebra k G) = 2 * Fintype.card V + 2 * G.edgeFinset.card

Dimension of the public zigzag algebra. For every finite simple graph, including graphs with isolated vertices, dim Z(G) = 2|V| + 2|E|. Every vertex contributes an idempotent and a volume class, while every undirected edge contributes its two orientations.

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theorem TauCeti.finrank_zigzagAlgebra_A1 (k : Type w) [CommRing k] [Nontrivial k] :
Module.finrank k ↑(zigzagAlgebra k ⊥) = 2

The public zigzag algebra of the one-vertex graph has dimension two. This is the dimension check for the A₁ convention: its unique component is the dual numbers, not the coefficient field produced by the uniform path quotient.