Homogeneous vertex idempotents of the componentwise zigzag algebra

The vertex vectors form a complete orthogonal family of idempotents. They span the degree-zero piece of the signed grading, including on isolated-vertex dual-number factors. These are the homogeneous generators of the vertex projectives.

The proofs use the existing componentwise multiplication table and homogeneous vertex–arrow–volume basis, together with Mathlib's graded projections.

theorem TauCeti.completeOrthogonalIdempotents_zigzagAlgebraBasis_inl (k : Type w) [CommRing k] {V : Type u} [Finite V] (G : SimpleGraph V) [Fintype V] : CompleteOrthogonalIdempotents fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i)

The vertex vectors of the public algebra are a complete orthogonal family of idempotents.

theorem TauCeti.zigzagAlgebraIntegerGrade_zero_eq_span_range (k : Type w) [CommRing k] {V : Type u} [Finite V] (G : SimpleGraph V) : zigzagAlgebraIntegerGrade k G 0 = Submodule.span k (Set.range fun (i : V) => (zigzagAlgebraBasis k G) (Sum.inl i))

The signed degree-zero piece of the public algebra is spanned by the vertex idempotents.