The q-Euler form of a path algebra and its graded Cartan matrix

Let Q be a finite quiver whose path algebra kQ is finite-dimensional, graded by path length. Two computations of the same lattice meet here. The categorical q-Euler form on G₀^gr(mod kQ) has matrix 1 - q⁻¹ A in the basis of graded vertex simples, where the (i, j) entry of A counts arrows i ⟶ j (TauCeti.PathAlgebra.gradedExtEulerForm_basis). The graded Cartan matrix C, computed by counting paths, satisfies (C.map invert)ᵀ (1 - q⁻¹ A) = 1 (TauCeti.PathAlgebra.gradedCartanMatrix_map_invert_transpose_mul_one_sub_T_neg_smul_eq_one). Together they identify χ_q(Sᵢ, Sⱼ) with the (i, j) entry of ((C.map invert)ᵀ)⁻¹, the path-algebra case of TauCeti.gradedExtEuler_eq_inv_gradedCartanMatrix_map_invert_transpose, here obtained from the explicit formulas on both sides.

Main results

• TauCeti.PathAlgebra.gradedExtEulerForm_gradedVertexSimpleClassBasis_eq_inv: the q-Euler matrix of the graded vertex simples is the inverse of the conjugate-transposed graded Cartan matrix.

References

• I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras I, Chapter III, Proposition 3.13, for the ungraded relation between the Cartan matrix and the Euler form.
• Z. Dancso and A. Licata, "Koszul algebras and flow lattices", Sections 2.2 and 3.1--3.2, for the q-antilinear convention and the conjugate-transpose relation.

theorem TauCeti.PathAlgebra.gradedExtEulerForm_gradedVertexSimpleClassBasis_eq_inv (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [DecidableEq Q] [Module.Finite k (pathAlgebra k Q)] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] (i j : Q) : ((gradedExtEulerForm k Q) ((gradedVertexSimpleClassBasis k Q) i)) ((gradedVertexSimpleClassBasis k Q) j) = ((gradedCartanMatrix (integerGrade k Q) (gradedVertexProjectiveClassBasis k Q) (gradedVertexSimpleClassBasis k Q)).map ⇑LaurentPolynomial.invert).transpose⁻¹ i j

The q-Euler matrix of a path algebra is the inverse of its conjugate-transposed graded Cartan matrix: χ_q(Sᵢ, Sⱼ) is the (i, j) entry of ((C.map invert)ᵀ)⁻¹, where C is the graded Cartan matrix in the bases of graded vertex projectives and graded vertex simples.