Classification of rank-two Frobenius algebras with ker(m) ≅ A

Classify or describe concretely all rank-two commutative Frobenius algebras A over a Dedekind domain O for which the kernel of the multiplication map m: A ⊗_O A → A is isomorphic to A as an A-module. In this setting, A is an O-algebra of the form A = O · 1 ⊕ μ · X with μ a rank-one projective O-module (an ideal of order two satisfying μ^2 = (z)), and the multiplication on μX is given by (u_1X)(u_2X) = (u_1u_2/z)(aX + b) for u_1,u_2 ∈ μ. Determine necessary and sufficient conditions on the parameters (a, b, ε(1), ε(X)) and the ideal μ under which ker(m) ≅ A holds.

Background

The paper studies two-dimensional TQFTs over Dedekind domains through rank-two commutative Frobenius algebras A that are projective but not free as O-modules. Such algebras are presented as A = O * 1 ⊕ μ * X, where μ is a rank-one projective O-module of order two in the ideal class group, with μ^2 = (z). Multiplication and trace data are parameterized by elements a ∈ z^{-1}μ, b ∈ z^{-1}O, ε(1) ∈ O, and ε(X) ∈ z^{-1}μ, subject to integrality and nondegeneracy constraints.

In the link-homology context developed in Section 4, the authors analyze the multiplication map m: A ⊗_O A → A and identify the kernel explicitly. They show that ker(m) ≅ A ⊗_O μ as O-modules and investigate when ker(m) ≅ A as A-modules—a property that would facilitate invariance under the Reidemeister I move in the integral setup. While sufficient conditions are provided in special cases (e.g., ε(X) = 0 or invertibility of certain combinations of parameters), a general classification or simple characterization of all Frobenius algebras achieving ker(m) ≅ A is not known.