Interpretation of the cohomology class Jac(A) in H^1(Aut(A), A^×)

Determine the mathematical meaning and interpretive content of the class Jac(A) in the non-abelian group cohomology H^1(Aut(A), A^×) that is attached to a Frobenius algebra A via the Nakayama Jacobian 1‑cocycle jac^~_σ: Aut(A) → A^×. Develop methods to compute Jac(A) in non‑trivial examples and identify what structural or representation-theoretic information about A the class encodes.

Background

The paper defines, for a Frobenius algebra A with Nakayama automorphism σ, a 1‑cocycle jac^~_σ: Aut(A) → A^× whose cohomology class Jac(A) ∈ H^1(Aut(A), A^×) is independent of the choice of Frobenius form. This class captures information derived from the Nakayama automorphism through Jacobians of automorphisms, and acts as an invariant of A.

While several structural properties are established (e.g., restrictions to Inn(A), relations to symmetry, and potential connections to Out(A) and Pic(A)), the authors explicitly note that the meaning and interpretational significance of Jac(A) are unknown, and that non-abelian cohomology H^1(Aut(A), A^×) is hard to compute in non-trivial examples. This motivates a precise characterization and workable computational framework for Jac(A).