DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Frobenius functionals and associative pairings

Definition

Let $k$ be a field and $A$ a finite-dimensional unital associative $k$ -algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms). A bilinear form $β : A \times A \to k$ is associative when $β (a b, c) = β (a, b c)$ for all $a, b, c \in A$ . It is nondegenerate when both linear maps

A \to A^{*}, a \mapsto [b \mapsto β (a, b)], b \mapsto [a \mapsto β (a, b)]

are isomorphisms. A linear functional $λ : A \to k$ is a Frobenius functional when $β_{λ} (a, b) : = λ (a b)$ is nondegenerate. Associativity of $β_{λ}$ follows from associativity of multiplication. The algebra $A$ is Frobenius when it has at least one Frobenius functional.

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Paolo Saracco, Notes on Hopf Modules, Integrals and Frobenius Hopf Algebras (course notes, 2019) (standard reference, not scraped)
Erasmus Mini-Course on Hopf Algebras (lecture notes) (standard reference, not scraped)