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.

The finite dual of an associative algebra

Definition

Let $k$ be a field (Field) and let $A$ be a unital associative $k$ -algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms). Write $A^{*} = {Hom}_{k} (A, k)$ for its algebraic dual (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ), and use the $k$ -tensor product (Scalars, tensor powers, the empty tensor, opposite algebras and finite sums). Its finite dual (also called its restricted dual) is the subset

A^{\circ} : = {f \in A^{*} : there is a two-sided ideal I ⊴ A with \dim_{k} (A / I) < \infty and f (I) = 0} .

Equivalently, $f \in A^{\circ}$ exactly when it factors through a finite-dimensional quotient algebra $A / I$ . The theorem The finite dual is a coalgebra and descends through bialgebras and Hopf algebras ↗ proves that this subset is a vector subspace and that the transposed multiplication $m_{A}^{*} (f)$ , defined by $m_{A}^{*} (f) (a \otimes b) = f (a b)$ , corestricts to $A^{\circ} \otimes A^{\circ}$ .

Depends on

Used by

Dependency tree · two levels

46 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

Kalle Kytölä, Introduction to Hopf algebras and representations, §3.9 (standard reference, not scraped)