DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Group-like elements and primitives relative to a group-like element

Definition

Let $C$ be a coalgebra over $k$ . An element $g \in C$ is group-like when

Δ (g) = g \otimes g, ε (g) = 1.

The counit condition excludes the zero vector and fixes the normalization.

Given a specified group-like element $g$ , an element $x \in C$ is primitive relative to $g$ when

Δ (x) = x \otimes g + g \otimes x, ε (x) = 0.

This is the $(g, g)$ -skew-primitive condition. In a bialgebra with distinguished group-like unit $1$ , the usual primitive condition is the special case $g = 1$ ; an arbitrary coalgebra has no distinguished unit unless one is supplied.

Depends on

Used by

Dependency tree · two levels

13 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

William Chin, A Brief Introduction to Coalgebra Representation Theory (standard reference, not scraped)
Benedikt Stufler, Hopf Algebras lecture notes, University of Zurich course, 2018 (standard reference, not scraped)