{"paper":{"title":"Weak Multiplier Hopf Algebras. Preliminaries, motivation and basic examples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RA","authors_text":"Alfons Van Daele, Shuanhong Wang","submitted_at":"2012-10-15T09:38:09Z","abstract_excerpt":"Let $G$ be a {\\it finite group}. Consider the algebra $A$ of all complex functions on G (with pointwise product). Define a coproduct $\\Delta$ on A by $\\Delta(f)(p,q)=f(pq)$ where $f\\in A$ and $p,q\\in G$. Then $(A,\\Delta)$ is a Hopf algebra. If $G$ is only a {\\it groupoid}, so that the product of two elements is not always defined, one still can consider $A$ and define $\\Delta(f)(p,q)$ as above when $pq$ is defined. If we let $\\Delta(f)(p,q)=0$ otherwise, we still get a coproduct on $A$, but $\\Delta(1)$ will no longer be the identity in $A\\ot A$. The pair $(A,\\Delta)$ is not a Hopf algebra but "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3954","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}