{"paper":{"title":"Weak multiplier Hopf algebras III. Integrals and duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Alfons Van Daele, Shuanhong Wang","submitted_at":"2017-01-18T05:28:44Z","abstract_excerpt":"Let $(A,\\Delta)$ be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\\Delta$ on $A$, satisfying certain properties. The main difference with multiplier Hopf algebras is that now, the canonical maps $T_1$ and $T_2$ on $A\\otimes A$, defined by $$T_1(a\\otimes b)=\\Delta(a)(1\\otimes b) \\qquad\\quad\\text{and}\\qquad\\quad T_2(c\\otimes a)=(c\\otimes 1)\\Delta(a),$$ are no longer assumed to be bijective. Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from $A$ to itself.\n  In this pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04951","kind":"arxiv","version":4},"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"}