{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:31:52Z","timestamp":1772253112388,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,30]],"date-time":"2020-11-30T00:00:00Z","timestamp":1606694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871144"],"award-info":[{"award-number":["11871144"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let (A,\u0394) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct \u0394:A\u27f6M(A\u2297A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps \u03b5s and \u03b5t are studied, and their symmetric pair of images, the source algebra and the target algebra \u03b5s(A) and \u03b5t(A), are also investigated. We show that the canonical idempotent E (which is eventually \u0394(1)) belongs to the multiplier algebra M(B\u2297C), where (B=\u03b5s(A), C=\u03b5t(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,\u0394), it is possible to make C\u2297B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the \u2019Hopf algebra part\u2019 of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E\u2208M(B\u2297C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known \u2019quantization\u2019 of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).<\/jats:p>","DOI":"10.3390\/sym12121975","type":"journal-article","created":{"date-parts":[[2020,11,29]],"date-time":"2020-11-29T21:55:31Z","timestamp":1606686931000},"page":"1975","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Weak Multiplier Hopf Algebras II: Source and Target Algebras"],"prefix":"10.3390","volume":"12","author":[{"given":"Alfons Van","family":"Daele","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium"}]},{"given":"Shuanhong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Southeast University, Nanjing 210096, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1090\/S0002-9947-1994-1220906-5","article-title":"Multiplier Hopf algebras","volume":"342","year":"1994","journal-title":"Trans. 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