[{"doi":"10.1007/s00031-020-09572-6","month":"12","scopus_import":"1","language":[{"iso":"eng"}],"article_type":"original","date_published":"2020-12-01T00:00:00Z","volume":25,"ec_funded":1,"project":[{"call_identifier":"FP7","name":"Arithmetic and physics of Higgs moduli spaces","grant_number":"320593","_id":"25E549F4-B435-11E9-9278-68D0E5697425"}],"title":"The PBW theorem for affine Yangians","year":"2020","date_updated":"2025-07-10T11:54:50Z","publication_identifier":{"eissn":["1531-586X"],"issn":["1083-4362"]},"type":"journal_article","arxiv":1,"abstract":[{"text":"We prove that the Yangian associated to an untwisted symmetric affine Kac–Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed in [YZ14] as an algebraic formalism of cohomological Hall algebras. As a consequence, we obtain the Poincare–Birkhoff–Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay, Regelskis, and Wendlandt [GRW18].","lang":"eng"}],"citation":{"short":"Y. Yang, G. Zhao, Transformation Groups 25 (2020) 1371–1385.","chicago":"Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” <i>Transformation Groups</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s00031-020-09572-6\">https://doi.org/10.1007/s00031-020-09572-6</a>.","ista":"Yang Y, Zhao G. 2020. The PBW theorem for affine Yangians. Transformation Groups. 25, 1371–1385.","ieee":"Y. Yang and G. Zhao, “The PBW theorem for affine Yangians,” <i>Transformation Groups</i>, vol. 25. Springer Nature, pp. 1371–1385, 2020.","mla":"Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” <i>Transformation Groups</i>, vol. 25, Springer Nature, 2020, pp. 1371–85, doi:<a href=\"https://doi.org/10.1007/s00031-020-09572-6\">10.1007/s00031-020-09572-6</a>.","ama":"Yang Y, Zhao G. The PBW theorem for affine Yangians. <i>Transformation Groups</i>. 2020;25:1371-1385. doi:<a href=\"https://doi.org/10.1007/s00031-020-09572-6\">10.1007/s00031-020-09572-6</a>","apa":"Yang, Y., &#38; Zhao, G. (2020). The PBW theorem for affine Yangians. <i>Transformation Groups</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00031-020-09572-6\">https://doi.org/10.1007/s00031-020-09572-6</a>"},"oa_version":"Preprint","author":[{"first_name":"Yaping","last_name":"Yang","full_name":"Yang, Yaping","id":"360D8648-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Zhao, Gufang","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","first_name":"Gufang","last_name":"Zhao"}],"acknowledgement":"Gufang Zhao is affiliated to IST Austria, Hausel group until July of 2018. Supported by the Advanced Grant Arithmetic and Physics of Higgs moduli spaces No. 320593 of the European Research Council.","_id":"7940","article_processing_charge":"No","isi":1,"quality_controlled":"1","external_id":{"isi":["000534874300003"],"arxiv":["1804.04375"]},"publication":"Transformation Groups","status":"public","oa":1,"publisher":"Springer Nature","department":[{"_id":"TaHa"}],"page":"1371-1385","intvolume":"        25","publication_status":"published","day":"01","date_created":"2020-06-07T22:00:55Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1804.04375"}]},{"oa_version":"Preprint","author":[{"first_name":"Yaping","last_name":"Yang","full_name":"Yang, Yaping","id":"360D8648-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Zhao","first_name":"Gufang","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","full_name":"Zhao, Gufang"}],"place":"Cham","abstract":[{"lang":"eng","text":"These lecture notes are based on Yang’s talk at the MATRIX program Geometric R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec. 18–22, 2017, and Zhao’s talk at Perimeter Institute for Theoretical Physics in January 2018. We give an introductory survey of the results in Yang and Zhao (Quiver varieties and elliptic quantum groups, 2017. arxiv1708.01418). We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve."}],"citation":{"chicago":"Yang, Yaping, and Gufang Zhao. “How to Sheafify an Elliptic Quantum Group.” In <i>2017 MATRIX Annals</i>, 2:675–91. MXBS. Cham: Springer International Publishing, 2019. <a href=\"https://doi.org/10.1007/978-3-030-04161-8_54\">https://doi.org/10.1007/978-3-030-04161-8_54</a>.","ista":"Yang Y, Zhao G. 2019.How to Sheafify an Elliptic Quantum Group. In: 2017 MATRIX Annals. MATRIX Book Series, vol. 2, 675–691.","ieee":"Y. Yang and G. Zhao, “How to Sheafify an Elliptic Quantum Group,” in <i>2017 MATRIX Annals</i>, vol. 2, Cham: Springer International Publishing, 2019, pp. 675–691.","mla":"Yang, Yaping, and Gufang Zhao. “How to Sheafify an Elliptic Quantum Group.” <i>2017 MATRIX Annals</i>, vol. 2, Springer International Publishing, 2019, pp. 675–91, doi:<a href=\"https://doi.org/10.1007/978-3-030-04161-8_54\">10.1007/978-3-030-04161-8_54</a>.","short":"Y. Yang, G. Zhao, in:, 2017 MATRIX Annals, Springer International Publishing, Cham, 2019, pp. 675–691.","apa":"Yang, Y., &#38; Zhao, G. (2019). How to Sheafify an Elliptic Quantum Group. In <i>2017 MATRIX Annals</i> (Vol. 2, pp. 675–691). Cham: Springer International Publishing. <a href=\"https://doi.org/10.1007/978-3-030-04161-8_54\">https://doi.org/10.1007/978-3-030-04161-8_54</a>","ama":"Yang Y, Zhao G. How to Sheafify an Elliptic Quantum Group. In: <i>2017 MATRIX Annals</i>. Vol 2. MXBS. Cham: Springer International Publishing; 2019:675-691. doi:<a href=\"https://doi.org/10.1007/978-3-030-04161-8_54\">10.1007/978-3-030-04161-8_54</a>"},"date_updated":"2025-09-23T11:59:52Z","publication_identifier":{"isbn":["9783030041601"],"issn":["2523-3041"],"eissn":["2523-305X"],"eisbn":["9783030041618"]},"arxiv":1,"type":"book_chapter","title":"How to Sheafify an Elliptic Quantum Group","year":"2019","volume":2,"date_published":"2019-03-25T00:00:00Z","language":[{"iso":"eng"}],"doi":"10.1007/978-3-030-04161-8_54","month":"03","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1803.06627"}],"date_created":"2025-07-10T13:31:38Z","day":"25","intvolume":"         2","publication_status":"published","oa":1,"series_title":"MXBS","publisher":"Springer International Publishing","page":"675-691","department":[{"_id":"TaHa"}],"publication":"2017 MATRIX Annals","alternative_title":["MATRIX Book Series"],"status":"public","OA_place":"repository","external_id":{"arxiv":["1803.06627"]},"quality_controlled":"1","acknowledgement":"Y.Y. would like to thank the organizers of the MATRIX program Geometric R-Matrices: from Geometry to Probability for their kind invitation, and many participants of the program for useful discussions, including Vassily Gorbounov, Andrei Okounkov, Allen Knutson, Hitoshi Konno, Paul Zinn-Justin. Proposition 1 and Sect. 3.3 are new, for which we thank Hitoshi Konno for interesting discussions and communications. These notes were written when both authors were visiting the Perimeter Institute for Theoretical Physics (PI). We are grateful to PI for the hospitality.","article_processing_charge":"No","_id":"19987","OA_type":"green"}]