[{"publisher":"Springer Nature","publication_status":"epub_ahead","title":"The PBW theorem for affine Yangians","doi":"10.1007/s00031-020-09572-6","department":[{"_id":"TaHa"}],"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"}],"publication":"Transformation Groups","type":"journal_article","main_file_link":[{"url":"https://arxiv.org/abs/1804.04375","open_access":"1"}],"date_published":"2020-05-22T00:00:00Z","project":[{"_id":"25E549F4-B435-11E9-9278-68D0E5697425","name":"Arithmetic and physics of Higgs moduli spaces","call_identifier":"FP7","grant_number":"320593"}],"author":[{"full_name":"Yang, Yaping","first_name":"Yaping","last_name":"Yang","id":"360D8648-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Zhao, Gufang","first_name":"Gufang","last_name":"Zhao","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87"}],"article_type":"original","status":"public","date_created":"2020-06-07T22:00:55Z","article_processing_charge":"No","day":"22","year":"2020","publication_identifier":{"eissn":["1531586X"],"issn":["10834362"]},"citation":{"ista":"Yang Y, Zhao G. 2020. The PBW theorem for affine Yangians. Transformation Groups.","mla":"Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” *Transformation Groups*, Springer Nature, 2020, doi:10.1007/s00031-020-09572-6.","ama":"Yang Y, Zhao G. The PBW theorem for affine Yangians. *Transformation Groups*. 2020. doi:10.1007/s00031-020-09572-6","short":"Y. Yang, G. Zhao, Transformation Groups (2020).","apa":"Yang, Y., & Zhao, G. (2020). The PBW theorem for affine Yangians. *Transformation Groups*. https://doi.org/10.1007/s00031-020-09572-6","chicago":"Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” *Transformation Groups*, 2020. https://doi.org/10.1007/s00031-020-09572-6.","ieee":"Y. Yang and G. Zhao, “The PBW theorem for affine Yangians,” *Transformation Groups*, 2020."},"oa_version":"Preprint","month":"05","language":[{"iso":"eng"}],"external_id":{"arxiv":["1804.04375"]},"quality_controlled":"1","_id":"7940","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2020-06-08T06:54:19Z"},{"author":[{"last_name":"Rapcak","full_name":"Rapcak, Miroslav","first_name":"Miroslav"},{"first_name":"Yan","full_name":"Soibelman, Yan","last_name":"Soibelman"},{"full_name":"Yang, Yaping","first_name":"Yaping","last_name":"Yang"},{"full_name":"Zhao, Gufang","first_name":"Gufang","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","last_name":"Zhao"}],"date_published":"2019-09-21T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1810.10402"}],"project":[{"_id":"25E549F4-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Arithmetic and physics of Higgs moduli spaces","grant_number":"320593"}],"publication":"Communications in Mathematical Physics","department":[{"_id":"TaHa"}],"type":"journal_article","abstract":[{"lang":"eng","text":"We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr1,r2,r3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi–Yau categories, including those associated with toric Calabi–Yau 3-folds."}],"doi":"10.1007/s00220-019-03575-5","title":"Cohomological Hall algebras, vertex algebras and instantons","publication_status":"epub_ahead","publisher":"Springer Nature","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"7004","oa":1,"date_updated":"2020-01-16T12:38:16Z","language":[{"iso":"eng"}],"month":"09","quality_controlled":"1","external_id":{"arxiv":["1810.10402"]},"oa_version":"Preprint","publication_identifier":{"issn":["0010-3616"],"eissn":["1432-0916"]},"citation":{"chicago":"Rapcak, Miroslav, Yan Soibelman, Yaping Yang, and Gufang Zhao. “Cohomological Hall Algebras, Vertex Algebras and Instantons.” *Communications in Mathematical Physics*, 2019. https://doi.org/10.1007/s00220-019-03575-5.","ieee":"M. Rapcak, Y. Soibelman, Y. Yang, and G. Zhao, “Cohomological Hall algebras, vertex algebras and instantons,” *Communications in Mathematical Physics*, 2019.","ama":"Rapcak M, Soibelman Y, Yang Y, Zhao G. Cohomological Hall algebras, vertex algebras and instantons. *Communications in Mathematical Physics*. 2019. doi:10.1007/s00220-019-03575-5","mla":"Rapcak, Miroslav, et al. “Cohomological Hall Algebras, Vertex Algebras and Instantons.” *Communications in Mathematical Physics*, Springer Nature, 2019, doi:10.1007/s00220-019-03575-5.","ista":"Rapcak M, Soibelman Y, Yang Y, Zhao G. 2019. Cohomological Hall algebras, vertex algebras and instantons. Communications in Mathematical Physics.","short":"M. Rapcak, Y. Soibelman, Y. Yang, G. Zhao, Communications in Mathematical Physics (2019).","apa":"Rapcak, M., Soibelman, Y., Yang, Y., & Zhao, G. (2019). Cohomological Hall algebras, vertex algebras and instantons. *Communications in Mathematical Physics*. https://doi.org/10.1007/s00220-019-03575-5"},"date_created":"2019-11-12T14:01:27Z","status":"public","article_type":"original","year":"2019","day":"21"},{"issue":"5","citation":{"ieee":"Y. Yang and G. Zhao, “The cohomological Hall algebra of a preprojective algebra,” *Proceedings of the London Mathematical Society*, vol. 116, no. 5, pp. 1029–1074, 2018.","chicago":"Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective Algebra.” *Proceedings of the London Mathematical Society* 116, no. 5 (2018): 1029–74. https://doi.org/10.1112/plms.12111.","apa":"Yang, Y., & Zhao, G. (2018). The cohomological Hall algebra of a preprojective algebra. *Proceedings of the London Mathematical Society*, *116*(5), 1029–1074. https://doi.org/10.1112/plms.12111","ama":"Yang Y, Zhao G. The cohomological Hall algebra of a preprojective algebra. *Proceedings of the London Mathematical Society*. 2018;116(5):1029-1074. doi:10.1112/plms.12111","ista":"Yang Y, Zhao G. 2018. The cohomological Hall algebra of a preprojective algebra. Proceedings of the London Mathematical Society. 116(5), 1029–1074.","mla":"Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective Algebra.” *Proceedings of the London Mathematical Society*, vol. 116, no. 5, Oxford University Press (OUP), 2018, pp. 1029–74, doi:10.1112/plms.12111.","short":"Y. Yang, G. Zhao, Proceedings of the London Mathematical Society 116 (2018) 1029–1074."},"oa_version":"Preprint","publication_identifier":{"issn":["0024-6115"]},"page":"1029-1074","date_created":"2019-02-14T13:14:22Z","volume":116,"status":"public","intvolume":" 116","year":"2018","day":"01","_id":"5999","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"date_updated":"2019-08-02T12:39:07Z","language":[{"iso":"eng"}],"month":"05","quality_controlled":"1","external_id":{"arxiv":["1407.7994"]},"type":"journal_article","publication":"Proceedings of the London Mathematical Society","department":[{"_id":"TaHa"}],"abstract":[{"text":"We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When A is the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups.\r\nWe construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. As applications, we obtain a shuffle description of the Yangian. ","lang":"eng"}],"doi":"10.1112/plms.12111","publication_status":"published","title":"The cohomological Hall algebra of a preprojective algebra","publisher":"Oxford University Press (OUP)","author":[{"last_name":"Yang","first_name":"Yaping","full_name":"Yang, Yaping"},{"full_name":"Zhao, Gufang","first_name":"Gufang","last_name":"Zhao","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87"}],"date_published":"2018-05-01T00:00:00Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1407.7994"}]}]