[{"status":"public","day":"16","type":"book_chapter","series_title":"TM","publication":"Representation Theory and Algebraic Geometry","page":"347-392","publisher":"Springer Nature; Birkhäuser","scopus_import":"1","edition":"1","language":[{"iso":"eng"}],"month":"06","date_published":"2022-06-16T00:00:00Z","date_created":"2023-01-16T10:06:41Z","department":[{"_id":"TaHa"}],"abstract":[{"text":"We construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” GP(Q,A). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus D⊆G2m gives a quantization GPD(Q,A). This construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program (Mirković, Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic quantum groups, preprint. arxiv1708.01418).","lang":"eng"}],"author":[{"full_name":"Mirković, Ivan","last_name":"Mirković","first_name":"Ivan"},{"first_name":"Yaping","last_name":"Yang","full_name":"Yang, Yaping"},{"first_name":"Gufang","last_name":"Zhao","full_name":"Zhao, Gufang","id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87"}],"publication_status":"published","citation":{"mla":"Mirković, Ivan, et al. “Loop Grassmannians of Quivers and Affine Quantum Groups.” <i>Representation Theory and Algebraic Geometry</i>, edited by Vladimir Baranovskky et al., 1st ed., Springer Nature; Birkhäuser, 2022, pp. 347–92, doi:<a href=\"https://doi.org/10.1007/978-3-030-82007-7_8\">10.1007/978-3-030-82007-7_8</a>.","ama":"Mirković I, Yang Y, Zhao G. Loop Grassmannians of Quivers and Affine Quantum Groups. In: Baranovskky V, Guay N, Schedler T, eds. <i>Representation Theory and Algebraic Geometry</i>. 1st ed. TM. Cham: Springer Nature; Birkhäuser; 2022:347-392. doi:<a href=\"https://doi.org/10.1007/978-3-030-82007-7_8\">10.1007/978-3-030-82007-7_8</a>","ista":"Mirković I, Yang Y, Zhao G. 2022.Loop Grassmannians of Quivers and Affine Quantum Groups. In: Representation Theory and Algebraic Geometry. Trends in Mathematics, , 347–392.","short":"I. Mirković, Y. Yang, G. Zhao, in:, V. Baranovskky, N. Guay, T. Schedler (Eds.), Representation Theory and Algebraic Geometry, 1st ed., Springer Nature; Birkhäuser, Cham, 2022, pp. 347–392.","ieee":"I. Mirković, Y. Yang, and G. Zhao, “Loop Grassmannians of Quivers and Affine Quantum Groups,” in <i>Representation Theory and Algebraic Geometry</i>, 1st ed., V. Baranovskky, N. Guay, and T. Schedler, Eds. Cham: Springer Nature; Birkhäuser, 2022, pp. 347–392.","apa":"Mirković, I., Yang, Y., &#38; Zhao, G. (2022). Loop Grassmannians of Quivers and Affine Quantum Groups. In V. Baranovskky, N. Guay, &#38; T. Schedler (Eds.), <i>Representation Theory and Algebraic Geometry</i> (1st ed., pp. 347–392). Cham: Springer Nature; Birkhäuser. <a href=\"https://doi.org/10.1007/978-3-030-82007-7_8\">https://doi.org/10.1007/978-3-030-82007-7_8</a>","chicago":"Mirković, Ivan, Yaping Yang, and Gufang Zhao. “Loop Grassmannians of Quivers and Affine Quantum Groups.” In <i>Representation Theory and Algebraic Geometry</i>, edited by Vladimir Baranovskky, Nicolas Guay, and Travis Schedler, 1st ed., 347–92. TM. Cham: Springer Nature; Birkhäuser, 2022. <a href=\"https://doi.org/10.1007/978-3-030-82007-7_8\">https://doi.org/10.1007/978-3-030-82007-7_8</a>."},"_id":"12303","publication_identifier":{"eissn":["2297-024X"],"eisbn":["9783030820077"],"isbn":["9783030820060"],"issn":["2297-0215"]},"acknowledgement":"I.M. thanks Zhijie Dong for long-term discussions on the material that entered this work. We thank Misha Finkelberg for pointing out errors in earlier versions. His advice and his insistence have led to a much better paper. A part of the writing was done at the conference at IST (Vienna) attended by all coauthors. We therefore thank the organizers of the conference and the support of ERC Advanced Grant Arithmetic and Physics of Higgs moduli spaces No. 320593. The work of I.M. was partially supported by NSF grants. The work of Y.Y. was partially supported by the Australian Research Council (ARC) via the award DE190101231. The work of G.Z. was partially supported by ARC via the award DE190101222.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","oa_version":"Preprint","project":[{"grant_number":"320593","call_identifier":"FP7","_id":"25E549F4-B435-11E9-9278-68D0E5697425","name":"Arithmetic and physics of Higgs moduli spaces"}],"editor":[{"first_name":"Vladimir","full_name":"Baranovskky, Vladimir","last_name":"Baranovskky"},{"first_name":"Nicolas","full_name":"Guay, Nicolas","last_name":"Guay"},{"full_name":"Schedler, Travis","last_name":"Schedler","first_name":"Travis"}],"arxiv":1,"date_updated":"2023-01-27T07:07:31Z","oa":1,"article_processing_charge":"No","external_id":{"arxiv":["1810.10095"]},"title":"Loop Grassmannians of Quivers and Affine Quantum Groups","place":"Cham","year":"2022","doi":"10.1007/978-3-030-82007-7_8","ec_funded":1,"alternative_title":["Trends in Mathematics"],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1810.10095","open_access":"1"}]},{"date_created":"2020-06-07T22:00:55Z","department":[{"_id":"TaHa"}],"language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","date_published":"2020-12-01T00:00:00Z","article_type":"original","month":"12","page":"1371-1385","publication":"Transformation Groups","status":"public","intvolume":"        25","type":"journal_article","day":"01","main_file_link":[{"url":"https://arxiv.org/abs/1804.04375","open_access":"1"}],"isi":1,"external_id":{"arxiv":["1804.04375"],"isi":["000534874300003"]},"title":"The PBW theorem for affine Yangians","ec_funded":1,"year":"2020","doi":"10.1007/s00031-020-09572-6","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","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.","project":[{"name":"Arithmetic and physics of Higgs moduli spaces","_id":"25E549F4-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"320593"}],"oa_version":"Preprint","quality_controlled":"1","_id":"7940","publication_identifier":{"issn":["10834362"],"eissn":["1531586X"]},"volume":25,"date_updated":"2023-08-21T07:06:21Z","oa":1,"article_processing_charge":"No","arxiv":1,"author":[{"id":"360D8648-F248-11E8-B48F-1D18A9856A87","full_name":"Yang, Yaping","last_name":"Yang","first_name":"Yaping"},{"id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","last_name":"Zhao","full_name":"Zhao, Gufang","first_name":"Gufang"}],"abstract":[{"lang":"eng","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]."}],"publication_status":"published","citation":{"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>","ieee":"Y. Yang and G. Zhao, “The PBW theorem for affine Yangians,” <i>Transformation Groups</i>, vol. 25. Springer Nature, pp. 1371–1385, 2020.","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>.","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>","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>.","ista":"Yang Y, Zhao G. 2020. The PBW theorem for affine Yangians. Transformation Groups. 25, 1371–1385.","short":"Y. Yang, G. Zhao, Transformation Groups 25 (2020) 1371–1385."}},{"page":"663-671","publication":"Annales Scientifiques de l'Ecole Normale Superieure","issue":"3","type":"journal_article","day":"01","status":"public","intvolume":"        53","department":[{"_id":"TaHa"}],"date_created":"2020-09-20T22:01:38Z","date_published":"2020-06-01T00:00:00Z","article_type":"original","month":"06","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Société Mathématique de France","article_processing_charge":"No","volume":53,"date_updated":"2023-08-22T09:27:57Z","oa":1,"arxiv":1,"quality_controlled":"1","oa_version":"Preprint","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication_identifier":{"issn":["0012-9593"]},"_id":"8539","citation":{"ista":"Su C, Zhao G, Zhong C. 2020. On the K-theory stable bases of the springer resolution. Annales Scientifiques de l’Ecole Normale Superieure. 53(3), 663–671.","short":"C. Su, G. Zhao, C. Zhong, Annales Scientifiques de l’Ecole Normale Superieure 53 (2020) 663–671.","ama":"Su C, Zhao G, Zhong C. On the K-theory stable bases of the springer resolution. <i>Annales Scientifiques de l’Ecole Normale Superieure</i>. 2020;53(3):663-671. doi:<a href=\"https://doi.org/10.24033/asens.2431\">10.24033/asens.2431</a>","mla":"Su, C., et al. “On the K-Theory Stable Bases of the Springer Resolution.” <i>Annales Scientifiques de l’Ecole Normale Superieure</i>, vol. 53, no. 3, Société Mathématique de France, 2020, pp. 663–71, doi:<a href=\"https://doi.org/10.24033/asens.2431\">10.24033/asens.2431</a>.","chicago":"Su, C., Gufang Zhao, and C. Zhong. “On the K-Theory Stable Bases of the Springer Resolution.” <i>Annales Scientifiques de l’Ecole Normale Superieure</i>. Société Mathématique de France, 2020. <a href=\"https://doi.org/10.24033/asens.2431\">https://doi.org/10.24033/asens.2431</a>.","apa":"Su, C., Zhao, G., &#38; Zhong, C. (2020). On the K-theory stable bases of the springer resolution. <i>Annales Scientifiques de l’Ecole Normale Superieure</i>. Société Mathématique de France. <a href=\"https://doi.org/10.24033/asens.2431\">https://doi.org/10.24033/asens.2431</a>","ieee":"C. Su, G. Zhao, and C. Zhong, “On the K-theory stable bases of the springer resolution,” <i>Annales Scientifiques de l’Ecole Normale Superieure</i>, vol. 53, no. 3. Société Mathématique de France, pp. 663–671, 2020."},"publication_status":"published","author":[{"full_name":"Su, C.","last_name":"Su","first_name":"C."},{"id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","first_name":"Gufang","last_name":"Zhao","full_name":"Zhao, Gufang"},{"full_name":"Zhong, C.","last_name":"Zhong","first_name":"C."}],"abstract":[{"lang":"eng","text":"Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions."},{"lang":"fre","text":"Les bases stables cohomologiques et K-théoriques proviennent de l’étude de la cohomologie quantique et de la K-théorie quantique. La formule de restriction pour les bases stables cohomologiques a joué un rôle important dans le calcul de la connexion quantique du fibré cotangent de variétés de drapeaux partielles. Dans cet article, nous étudions les bases stables K-théoriques de fibré cotangents des variétés de drapeaux. Nous décrivons ces bases en fonction de l’action de l’algèbre de Hecke affine et de l’algèbre de Kostant-Kumar. En utilisant cette description algébrique et la méthode des polynômes de racine, nous donnons une formule de restriction des bases stables. Nous l’appliquons\r\npour obtenir la formule de restriction pour les variétés de drapeaux partielles. Nous construisons également une relation entre la base stable et la base de Casselman dans les représentations de la série principale du groupe dual de Langlands p-adique. Comme une application, nous donnons une formule close pour la matrice de transition entre la base de Casselman et les fonctions caractéristiques. "}],"isi":1,"main_file_link":[{"url":"https://arxiv.org/abs/1708.08013","open_access":"1"}],"year":"2020","doi":"10.24033/asens.2431","external_id":{"arxiv":["1708.08013"],"isi":["000592182600004"]},"title":"On the K-theory stable bases of the springer resolution"},{"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."}],"author":[{"full_name":"Rapcak, Miroslav","last_name":"Rapcak","first_name":"Miroslav"},{"first_name":"Yan","full_name":"Soibelman, Yan","last_name":"Soibelman"},{"full_name":"Yang, Yaping","last_name":"Yang","first_name":"Yaping"},{"id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","first_name":"Gufang","last_name":"Zhao","full_name":"Zhao, Gufang"}],"publication_status":"published","citation":{"ista":"Rapcak M, Soibelman Y, Yang Y, Zhao G. 2020. Cohomological Hall algebras, vertex algebras and instantons. Communications in Mathematical Physics. 376, 1803–1873.","short":"M. Rapcak, Y. Soibelman, Y. Yang, G. Zhao, Communications in Mathematical Physics 376 (2020) 1803–1873.","mla":"Rapcak, Miroslav, et al. “Cohomological Hall Algebras, Vertex Algebras and Instantons.” <i>Communications in Mathematical Physics</i>, vol. 376, Springer Nature, 2020, pp. 1803–73, doi:<a href=\"https://doi.org/10.1007/s00220-019-03575-5\">10.1007/s00220-019-03575-5</a>.","ama":"Rapcak M, Soibelman Y, Yang Y, Zhao G. Cohomological Hall algebras, vertex algebras and instantons. <i>Communications in Mathematical Physics</i>. 2020;376:1803-1873. doi:<a href=\"https://doi.org/10.1007/s00220-019-03575-5\">10.1007/s00220-019-03575-5</a>","chicago":"Rapcak, Miroslav, Yan Soibelman, Yaping Yang, and Gufang Zhao. “Cohomological Hall Algebras, Vertex Algebras and Instantons.” <i>Communications in Mathematical Physics</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s00220-019-03575-5\">https://doi.org/10.1007/s00220-019-03575-5</a>.","ieee":"M. Rapcak, Y. Soibelman, Y. Yang, and G. Zhao, “Cohomological Hall algebras, vertex algebras and instantons,” <i>Communications in Mathematical Physics</i>, vol. 376. Springer Nature, pp. 1803–1873, 2020.","apa":"Rapcak, M., Soibelman, Y., Yang, Y., &#38; Zhao, G. (2020). Cohomological Hall algebras, vertex algebras and instantons. <i>Communications in Mathematical Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00220-019-03575-5\">https://doi.org/10.1007/s00220-019-03575-5</a>"},"_id":"7004","publication_identifier":{"issn":["0010-3616"],"eissn":["1432-0916"]},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","project":[{"call_identifier":"FP7","_id":"25E549F4-B435-11E9-9278-68D0E5697425","name":"Arithmetic and physics of Higgs moduli spaces","grant_number":"320593"}],"quality_controlled":"1","oa_version":"Preprint","arxiv":1,"oa":1,"date_updated":"2023-08-17T14:02:59Z","volume":376,"article_processing_charge":"No","external_id":{"arxiv":["1810.10402"],"isi":["000536255500004"]},"title":"Cohomological Hall algebras, vertex algebras and instantons","year":"2020","doi":"10.1007/s00220-019-03575-5","ec_funded":1,"isi":1,"main_file_link":[{"url":"https://arxiv.org/abs/1810.10402","open_access":"1"}],"intvolume":"       376","status":"public","day":"01","type":"journal_article","publication":"Communications in Mathematical Physics","page":"1803-1873","publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"06","article_type":"original","date_published":"2020-06-01T00:00:00Z","date_created":"2019-11-12T14:01:27Z","department":[{"_id":"TaHa"}]},{"arxiv":1,"volume":116,"date_updated":"2023-09-19T14:37:19Z","oa":1,"article_processing_charge":"No","_id":"5999","publication_identifier":{"issn":["0024-6115"]},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"Preprint","quality_controlled":"1","publication_status":"published","citation":{"short":"Y. Yang, G. Zhao, Proceedings of the London Mathematical Society 116 (2018) 1029–1074.","ista":"Yang Y, Zhao G. 2018. The cohomological Hall algebra of a preprojective algebra. Proceedings of the London Mathematical Society. 116(5), 1029–1074.","ama":"Yang Y, Zhao G. The cohomological Hall algebra of a preprojective algebra. <i>Proceedings of the London Mathematical Society</i>. 2018;116(5):1029-1074. doi:<a href=\"https://doi.org/10.1112/plms.12111\">10.1112/plms.12111</a>","mla":"Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective Algebra.” <i>Proceedings of the London Mathematical Society</i>, vol. 116, no. 5, Oxford University Press, 2018, pp. 1029–74, doi:<a href=\"https://doi.org/10.1112/plms.12111\">10.1112/plms.12111</a>.","chicago":"Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective Algebra.” <i>Proceedings of the London Mathematical Society</i>. Oxford University Press, 2018. <a href=\"https://doi.org/10.1112/plms.12111\">https://doi.org/10.1112/plms.12111</a>.","apa":"Yang, Y., &#38; Zhao, G. (2018). The cohomological Hall algebra of a preprojective algebra. <i>Proceedings of the London Mathematical Society</i>. Oxford University Press. <a href=\"https://doi.org/10.1112/plms.12111\">https://doi.org/10.1112/plms.12111</a>","ieee":"Y. Yang and G. Zhao, “The cohomological Hall algebra of a preprojective algebra,” <i>Proceedings of the London Mathematical Society</i>, vol. 116, no. 5. Oxford University Press, pp. 1029–1074, 2018."},"abstract":[{"lang":"eng","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. "}],"author":[{"full_name":"Yang, Yaping","last_name":"Yang","first_name":"Yaping"},{"id":"2BC2AC5E-F248-11E8-B48F-1D18A9856A87","first_name":"Gufang","full_name":"Zhao, Gufang","last_name":"Zhao"}],"isi":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1407.7994"}],"year":"2018","doi":"10.1112/plms.12111","external_id":{"isi":["000431506400001"],"arxiv":["1407.7994"]},"title":"The cohomological Hall algebra of a preprojective algebra","issue":"5","publication":"Proceedings of the London Mathematical Society","page":"1029-1074","day":"01","type":"journal_article","intvolume":"       116","status":"public","department":[{"_id":"TaHa"}],"date_created":"2019-02-14T13:14:22Z","month":"05","date_published":"2018-05-01T00:00:00Z","publisher":"Oxford University Press","scopus_import":"1","language":[{"iso":"eng"}]}]