@inbook{12303, abstract = {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).}, author = {Mirković, Ivan and Yang, Yaping and Zhao, Gufang}, booktitle = {Representation Theory and Algebraic Geometry}, editor = {Baranovskky, Vladimir and Guay, Nicolas and Schedler, Travis}, isbn = {9783030820060}, issn = {2297-024X}, pages = {347--392}, publisher = {Springer Nature; Birkhäuser}, title = {{Loop Grassmannians of Quivers and Affine Quantum Groups}}, doi = {10.1007/978-3-030-82007-7_8}, year = {2022}, } @article{7004, abstract = {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 = {Rapcak, Miroslav and Soibelman, Yan and Yang, Yaping and Zhao, Gufang}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {1803--1873}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras, vertex algebras and instantons}}, doi = {10.1007/s00220-019-03575-5}, volume = {376}, year = {2020}, } @article{7940, abstract = {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].}, author = {Yang, Yaping and Zhao, Gufang}, issn = {1531586X}, journal = {Transformation Groups}, pages = {1371--1385}, publisher = {Springer Nature}, title = {{The PBW theorem for affine Yangians}}, doi = {10.1007/s00031-020-09572-6}, volume = {25}, year = {2020}, } @article{8539, abstract = {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.}, author = {Su, C. and Zhao, Gufang and Zhong, C.}, issn = {0012-9593}, journal = {Annales Scientifiques de l'Ecole Normale Superieure}, number = {3}, pages = {663--671}, publisher = {Société Mathématique de France}, title = {{On the K-theory stable bases of the springer resolution}}, doi = {10.24033/asens.2431}, volume = {53}, year = {2020}, } @article{5999, abstract = {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. We 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 = {Yang, Yaping and Zhao, Gufang}, issn = {0024-6115}, journal = {Proceedings of the London Mathematical Society}, number = {5}, pages = {1029--1074}, publisher = {Oxford University Press}, title = {{The cohomological Hall algebra of a preprojective algebra}}, doi = {10.1112/plms.12111}, volume = {116}, year = {2018}, }