Loop Grassmannians of Quivers and Affine Quantum Groups

Abstract

AbstractWe construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” \({\mathcal G}^P(Q,A)\). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus \({\mathcal D}{ \subseteq } \mathbb {G}_m^2\) gives a quantization \({\mathcal G}^P_{\mathcal D}(Q,A)\). 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).