Loop Grassmannians of quivers and affine quantum groups

Abstract

We 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 $\mathcal{D}\subset {\mathbb{G}}_m^2$ gives a quantization ${\mathcal{G}}_{\mathcal{D}}^P(Q,A)$. The construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program ['Some extensions of the notion of loop Grassmannians', Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardesic issue. No. 532, (2017) 53--74.] and on the construction of affine quantum groups from generalized cohomologies ArXiv: 1708.01418.

v2: 32 pages, an error in Section 3.5 corrected, more details added throughout