Loop Grassmannians of quivers and affine quantum groups

TL;DR

The paper introduces a generalized construction of loop Grassmannians for quivers, cohomology theories, and posets, extending classical cases and connecting to affine quantum groups and the Geometric Langlands program.

Contribution

It constructs a new class of loop Grassmannians associated with quivers, cohomology theories, and posets, and introduces a quantization via a dilation torus, advancing the geometric framework for affine quantum groups.

Findings

Generalizes classical loop Grassmannians to quivers and posets

Defines a quantization of these Grassmannians using a dilation torus

Links the construction to affine quantum groups and the Geometric Langlands program

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 $\mathcal{G}^P_\mathcal{D}(Q,A)$. 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.