Algebraic study of quantum configuration spaces of decorated flags

Abstract

Let $G$ be a connected, simply connected complex simple algebraic group and ${\mathcal{A}}_G=G/U^{+}$ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags ${\mathcal{O}}_q({\mathrm{Conf}}_K{\mathcal{A}}_G)$ and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra ${\mathcal{O}}_q({\mathrm{Conf}}_K{\mathcal{A}}_G)$ gives a quantum analogue of the configuration space ${\mathrm{Conf}}_K{\mathcal{A}}_G$ of $K$ decorated flags, which provides local building blocks for the Fock--Goncharov moduli space ${\mathcal{A}}_{G,\Sigma }$ of decorated twisted $G$-local systems on a marked surface $\Sigma$. We establish basic algebraic properties of ${\mathcal{O}}_q({\mathrm{Conf}}_K{\mathcal{A}}_G)$ such as quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of ${\mathcal{A}}_{G,\Sigma }$. Moreover, we construct quantum seeds for ${\mathcal{O}}_q({\mathrm{Conf}}_4{\mathcal{A}}_G)$ by transporting the Berenstein--Zelevinsky quantum cluster structure on ${\mathcal{O}}_q(G)$ via quantum Wilson lines, and prove that ${\mathcal{O}}_q({\mathrm{Conf}}_4{\mathcal{A}}_G)$ coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov--Shen exchange matrices. We also show that quantum seeds for ${\mathcal{O}}_q({\mathrm{Conf}}_4{\mathcal{A}}_G)$ restrict to those for ${\mathcal{O}}_q({\mathrm{Conf}}_3{\mathcal{A}}_G)$. Finally, up to a natural conjecture, we construct quantum seeds for ${\mathcal{O}}_q({\mathrm{Conf}}_K{\mathcal{A}}_G)$, $K\ge 5$, and prove that the corresponding quantum cluster algebras contain the quantum configuration space ${\mathcal{O}}_q({\mathrm{Conf}}_K{\mathcal{A}}_G)$.

92 pages