Global bases for Bosonic extensions of quantum unipotent coordinate rings

TL;DR

This paper develops a global basis theory for bosonic extensions of quantum unipotent coordinate rings associated with arbitrary generalized Cartan matrices, connecting it to quantum Grothendieck rings in specific cases.

Contribution

It introduces a global basis framework for bosonic extensions of quantum coordinate rings and links it to quantum Grothendieck rings for simply-laced finite types.

Findings

Established global basis theory for bosonic extensions.

Connected global bases to $(t,q)$-characters of simple modules.

Identified isomorphism with quantum Grothendieck rings in specific cases.

Abstract

In the paper, we establish the global basis theory for the bosonic extension $\widehat{\mathcal{A}}$ associated with an arbitrary generalized Cartan matrix. When $\widehat{\mathcal{A}}$ is of simply-laced finite type, it is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over a quantum affine algebra. In this case, we show that the $(t,q)$-characters of simple modules in the Hernandez-Leclerc category correspond to the normalized global basis of $\widehat{\mathcal{A}}$.