Braid group action on quantum virtual Grothendieck ring through constructing presentations

TL;DR

This paper explores the structure of the quantum virtual Grothendieck ring for simple Lie algebras, establishing an isomorphism with quantum coordinate algebras, and demonstrates a braid group action on it.

Contribution

It constructs a presentation of the quantum virtual Grothendieck ring using categorification and quiver Hecke algebras, revealing its structure as a boson-extension and establishing a braid group action.

Findings

Established an isomorphism between subring of $rakK_q( ext{g})$ and $ ext{A}_q( ext{n})$

Presented $rakK_q( ext{g})$ as a boson-extension of $ ext{A}_q( ext{n})$

Proved automorphisms induce a braid group $B_ ext{g}$ action

Abstract

As a continuation of \cite{JLO1}, we investigate the quantum virtual Grothendieck ring $\frakK_q(\g)$ associated with a finite dimensional simple Lie algebra $\g$, especially of non-simply-laced type. We establish an isomorphism $\Uppsi_Q$ between the heart subring $\frakK_{q,Q}(\g)$ of $\frakK_q(\g)$ associated with a Dynkin quiver $Q$ of type $\g$ and the unipotent quantum coordinate algebra $\calA_q(\n)$ of type $\g$. This isomorphism and the categorification theory via quiver Hecke algebras enable us to obtain a presentation of $\frakK_q(\g)$, which reveals that $\frakK_q(\g)$ can be understood as a boson-extension of $\calA_q(\n)$. Then we show that the automorphisms, arising from the reflections on Dynkin quivers and the isomorphisms $\Uppsi_Q$, preserve the canonical basis $\sfL_q$ of $\frakK_q(\g)$. Finally, we prove that such automorphisms produce a braid group $B_\g$ action on $\frakK_q(\g)$.