Braid group action on quantum virtual Grothendieck ring through constructing presentations

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