PBW theory for Bosonic extensions of quantum groups

In this paper, we develop the PBW theory for the bosonic extension $\text{qbA}\text{g}$ of a quantum group ${\mathcal{U}}_q(\text{g})$ of \emph{any} finite type. When $\text{g}$ belongs to the class of \emph{simply-laced type}, the algebra $\text{qbA}\text{g}$ arises from the quantum Grothendieck ring of the Hernandez-Leclerc category over quantum affine algebras of untwisted affine types. We introduce and investigate a symmetric bilinear form $\text{pair},$ on $\text{qbA}\text{g}$ which is invariant under the braid group actions ${\text{bT}}_i$ on $\text{qbA}\text{g}$, and study the adjoint operators ${\text{Ep}}_{i,p}$ and ${\text{Es}}_{i,p}$ with respect to $\text{pair},$. It turns out that the adjoint operators ${\text{Ep}}_{i,p}$ and ${\text{Es}}_{i,p}$ are analogues of the $q$-derivations $e_i^{\mathrm{\prime }}$ and ${\text{es}}_i$ on the negative half ${\text{calU}}_q^{-}(\text{g})$ of ${\text{calU}}_q(\text{g})$. Following this, we introduce a new family of subalgebras denoted as $\text{qbA}\mathfrak{g}(\text{ttb})$ in $\text{qbA}\mathfrak{g}$. These subalgebras are defined for any elements $\text{ttb}$ in the positive submonoid ${\text{bg}}^{+}$ of the (generalized) braid group $\text{ttB}$ of $\text{g}$. We prove that $\text{qbA}\mathfrak{g}(\text{ttb})$ exhibits PBW root vectors and PBW bases defined by ${\text{bT}}_{\text{ii}}$ for any sequence $\text{ii}$ of $\text{ttb}$. The PBW root vectors satisfy a Levendorskii-Soibelman formula and the PBW bases are orthogonal with respect to $\text{pair},$. The algebras $\text{qbA}\text{g}(\text{ttb})$ can be understood as a natural extension of quantum unipotent coordinate rings.