PBW theory for Bosonic extensions of quantum groups

In this paper, we develop the PBW theory for the bosonic extension $\qbA{\g}$ of a quantum group $\mathcal{U}_q(\g)$ of \emph{any} finite type. When $\g$ belongs to the class of \emph{simply-laced type}, the algebra $\qbA{\g}$ 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 $\pair{\ , \ }$ on $\qbA{\g}$ which is invariant under the braid group actions $\bT_i$ on $\qbA{\g}$, and study the adjoint operators $\Ep_{i,p}$ and $\Es_{i,p}$ with respect to $\pair{\ , \ }$. It turns out that the adjoint operators $\Ep_{i,p}$ and $\Es_{i,p}$ are analogues of the $q$-derivations $e_i'$ and $\es_i$ on the negative half $\calU_q^-(\g)$ of $\calU_q(\g)$. Following this, we introduce a new family of subalgebras denoted as $\qbA{\mathfrak{g}}(\ttb)$ in $\qbA{\mathfrak{g}}$. These subalgebras are defined for any elements $\ttb$ in the positive submonoid $\bg^+$ of the (generalized) braid group $\ttB$ of $\g$. We prove that $\qbA{\mathfrak{g}}(\ttb)$ exhibits PBW root vectors and PBW bases defined by $\bT_\ii$ for any sequence $\ii$ of $\ttb$. The PBW root vectors satisfy a Levendorskii-Soibelman formula and the PBW bases are orthogonal with respect to $\pair{\ , \ }$. The algebras $\qbA{\g} (\ttb)$ can be understood as a natural extension of quantum unipotent coordinate rings.