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 $\qbA{\g}$ arisesfrom the quantum Grothendieck ring of the Hernandez-Leclerc category overquantum affine algebras of untwisted affine types. We introduce and investigatea symmetric bilinear form $\pair{\ , \ }$ on $\qbA{\g}$ which is invariantunder the braid group actions $\bT_i$ on $\qbA{\g}$, and study the adjointoperators $\Ep_{i,p}$ and $\Es_{i,p}$ with respect to $\pair{\ , \ }$. It turnsout 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 denotedas $\qbA{\mathfrak{g}}(\ttb)$ in $\qbA{\mathfrak{g}}$. These subalgebras aredefined 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 aLevendorskii-Soibelman formula and the PBW bases are orthogonal with respect to$\pair{\ , \ }$. The algebras $\qbA{\g} (\ttb)$ can be understood as a naturalextension of quantum unipotent coordinate rings.