PBW Theory for Bosonic Extensions of Quantum Groups

Abstract In this paper, we develop the Poincaré–Birkhoff–Witt (PBW) theory for the bosonic extension $\widehat{{\mathcal{A}}}_{\mathfrak{g}}$ of a quantum group $\mathcal{U}_{q}(\mathfrak{g})$ of any finite type. When $\mathfrak{g}$ belongs to the class of simply-laced type, the algebra $\widehat{{\mathcal{A}}}_{\mathfrak{g}}$ 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 $((\, , )) $ on $\widehat{{\mathcal{A}}}_{\mathfrak{g}}$, which is invariant under the braid group actions $\mathcal{T}_{i}$ on $\widehat{{\mathcal{A}}}_{\mathfrak{g}}$, and study the adjoint operators $E^{\prime}_{i,p}$ and $E^{\star }_{i,p}$ with respect to $((\, , ))$. It turns out that the adjoint operators $E^{\prime}_{i,p}$ and $E^{\star }_{i,p}$ are analogues of the $q$-derivations $e_{i}^{\prime}$ and $e^\star _{i}$ on the negative half $\mathcal{U}_{q}^{-}(\mathfrak{g})$ of $\mathcal{U}_{q}(\mathfrak{g})$. Following this, we introduce a new family of subalgebras denoted as $\widehat{{\mathcal{A}}}_{\mathfrak{g}}(\texttt{b})$ in $\widehat{{\mathcal{A}}}_{\mathfrak{g}}$. These subalgebras are defined for any elements $\texttt{b}$ in the positive submonoid $\texttt{B}^{+}$ of the (generalized) braid group $\texttt{B}$ of $\mathfrak{g}$. We prove that $\widehat{{\mathcal{A}}}_{\mathfrak{g}}(\texttt{b})$ exhibits PBW root vectors and PBW bases defined by $\mathcal{T}_{\boldsymbol{i}}$ for any sequence $\boldsymbol{i}$ of $\texttt{b}$. The PBW root vectors satisfy a Levendorskii–Soibelman formula and the PBW bases are orthogonal with respect to $((\, , ))$. The algebras $\widehat{{\mathcal{A}}}_{\mathfrak{g}} (\texttt{b})$ can be understood as a natural extension of quantum unipotent coordinate rings.