In this talk, we talk about the PBW theory for the bosonic extension \(\widehat{A}_{\mathfrak{g}}\) of a quantum group, which is a joint work with Se-jin Oh. When \(\mathfrak{g}\) belongs to the class of simply-laced finite type, the algebra \(\widehat{A}_{\mathfrak{g}}\) is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over quantum affine algebras. We introduce PBW vectors and PBW monomials using the braid group actions on \(\widehat{A}_{\mathfrak{g}}\), and define a new family of subalgebras, denoted by \(\widehat{A}_{\mathfrak{g}}(b)\), for any element \(b\) in the (generalized) Braid group corresponding to \(\mathfrak{g}\). We explain why the algebras \(\widehat{A}_{\mathfrak{g}}(b)\) can be understood as a natural extension of quantum unipotent coordinate rings \(A_q(\mathfrak{n}(w))\), and show that the PBW monomials form an orthogonal basis of \(\widehat{A}_{\mathfrak{g}}(b)\). This talk is based on arXiv:2401.04878.