PBW Theory for Bosonic Extensions of Quantum Groups

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