Shuffle Algebras and Their Integral Forms: Specialization Map Approach in Types B n and G 2

We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the positive subalgebras of quantum loop algebras of type $B_{n}$ and $G_{2}$, as well as their Lusztig and RTT (for type $B_{n}$ only) integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these ${\mathbb {Q}}(v)$-algebras (proved earlier in [26] by completely different tools) and generalize the latter to the above ${{\mathbb {Z}}}[v,v^{-1}]$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $B_{n}$ and $G_{2}$ Yangians and their Drinfeld-Gavarini duals. All of this generalizes the type $A_{n}$ results of [30].