Shuffle Algebras and Their Integral Forms: Specialization Map Approach in Types Bn and G2.

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 ]. [ABSTRACT FROM AUTHOR]