Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$

We construct a family of PBWD 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 $Q (v)$ -algebras (proved earlier in arXiv:2102.11269 by completely different tools) and generalize the latter to the above $Z [v, v^{- 1}]$ -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 arXiv:1808.09536 by the second author.

Shuffle algebras and their integral forms: specialization map approach in types $B_{n}$ and $G_{2}$

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Shuffle algebras and their integral forms: specialization map approach in types $B_{n}$ and $G_{2}$

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Shuffle algebras and their integral forms: specialization map approach in types BnBn​ and G2G2​

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Shuffle algebras and their integral forms: specialization map approach in types BnBn​ and G2G2​

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Shuffle algebras and their integral forms: specialization map approach in types $B_{n}$ and $G_{2}$

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Shuffle algebras and their integral forms: specialization map approach in types $B_{n}$ and $G_{2}$

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