Shuffle algebras and their integral forms: specialization map approach in types $C_n$ and $D_n$

Yue Hu; Alexander Tsymbaliuk

doi:10.1016/j.jalgebra.2025.10.036

paper

Shuffle algebras and their integral forms: specialization map approach in types $C_{n}$ and $D_{n}$

arXiv:2503.22535 · doi:10.1016/j.jalgebra.2025.10.036

Abstract

We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type $C_{n}$ and $D_{n}$ , as well as their Lusztig and RTT 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 $C_{n}$ and $D_{n}$ Yangians and their Drinfeld-Gavarini duals. While this naturally generalizes our earlier treatment of the classical type $B_{n}$ in arXiv:2305.00810 and $A_{n}$ in arXiv:1808.09536, the specialization maps in the present setup are more compelling.

v2: 56 pages, minor corrections, some details added. v1: 55 pages, comments are welcome! arXiv admin note: text overlap with arXiv:2305.00810

Shuffle algebras and their integral forms: specialization map approach in types Cn​ and Dn​

Abstract

Shuffle algebras and their integral forms: specialization map approach in types $C_{n}$ and $D_{n}$