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

Yue Hu; Alexander Tsymbaliuk

doi:10.1093/imrn/rnae029

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

Quantum Algebra 2024-04-10 v2 Rings and Algebras Representation Theory

Authors: Yue Hu , Alexander Tsymbaliuk

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Abstract

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 $\mathbb{Q}(v)$ -algebras (proved earlier in arXiv:2102.11269 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 arXiv:1808.09536 by the second author.

Keywords

shuffle algebras quantum loop algebras pbwd bases integral forms yangians type b_n type g_2

Cite

@article{arxiv.2305.00810,
  title  = {Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$},
  author = {Yue Hu and Alexander Tsymbaliuk},
  journal= {arXiv preprint arXiv:2305.00810},
  year   = {2024}
}

Comments

This is a majorly corrected version of arXiv:2206.12806 by the first author, with a new generalization to integral forms and Yangian counterparts. v1: 46 pages, comments are welcome! v2: 47 pages, minor corrections, details added

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

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