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

Yue Hu, Alexander Tsymbaliuk

TL;DR

The paper advances the theory of shuffle algebra realizations for classical types by constructing PBWD bases for the positive subalgebras $U_v^{>}(L\mathfrak{g})$ with $\mathfrak{g}=\mathfrak{sp}_{2n}$ or $\mathfrak{so}_{2n}$ (types $C_n$ and $D_n$), along with Lusztig and RTT integral forms and their PBWD bases. It introduces and analyzes specialization maps to connect shuffle realizations with root-vector constructions, ensuring they remain compatible with wheel/pole constraints and Kostant partitions; this yields explicit, integral bases and isomorphisms to shuffle algebras $S$ and RTT counterparts $\mathcal{S}$. The results extend prior work in types $A_n$, $B_n$, and $G_2$ to the remaining classical types and their Yangian analogues, providing a unified framework for positive subalgebras, their integral forms, and associated duals. The methodology—combining convex Lyndon-based root vectors, two integral forms, and careful specialization analysis—facilitates effective treatment of high-degree noncommutative elements and paves the way for applications in representation theory, integrable systems, and knot invariants. Overall, the work delivers a comprehensive, canonical shuffle-algebra realization and PBWD-basis framework for types $C_n$ and $D_n$, including their rational (Yangian) extensions and duals, with a clear path for further generalizations.

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 $\mathbb{Q}(v)$-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 $C_n$ 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.

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

TL;DR

The paper advances the theory of shuffle algebra realizations for classical types by constructing PBWD bases for the positive subalgebras with or (types and ), along with Lusztig and RTT integral forms and their PBWD bases. It introduces and analyzes specialization maps to connect shuffle realizations with root-vector constructions, ensuring they remain compatible with wheel/pole constraints and Kostant partitions; this yields explicit, integral bases and isomorphisms to shuffle algebras and RTT counterparts . The results extend prior work in types , , and to the remaining classical types and their Yangian analogues, providing a unified framework for positive subalgebras, their integral forms, and associated duals. The methodology—combining convex Lyndon-based root vectors, two integral forms, and careful specialization analysis—facilitates effective treatment of high-degree noncommutative elements and paves the way for applications in representation theory, integrable systems, and knot invariants. Overall, the work delivers a comprehensive, canonical shuffle-algebra realization and PBWD-basis framework for types and , including their rational (Yangian) extensions and duals, with a clear path for further generalizations.

Abstract

We construct a family of PBWD bases for the positive subalgebras of quantum loop algebras of type and , as well as their Lusztig and RTT integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these -algebras (proved earlier in arXiv:2102.11269 by completely different tools) and generalize the latter to the above -forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type and Yangians and their Drinfeld-Gavarini duals. While this naturally generalizes our earlier treatment of the classical type in arXiv:2305.00810 and in arXiv:1808.09536, the specialization maps in the present setup are more compelling.

Paper Structure

This paper contains 25 sections, 49 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Summary.
  3. Outline of the paper.
  4. Acknowledgements
  5. Preliminaries
  6. Quantum loop algebras and shuffle algebras in types $C_n$ and $D_n$.
  7. Root vectors and PBWD bases in types $C_n,D_n$
  8. Two integral forms in types $C_n$ and $D_n$
  9. Specialization maps in types $C_{n}$ and $D_n$
  10. Shuffle algebra and its integral forms in type $C_n$
  11. $U_{v}^{>}(L\mathfrak{sp}_{2n})$ and its shuffle algebra realization
  12. Shuffle algebra realization of the Lusztig integral form in type $C$
  13. Shuffle algebra realization of the RTT integral form $\mathcal{U}_{v}^{>}(L\mathfrak{sp}_{2n})$
  14. Shuffle algebra and its integral forms in type $D_n$
  15. $U_{v}^{>}(L\mathfrak{o}_{2n})$ and its shuffle algebra realization
  16. ...and 10 more sections

Key Result

Theorem 2.2

The assignment $e_{i,r}\mapsto x_{i,1}^{r}\in S_{\mathbf{1}_i} \ (i\in I, r\in{\mathbb{Z}})$, where $\mathbf{1}_i=(0,\ldots,1,\ldots,0)$ with $1$ at the $i$-th coordinate, gives rise to a ${\mathbb{Q}}(v)$-algebra isomorphism

Theorems & Definitions (75)

  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Lemma 2.9
  • Lemma 2.10
  • Lemma 3.1
  • proof
  • ...and 65 more