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A new new coproduct on quantum loop algebras

Andrei Neguţ

TL;DR

This work constructs a general, topological coproduct Δ_{{oldsymbol{p}}} on quantum loop algebras, formulating half-subalgebras whose Drinfeld double recovers the full algebra and recovers the Drinfeld–Jimbo coproduct in the quantum affine case. Using shuffle algebra realizations, pairings, and completions, it develops a tensor category structure on category ${oldsymbol{O}}$, introduces universal $R$-matrices and their factorizations, and analyzes slope-based decompositions and R-matrix intertwiners. The representation theory is developed for loop weights, including simple modules $L^{oldsymbol{p}}({oldsymbol{oldsymbol{ igth)}}})$, rational loop weights, and shifted algebras ${oldsymbol{U}}^{oldsymbol{r}}$, with multiplicativity of $q$-characters and explicit decompositions. The framework unifies multiple instances such as quantum affine algebras, K-theoretic Hall algebras, and BPS algebras, and provides tools for studying transfer matrices, integrable systems, and related algebraic structures. Overall, it lays a comprehensive foundation for tensor products, R-matrices, and representation theory across general quantum loop algebras via a robust shuffle- and double-structure approach.

Abstract

Quantum loop algebras generalize $U_q(\widehat{\mathfrak{g}})$ for simple Lie algebras $\mathfrak{g}$, and they include examples such as quantum affinizations of Kac-Moody Lie algebras, K-theoretic Hall algebras of quivers, and BPS algebras for toric Calabi-Yau threefolds. In the present paper, we define a coproduct on general quantum loop algebras, which coincides with the Drinfeld-Jimbo coproduct in the particular case of $U_q(\widehat{\mathfrak{g}})$ . We investigate the consequences of our construction for the representation theory of quantum loop algebras, particularly for tensor products of modules and R-matrices.

A new new coproduct on quantum loop algebras

TL;DR

This work constructs a general, topological coproduct Δ_{{oldsymbol{p}}} on quantum loop algebras, formulating half-subalgebras whose Drinfeld double recovers the full algebra and recovers the Drinfeld–Jimbo coproduct in the quantum affine case. Using shuffle algebra realizations, pairings, and completions, it develops a tensor category structure on category , introduces universal -matrices and their factorizations, and analyzes slope-based decompositions and R-matrix intertwiners. The representation theory is developed for loop weights, including simple modules , rational loop weights, and shifted algebras , with multiplicativity of -characters and explicit decompositions. The framework unifies multiple instances such as quantum affine algebras, K-theoretic Hall algebras, and BPS algebras, and provides tools for studying transfer matrices, integrable systems, and related algebraic structures. Overall, it lays a comprehensive foundation for tensor products, R-matrices, and representation theory across general quantum loop algebras via a robust shuffle- and double-structure approach.

Abstract

Quantum loop algebras generalize for simple Lie algebras , and they include examples such as quantum affinizations of Kac-Moody Lie algebras, K-theoretic Hall algebras of quivers, and BPS algebras for toric Calabi-Yau threefolds. In the present paper, we define a coproduct on general quantum loop algebras, which coincides with the Drinfeld-Jimbo coproduct in the particular case of . We investigate the consequences of our construction for the representation theory of quantum loop algebras, particularly for tensor products of modules and R-matrices.
Paper Structure (42 sections, 19 theorems, 357 equations)

This paper contains 42 sections, 19 theorems, 357 equations.

Table of Contents

  1. Introduction
  2. The title
  3. Algebras
  4. Simple modules
  5. Coproducts
  6. R-matrices
  7. Techniques
  8. Acknowledgements
  9. Quantum loop and shuffle algebras
  10. Basic notations
  11. The pre-quantum loop algebra
  12. The shuffle algebra
  13. Extended algebras
  14. Gradings and completions
  15. The topological coproduct
  16. ...and 27 more sections

Key Result

Theorem 1.5

(subsumed by Theorem thm:main) There is a topological coproduct (see Subsection sub:gradings for the definition of $\stackrel{\mathsf{H}}{\otimes}$) which preserves the subalgebra ${\mathcal{A}}^{\geq {\boldsymbol{p}}}$.

Theorems & Definitions (50)

  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.3
  • Definition 2.5
  • Definition 2.10
  • Remark 2.11
  • Remark 2.13
  • Lemma 2.15
  • proof
  • Definition 2.17
  • ...and 40 more