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Standard Lyndon loop words: weighted orders

Severyn Khomych, Nazar Korniichuk, Kostiantyn Molokanov, Alexander Tsymbaliuk

TL;DR

The paper extends the combinatorial Lyndon word framework to weighted (or twisted) loop alphabets by introducing exponent-tightness, enabling a robust PBW-type description for quantum loop algebras. It develops standard Lyndon loop words via a finite-filtration approach, proves exponent-tightness and stabilization, and establishes periodicity through a map Υ that shifts exponents in a controlled way. It then connects these combinatorics to affine Weyl groups, via reduced decompositions, and constructs PBW-type bases for quantum loop algebras, including a quarter-subalgebra formulation and a Beĭck-type isomorphism with quantum affine algebras. The generalized_orders section broadens the construction to more flexible word orders, preserving many core properties while highlighting where periodicity may fail, thereby extending the reach of Lyndon-based PBW bases in the quantum loop/affine setting.

Abstract

We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109482] to a more general class of orders on the underlying alphabet, as suggested in Remark 3.15 of loc.cit. The main new ingredient is the exponent-tightness of these words, which also allows to generalize the construction of PBW bases of the untwisted quantum loop algebra via the combinatorics of loop words.

Standard Lyndon loop words: weighted orders

TL;DR

The paper extends the combinatorial Lyndon word framework to weighted (or twisted) loop alphabets by introducing exponent-tightness, enabling a robust PBW-type description for quantum loop algebras. It develops standard Lyndon loop words via a finite-filtration approach, proves exponent-tightness and stabilization, and establishes periodicity through a map Υ that shifts exponents in a controlled way. It then connects these combinatorics to affine Weyl groups, via reduced decompositions, and constructs PBW-type bases for quantum loop algebras, including a quarter-subalgebra formulation and a Beĭck-type isomorphism with quantum affine algebras. The generalized_orders section broadens the construction to more flexible word orders, preserving many core properties while highlighting where periodicity may fail, thereby extending the reach of Lyndon-based PBW bases in the quantum loop/affine setting.

Abstract

We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109482] to a more general class of orders on the underlying alphabet, as suggested in Remark 3.15 of loc.cit. The main new ingredient is the exponent-tightness of these words, which also allows to generalize the construction of PBW bases of the untwisted quantum loop algebra via the combinatorics of loop words.
Paper Structure (32 sections, 32 theorems, 200 equations)

This paper contains 32 sections, 32 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Summary
  3. Outline
  4. Acknowledgement
  5. Combinatorial approach to Lie algebras
  6. Lyndon words
  7. Standard Lyndon words
  8. Application to simple Lie algebras
  9. Results of Leclerc
  10. Standard Lyndon loop words
  11. Filtration and basic properties
  12. Exponent tightness
  13. Stabilization
  14. Periodicity
  15. Convexity and minimality
  16. ...and 17 more sections

Key Result

Lemma 2.4

If $\ell_1 < \ell_2$ are Lyndon, then $\ell_1\ell_2$ is also Lyndon, and so $\ell_1 \ell_2 < \ell_2 \ell_1$.

Theorems & Definitions (66)

  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • Proposition 2.11
  • Theorem 2.12
  • ...and 56 more