Standard Lyndon loop words: weighted orders

TL;DR

This paper extends the theory of standard Lyndon loop words to broader orders on alphabets, introducing exponent-tightness and enabling generalized PBW basis constructions for quantum loop algebras.

Contribution

It generalizes the concept of Lyndon loop words to more flexible orders and introduces exponent-tightness, advancing the combinatorial construction of PBW bases.

Findings

Generalization of Lyndon loop words to broader alphabet orders

Introduction of exponent-tightness property

Construction of PBW bases for quantum loop algebras

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.