Standard Lyndon Loop Words: Weighted Orders

Abstract We generalize the study of standard Lyndon loop words from [16] to a more general class of orders on the underlying alphabet, as suggested in [16, Remark 3.15]. 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 $U_{q}(L{{\mathfrak{g}}})$ via the combinatorics of loop words.