Unbounded $$\mathfrak {sl}_3$$-laminations around punctures

Abstract We continue to study the unbounded $$\mathfrak {sl}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> -laminations [16], with a focus on their structures at punctures. A key ingredient is their relation to the root data of $$\mathfrak {sl}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> . After giving a classification of signed $$\mathfrak {sl}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> -webs around a puncture, we describe the tropicalization of the Goncharov–Shen’s Weyl group action in detail. We also clarify the relationship with several other approaches by Shen–Sun–Weng [28] and Fraser–Pylyavskyy [10]. Finally, we discuss a formulation of unbounded $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -laminations for a general semisimple Lie algebra $$\mathfrak {g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> in brief.