Langlands duality and Poisson–Lie duality via cluster theory and tropicalization
| العنوان: | Langlands duality and Poisson–Lie duality via cluster theory and tropicalization |
|---|---|
| المؤلفون: | Anton Alekseev, Benjamin Hoffman, Arkady Berenstein, Yanpeng Li |
| المصدر: | Selecta Mathematica. 27 |
| بيانات النشر: | Springer Science and Business Media LLC, 2021. |
| سنة النشر: | 2021 |
| مصطلحات موضوعية: | Combinatorics, General Mathematics, Poisson manifold, Structure (category theory), General Physics and Astronomy, Lie group, Duality (optimization), Cone (category theory), Isomorphism, Langlands dual group, Mathematics::Representation Theory, Mathematics, Symplectic geometry |
| الوصف: | Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group $$G^\vee $$ , and its Poisson–Lie dual group $$G^*$$ , respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell $$G^{\vee ; w_0, e} \subset G^\vee $$ is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of $$K^* \subset G^*$$ (the Poisson–Lie dual of the compact form $$K \subset G$$ ). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of $$K^*$$ are equal to symplectic volumes of the corresponding coadjoint orbits in $${{\,\mathrm{Lie}\,}}(K)^*$$ . To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Super (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells $$G^{w_0, e} \subset G$$ and $$G^{\vee ; w_0, e} \subset G^\vee $$ . |
| تدمد: | 1420-9020 1022-1824 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_________::3780b3207528355734c0c813fac8aa3c https://doi.org/10.1007/s00029-021-00682-x |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi...........3780b3207528355734c0c813fac8aa3c |
| قاعدة البيانات: | OpenAIRE |
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