English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $G$ be a connected semisimple Lie group. There are two natural dualityconstructions that assign to it the Langlands dual group $G^\vee$ and thePoisson-Lie dual group $G^*$. The main result of this paper is the followingrelation between these two objects: the integral cone defined by the clusterstructure and the Berenstein-Kazhdan potential on the double Bruhat cell$G^{\vee; w_0, e} \subset G^\vee$ is isomorphic to the integral Bohr-Sommerfeldcone 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 [5],the first cone parametrizes the canonical bases of irreducible $G$-modules. Thecorresponding points in the second cone belong to integral symplectic leaves ofthe partial tropicalization labeled by the highest weight of therepresentation. As a by-product of our construction, we show that symplecticvolumes of generic symplectic leaves in the partial tropicalization of $K^*$are equal to symplectic volumes of the corresponding coadjoint orbits in$\operatorname{Lie}(K)^*$.  To achieve these goals, we make use of (Langlands dual) double clustervarieties defined by Fock and Goncharov [9]. These are pairs of clustervarieties whose seed matrices are transpose to each other. There is a naturallydefined isomorphism between their tropicalizations. The isomorphism between thecones described above is a particular instance of such an isomorphismassociated to the double Bruhat cells $G^{w_0, e} \subset G$ and $G^{\vee; w_0,e} \subset G^\vee$.

Langlands Duality and Poisson-Lie Duality via Cluster Theory and Tropicalization