Seminar of Algebra

Quantum affine algebra Uq(\hat{g}) is a Hopf algebra that is a $q$-deformation of the universal enveloping algebra of an affine Lie algebra \hat{g}. Hernandez and Leclerc in 2010 introduced a certain subcategory C_{\ell} of the category of finite dimensional Uq(\hat{g})-modules. They proved that K_0(C_{\ell}) has a cluster algebra structure and in the case of g=sl_k, K_0(C_{\ell}) is isomorphic to a quotient of Grassmannian cluster algebra. In joint work with Wen Chang, Bing Duan, and Chris Fraser, we proved that the dual canonical basis of a Grassmannian cluster algebra is parametrized by semistandard Young tableaux. Using results in representations of p-adic groups and quantum affine algebras, we gave a formula to compute elements in the dual canonical basis of a Grassmannian cluster algebra. In this talk, I will talk about joint work with Nick Early about a construction of prime modules of quantum affine algebras using Newton polytopes. We apply the results of prime modules to construct u-variables for Grassmannian cluster algebras which are useful in scattering amplitudes. I will also talk about joint work with James Drummond and Ömer Gürdoğan about tropicalization of quasi-automorphisms of cluster algebras. Using tropicalization, we study fixed points and orbits of Chris Fraser's braid group actions on Grassmannian cluster algebras.