| Résume | Shifted quantum affine algebras are a family of quantum groups parameterized by a coweight of the underlying Lie algebras. Originally introduced to study K theoretic Coulomb branches, shifted quantum affine algebras can now be appreciated also from the point of view of the monoidal categorification of cluster algebras. Hernandez has introduced a category O of representations of these algebras and later Geiss-Hernandez-Leclerc have proved that the Grothendieck ring of this category admits a cluster algebra structure. In the first part of the talk I will present shifted quantum affine algebras, some properties of their representations and the cluster algebra construction of Geiss-Hernandez-Leclerc. In the second part I will provide a quantization of such cluster algebra and present some applications regarding a quantization of QQ-systems (remarkable relations in the Grothendieck ring of the category O) and the quantum oscillator algebra. |
