Séminaires : Séminaire d'Algèbre

The Lambda-invariant for quantum affine algebras was introduced by Kashiwara-Kim-Oh-Park as an important tool in their study of the monoidal categorification of cluster algebras. At the level of the cluster algebra, it is related to a compatible Poisson structure, and it was recently shown to coincide with Peigen Cao’s tropical invariant.

In this talk, we interpret these invariants using the additive categorification of cluster algebras via Higgs categories in the sense of Yilin Wu. Whenever the relative Ginzburg algebra is proper, we show that the Higgs category admits a canonical quantum structure in the sense of Grabowski-Pressland, and we give a homological interpretation of the corresponding tropical invariant.

For certain finite-rank cluster algebras categorified by Kashiwara-Kim-Oh-Park, we show that the associated relative Ginzburg algebra is indeed proper, and that our additive Lambda-invariant agrees with their monoidal one.

This is joint work in progress with Geoffrey Janssens and Peigen Cao.

This talk will take place in hybrid mode at the Institut Henri Poincaré.