Séminaires : Séminaire d'Algèbre

We introduce new families of monoidal structures suited to categorifying important invariants arising in the representation theory of quantum affine algebras and in the theory of cluster algebras. Our framework builds on Auslander--Reiten theory and higher homological algebra; namely, for each finite acyclic quiver QQQ, we construct a monoidal category RQ\mathcal{R}_QRQ​ whose indecomposable objects are tensor products of modules over the path algebra of QQQ. We show that to each indecomposable object XXX in RQ\mathcal{R}_QRQ​ corresponds a (unique up to homotopy) chain complex C∙(X)C_{\bullet}(X)C∙​(X) satisfying remarkable homological properties. These complexes are constructed using some iterated mapping cone procedure, which yields a collection of distinguished triangles in the bounded homotopy category of RQ\mathcal{R}_QRQ​. We then prove that in the case where QQQ is a Dynkin quiver, the Euler characteristics of (the image under a suitable functor of) C∙(X)C_{\bullet}(X)C∙​(X) can be identified with the (truncated) qqq-character of a standard module over the quantum affine algebra Uq(gQ^)U_q(\widehat{\mathfrak{g}_Q})Uq​(gQ​​). We also show that the above mentioned distinguished triangles categorify certain exchange relations in the finite type cluster algebra AQ\mathcal{A}_QAQ​. If time allows, we will discuss perspectives towards monoidal categorifications of more general cluster structures.

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