BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Elie Casbi (U. Vienna) DTSTART:20261116T130000Z DTEND:20261116T140000Z DTSTAMP:20261114T031042Z UID:paris-algebra-seminar/233 DESCRIPTION:Title: Representation theory of quantum affine algebras via tenso r products of modules over hereditary algebras\nby Elie Casbi (U. Vien na) as part of Paris algebra seminar\n\n\nAbstract\nWe introduce new famil ies of monoidal structures suited to categorifying important invariants a rising in the representation theory of quantum affine algebras and in the theory of cluster algebras. Our framework builds on Auslander--Reiten the ory and higher homological algebra\; namely\, for each finite acyclic quiv er $Q$\, we construct a monoidal category $\\mathcal{R}_Q$ whose indecompo sable objects are tensor products of modules over the path algebra of $Q$. We show that to each indecomposable object $X$ in $\\mathcal{R}_Q$ corre sponds a (unique up to homotopy) chain complex $C_{\\bullet}(X)$ satisf ying remarkable homological properties. These complexes are constructed u sing some iterated mapping cone procedure\, which yields a collection of distinguished triangles in the bounded homotopy category of $\\mathcal{R}_ Q$. We then prove that in the case where $Q$ is a Dynkin quiver\, the Eul er characteristics of (the image under a suitable functor of) $C_{\\bullet }(X)$ can be identified with the (truncated) $q$-character of a standard m odule over the quantum affine algebra $U_q(\\widehat{\\mathfrak{g}_Q})$. W e also show that the above mentioned distinguished triangles categorify ce rtain exchange relations in the finite type cluster algebra $\\mathcal{A} _Q$. If time allows\, we will discuss perspectives towards monoidal catego rifications of more general cluster structures.\n \n\n\nThis talk will tak e place in hybrid mode at the Institut Henri Poincaré.\n LOCATION:https://researchseminars.org/talk/paris-algebra-seminar/233/ END:VEVENT END:VCALENDAR