We propose a new framework of monoidal categorifications of finite ADE types cluster algebras involving triangulated categories instead of abelian categories. Let Q be any Dynkin quiver and let A_Q denote the corresponding finite type cluster algebra (with a suitable choice of frozen variables). We define a certain additive symmetric monoidal category out of the Auslander-Reiten theory of Q and consider its (bounded) homotopy category K_Q. Using some iterated mapping cone technique, we construct a family of chain complexes in bijection with almost positive roots that are characterized by natural exactness conditions. We then prove that the Euler charasteristics of the chain complex associated to any positive root β coincides with the truncated q-character of the simple module categorifying the cluster variable x[β] in A_Q via Hernandez-Leclerc’s monoidal categorification. We also conjecture that all exchange relations in A_Q are categorified by distinguished triangles in K_Q. If time allows, we will discuss potential extensions of this construction to other cluster structures.

Abstract: To any object in a braided tensor category we can associate a Nichols algebra, either through braid group combinatorics or through a universal property. The main motivation for Nichols algebras is to systematically construct the quantum group from its Cartan part. This idea goes back to Lusztig, but I will present it in modern categorical language. Then I will sketch how Nichols algebras can be used to construct solutions of the Knishnik Zomolochikov differential equation from its respective abelian version, and to construct a conformal field theory associated to the (small) quantum group from a free field theory. This parallelism is a main ingredient in our recent proof of the logarithmic Kazhdan Lusztig correspondence, which is the nonsemisimple small-quantum-group-analog of the more familiar Kazhdan Lusztig correspondence. At the same time, the construction produces many new nonsemisimple braided tensor categories, topological field theories and conformal field theories, correspondingly.

We discuss how one can “categorify” the Temperley-Lieb, Hecke algebras, and their modules. We then discuss what is knowable (and unknowable!) about the representation theoretic structure of these categorifications.