Cluster Algebras and Quantum Affine Algebras, after B. Leclerc

This talk, based on [14], is a report on recent work by B. Leclerc on a new type of categorification for cluster algebras. Cluster algebras were invented by Fomin and Zelevinsky [8] at the beginning of this decade. Since then, a major effort has gone into their categorification (cf. for example [15] [1] [2] [3] [10]). Namely, in many cases, it was proved that for a given cluster algebra A, there exists a triangulated (or Frobenius) category C, such that • the cluster variables x of A correspond to certain indecomposables Tx of C, • two cluster variables x and y belong to the same cluster if and only if there are no non split extensions between the corresponding objects Tx and Ty, • the cluster monomial m = xy · · · z corresponds to the the object M = Tx ⊕ Ty ⊕ · · ·Tz of C, • the exchange relations xx∗ = m + m′ of A correspond to triangles Tx → M → Tx∗ → ΣTx and Tx∗ → M ′ → Tx → ΣTx∗