Cluster structures via representation theory: cluster ensembles, tropical duality, cluster characters and quantisation

We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category C and cluster-tilting subcategory T satisfying only mild finiteness conditions. We show that the structure theory of C and the representation theory of T give rise to the rich combinatorial structures of seed data and cluster ensembles, via Grothendieck groups and homological algebra. We demonstrate that there is a natural dictionary relating cluster-tilting subcategories and their tilting theory to A-side tropical cluster combinatorics and, dually, relating modules over T to the X-side; here T is the image of T in the triangulated stable category of C . Moreover, the exchange matrix associated to T arises from a natural map pT:K0(mod T )→K0( T ) closely related to taking projective resolutions. Via our approach, we categorify many key identities involving mutation, g -vectors and c -vectors, including in infinite rank cases and in the presence of loops and 2-cycles. We are also able to define A- and X-cluster characters, which yield A- and X-cluster variables when there are no loops or 2-cycles, and which enable representation-theoretic proofs of cluster-theoretical statements. Continuing with the same categorical philosophy, we give a definition of a quantum cluster category, as a cluster category together with the choice of a map closely related to the adjoint of pT. Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Geiß-Leclerc-Schröer.