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 \(\mathcal{C}\) and cluster-tilting subcategory \(\mathcal{T}\) satisfying only mild finiteness conditions. We show that the structure theory of \(\mathcal{C}\) and the representation theory of \(\mathcal{T}\) 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 \(\underline{\mathcal{T}}\) to the X-side; here \(\underline{\mathcal{T}}\) is the image of \(\mathcal{T}\) in the triangulated stable category of \(\mathcal{C}\). Moreover, the exchange matrix associated to \(\mathcal{T}\) arises from a natural map \(p_{\mathcal{T}}\colon\mathrm{K}_0(\operatorname{mod}\underline{\mathcal{T}})\to\mathrm{K}_0(\mathcal{T})\) 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 \(p_{\mathcal{T}}\). Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Geiß--Leclerc--Schr\"oer.