$c$-vectors of 2-Calabi--Yau categories and Borel subalgebras of ${\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}$

We develop a general framework for $c$-vectors of 2-Calabi–Yau categories, which deals with cluster tilting subcategories that are not reachable from each other and contain infinitely many indecomposable objects. It does not rely on iterative sequences of mutations. We prove a categorical (infinite-rank) generalization of the Nakanishi–Zelevinsky duality for $c$-vectors and establish two formulae for the effective computation of $c$-vectors – one in terms of indices and the other in terms of dimension vectors for cluster tilted algebras. In this framework, we construct a correspondence between the $c$-vectors of the cluster categories $\mathcal{C}(A_{\mathrm{\infty }})$ of type $A_{\mathrm{\infty }}$ due to Igusa–Todorov and the roots of the Borel subalgebras of ${\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}$. Contrary to the finite dimensional case, the Borel subalgebras of ${\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }}$ are not conjugate to each other. On the categorical side, the cluster tilting subcategories of $\mathcal{C}(A_{\mathrm{\infty }})$ exhibit different homological properties. The correspondence builds a bridge between the two classes of objects.