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

TL;DR

This paper develops a framework for $c$-vectors in 2-Calabi--Yau categories, generalizing duality and establishing a connection between $c$-vectors and roots of Borel subalgebras of ${\mathfrak{sl}}_{\infty}$, revealing new categorical and algebraic insights.

Contribution

It introduces a general framework for $c$-vectors in infinite-rank 2-Calabi--Yau categories, including duality and computational formulas, and links $c$-vectors to roots of Borel subalgebras of ${\mathfrak{sl}}_{\infty}$.

Findings

Established a categorical duality for $c$-vectors in infinite rank.

Derived formulas for computing $c$-vectors via indices and dimension vectors.

Connected $c$-vectors of ${\mathscr{C}}(A_{\infty})$ to roots of Borel subalgebras of ${\mathfrak{sl}}_{\infty}$.

Abstract

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 ${\mathscr{C}}(A_{\infty})$ of type $A_{\infty}$ due to Igusa--Todorov and the roots of the Borel subalgebras of ${\mathfrak{sl}}_{\infty}$. Contrary to the finite dimensional case, the Borel subalgebras of ${\mathfrak{sl}}_{\infty}$ are not conjugate to each other. On the categorical side, the cluster tilting subcategories of ${\mathscr{C}}(A_{\infty})$ exhibit different homological properties. The correspondence builds a bridge between the two classes of objects.