Categories of split filtrations and graded quiver varieties

TL;DR

This work generalizes the geometric and representation-theoretic framework of affine graded Nakajima quiver varieties to the $n$-fold setting, connecting $n$-fold tensor products of standard quantum affine modules with representation varieties of a new singular category $ ext{S}^{n ext{-filt}}$. The authors construct split filtrations, realize them as modules over a triangular matrix category, and prove a central equivalence between the stable category of Gorenstein projective modules and the derived category $D^b(k eflectbox{oldmath$ eflectbox{ extsf{A}}$}_n simes kQ)$, along with a stratification functor $ abla^n$ (and its variant) that ties filtrations to the stratification of graded quiver varieties. The framework extends Keller–Scherotzke's KS correspondence to the filtrated singular Nakajima category, and leverages mesh-category techniques, Kan extensions, and Happel embeddings to build a robust, compatible theory of filtrations, resolutions, and Gorenstein structures. The results provide a geometric realization of $n$-fold tensor products in the quantum affine setting and pave the way for deeper connections between singular Nakajima categories, graded quiver varieties, and quantum group representations.

Abstract

By the work of Hernandez-Leclerc, Leclerc-Plamondon, and Keller-Scherotzke, affine graded Nakajima quiver varieties associated with a Dynkin quiver $Q$ admit an algebraic description in terms of modules over the singular Nakajima category $\mathcal{S}$ and a stratification functor to the derived category of $Q$. In this paper, we extend this framework to Nakajima's $n$-fold affine graded tensor product varieties, which allow one to geometrically realize $n$-fold tensor products of standard modules over the quantum affine algebra. We introduce a category of filtrations with splitting of length $n$ of modules over a category and show that it is equivalent to the module category of a triangular matrix category. Applied to the singular Nakajima category, this yields a category $\mathcal{S}^{n\operatorname{-filt}}$ whose modules are parametrized by the points of the $n$-fold tensor product varieties. Generalizing the results of Keller-Scherotzke from $\mathcal{S}$ to $\mathcal{S}^{n\operatorname{-filt}}$, we prove that the stable category of finitely generated Gorenstein projective $\mathcal{S}^{n\operatorname{-filt}}$-modules is triangle equivalent to the derived category of the algebra of $n \times n$ upper triangular matrices over the path algebra of $Q$, and we obtain a corresponding stratification functor.

Theorems & Definitions (82)

• Theorem 1.1: Proposition \ref{['prop:characterization of Filt as a module category']}
• Theorem 1.2: Theorem \ref{['thm:equivalences for stable categories']}
• Remark 1.3
• Theorem 1.4: Corollary \ref{['cor:stratum of subquotients']}
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6: Happel88
• Definition 3.1
• Remark 3.2