Ozsvath-Szabo bordered algebras and subquotients of category O

TL;DR

This paper reveals that Ozsváth-Szabó's bordered algebra is a graded flat deformation of a quotient of parabolic category O, linking knot Floer homology computations to categorifications of quantum group representations.

Contribution

It establishes an explicit isomorphism between Ozsváth-Szabó's algebra and a quotient of Sartori's diagrammatic algebra, enabling new diagrammatic interpretations and categorifications.

Findings

Identifies Ozsváth-Szabó algebra as a graded flat deformation of a category O quotient.

Provides an explicit algebra isomorphism linking two algebraic frameworks.

Constructs bimodules categorifying quantum group actions on tensor powers of the vector representation.

Abstract

We show that Ozsv\'ath-Szab\'o's bordered algebra used to efficiently compute knot Floer homology is a graded flat deformation of the regular block of a $\mathfrak{q}$-presentable quotient of parabolic category $\mathcal{O}$. We identify the endomorphism algebra of a minimal projective generator for this block with an explicit quotient of the Ozsv\'ath-Szab\'o algebra using Sartori's diagrammatic formulation of the endomorphism algebra. Both of these algebras give rise to categorifications of tensor products of the vector representation $V^{\otimes n}$ for $U_q(\mathfrak{gl}(1|1))$. Our isomorphism allows us to transport a number of constructions between these two algebras, leading to a new (fully) diagrammatic reinterpretation of Sartori's algebra, new modules over Ozsv\'ath-Szab\'o's algebra lifting various bases of $V^{\otimes n}$, and bimodules over Ozsv\'ath-Szab\'o's algebra categorifying the action of the quantum group element $F$ and its dual on $V^{\otimes n}$.