Category O and sl(k) link invariants

TL;DR

This paper develops a new functorial invariant for oriented tangles using category O, categorifying tensor products of sl(k) representations and relating to HOMFLYPT polynomials, leading to a homological link invariant.

Contribution

It introduces a novel functorial tangle invariant based on category O that categorifies tensor products of fundamental sl(k) representations and connects to HOMFLYPT polynomials.

Findings

Constructed a functorial invariant for oriented tangles.

Categorified tensor products of fundamental sl(k) representations.

Derived a homological invariant for links.

Abstract

We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k) representations. Projective functors restricted to these categories give rise to a functorial action of the Lie algebra. On the derived category, Zuckerman functors categorify sl(k)- homomorphisms. Cones of natural transformations between the identity functor and Zuckerman functors are assigned to crossings and this assignment satisfies the appropriate relations. On the Grothendieck group, the functors assigned to the crossings satisfy the sl(k)- specialization of the two variable HOMFLYPT polynomial. For the special case of links, we get a homological invariant.