Hannah Keese's 2014 honours thesis in mathematics, on "Current algebras, categorification and annular Khovanov homology", supervised by
Dr. Anthony Licata, is available here.
Outline
Lie algebras and quantum groups have been widely studied in the context of representation theory, while invariants of knots and links are objects of significant importance in topology. There exist connections between these objects that can be of interest for both of these fields. One example of such a connection is Reshetikhin and Turaev's construction of link invariants generalising the Jones polynomial from representations of quantum groups [RT90]. The Reshetikhin-Turaev invariants can be enriched by categorification: replacing representations of the quantum group by categories equipped with a quantum group action and replacing polynomial invariants by homological link invariants. Our objective in this thesis is to describe several of the categorifications which are important in both representation theory and topology and to study some of the new structure that appears in the categorified world.
On the topological side, one of the triumphs of categorification is Khovanov homology, a categorification of the Jones polynomial to a homological link invariant first introduced by Mikhail Khovanov in the late 1990s [Kho00]. Khovanov homology is strictly stronger than its decategorified counterpart [BN02] and can detect the unknot [KM11]. Khovanov has also made a significant contribution to the categorification of representations of quantum groups [HK01] [HK06], and moreover of quantum groups themselves [KL09].
The added algebraic structure of homological link invariants can furthermore can make these invariants interesting objects from the perspective of representation theory in their own right. Another aim of this thesis is to demonstrate this relationship in the case of a homological invariant of annular links, which arises as representations of a particular Lie algebra, known as a current algebra.
Chapter 1 introduces notation, definitions and basic theorems and properties that will be used in later chapters. In particular, we give an overview of the representation theory of semisimple Lie algebras and quantum groups, following [FH91], [Hum72] and [Lus93]. This representation
theory is used extensively in the remaining chapters. The main result is a classification of the irreducible finite-dimensional representations of the Lie algebra $\mathfrak{sl}_2$, and a detailed proof of the complete reducibility of its finite-dimensional representations.
In chapter 2, we study particular types of current algebras and their representations, such as the polynomial current algebras examined in [CG07], and representation algebras $\mathfrak{g}(V)$. We give a description of the representations of particular examples of these algebras that arise in our study of annular Khovanov homology using quiver representations. We first give a proof of a theorem of Loupias [Lou72] on quiver representations of the Lie algebra $\mathfrak{sl}_2(V_1)$. We then state and prove an analogous result for the current algebra $\mathfrak{sl}_2^-(V_2)$ which reappears in chapter 4 in our discussion of annular Khovanov homology.
In chapter 3 we examine the work of Khovanov and Huerfano in [HK01] and [HK06], presenting their work on categorifications of representations of quantum groups. This chapter is largely independent of the preceding chapters, however some of the algebraic objects studied here are encountered in earlier sections. For example, the zigzag algebra plays a significant role in both the representation theory of current algebras in chapter 2 as well as the construction of a categorification of the adjoint representation in this chapter.
The principal objective of chapter 4 is to relate knot homology to the representation theory studied in previous chapters, giving a representation theoretic presentation of annular Khovanov homology, a homology theory of knots and links in the solid torus defined by Asaeda, Przytycki and Sikora in [APS04]. In particular, the main theorem 4.4.1 of this chapter defines an explicit action of the current algebra $\mathfrak{sl}_2^-(V_2)$
on annular Khovanov homology. We give a complete and independent proof of this theorem, originally due to Grigsby-Licata-Wehrli [GLW]. Of particular interest here is the relationship between the current algebra action and Lee's deformation of Khovanov homology, as seen in [Lee05].
