[Paper Review] Introduction to categorification

Alistair Savage|arXiv (Cornell University)|Jan 23, 2014

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper introduces categorification as a unifying framework in representation theory, focusing on weak and strong categorification of algebraic structures such as the Weyl algebra and Heisenberg algebra. It uses module categories, Grothendieck groups, and 2-categories to lift algebraic relations to categorical isomorphisms, with key results showing how polynomial and Fock space representations can be categorified via nilcoxeter algebras and symmetric function categories.

ABSTRACT

These are the notes for a two-week mini-course given at a winter school in January 2014 as part of the thematic semester New Directions in Lie Theory at the Centre de Recherches Mathématiques in Montréal. The goal of the course was to give an overview of the idea of categorification, with an emphasis on some examples where explicit computation is possible with minimal background. The notes begin with a very brief review of the representation theory of associative algebras, before introducing the concept of weak categorification with some simple examples. It then proceeds to a discussion of more sophisticated examples of categorification, including a weak categorification of the polynomial representation of the Weyl group and the Fock space representation of the Heisenberg algebra. The notes conclude with a discussion of strong categorification and a brief overview of some further directions in the field.

Motivation & Objective

• To provide a self-contained introduction to categorification for graduate students with basic algebra and category theory background.
• To demonstrate how algebraic structures like the Weyl algebra and Heisenberg algebra can be categorified using module categories and Grothendieck groups.
• To introduce the distinction between weak and strong categorification, with explicit examples from symmetric functions and nilcoxeter algebras.
• To lay the foundation for advanced topics such as the graphical Heisenberg category and 2-categorical structures in categorified representation theory.

Proposed method

• Uses Grothendieck groups to relate module categories to algebraic structures, enabling weak categorification.
• Employs the nilcoxeter algebra and its modules to weakly categorify the polynomial representation of the Weyl algebra.
• Applies symmetric functions and the Hopf algebra Sym to construct a categorification of the Fock space representation of the Heisenberg algebra.
• Introduces 2-categories and 2-functors to formalize strong categorification, particularly via the graphical Heisenberg category.
• Utilizes towers of algebras and the Heisenberg double to realize strong categorification of the Heisenberg algebra.
• Applies categorical tools such as idempotent completion, Karoubi envelopes, and additive 2-categories to enrich the structure of categorified algebras.

Experimental results

Research questions

• RQ1 How can the polynomial representation of the Weyl algebra be weakly categorified using module categories?
• RQ2 What is the role of the nilcoxeter algebra in realizing a weak categorification of the Weyl algebra’s action on polynomials?
• RQ3 How can the Fock space representation of the Heisenberg algebra be categorified through symmetric functions and module categories?
• RQ4 What structures (e.g. 2-categories, graphical categories) are required to achieve strong categorification of the Heisenberg algebra?
• RQ5 How do Grothendieck groups of module categories recover the original algebraic structures in categorification?

Key findings

• The polynomial representation of the Weyl algebra is weakly categorified via the nilcoxeter algebra and its modules, with the Grothendieck group isomorphic to the polynomial ring.
• The Fock space representation of the Heisenberg algebra is weakly categorified using the Hopf algebra Sym of symmetric functions, with the Grothendieck group of the category of modules realizing the Fock space.
• Strong categorification of the Heisenberg algebra is achieved through the graphical Heisenberg category, which provides a 2-categorical lift of the algebraic relations.
• The Heisenberg double construction realizes the strong categorification of the Heisenberg algebra using towers of algebras and bimodules.
• The Grothendieck group of the 2-category of modules over the graphical Heisenberg category recovers the original Heisenberg algebra.
• The categorified Fock space is realized as the split Grothendieck group of the category of modules over the symmetric function category, with the action lifted to functors between categories.