Categorifying fractional Euler characteristics, Jones-Wenzl projector and $3j$-symbols

TL;DR

This paper develops a categorification framework for quantum invariants using representation theory, diagrammatic methods, and algebraic structures, advancing the understanding of 3-manifold invariants and quantum algebra.

Contribution

It introduces new diagrammatic formulas for 3j-symbols, employs complete intersection rings for rational Euler characteristics, and categorifies key quantum invariants like the Jones-Wenzl projector.

Findings

Diagrammatic visualization of 3j-symbols as weighted signed line arrangements.

New binomial expressions for 3j-symbols realized as graded Euler characteristics.

Categorification of Jones-Wenzl projector and network evaluations using Ext-algebras.

Abstract

We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan-Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, Theta-networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants, \cite{TV}. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.