A diagrammatic approach to categorification of quantum groups I

TL;DR

This work constructs diagrammatic graded rings $R(\nu)$ from a loop-free graph $\Gamma$ and proves that the projective Grothendieck groups $K_0(R)$ categorify the integral form $_{\mathcal{A}}\mathbf{f}$ of the negative half of the quantum group $U^-_q(\mathfrak{g})$ for simply-laced $\mathfrak{g}$. It develops a faithful action of $R(\nu)$ on polynomial rings, analyzes the center $\mathrm{Sym}(\nu)$, and provides an explicit basis for $R(\nu)$ as a module over the center, establishing freeness and positivity of the grading. A twisted bialgebra homomorphism $\gamma: {}_{\mathcal{A}}\mathbf{f} \to K_0(R)$ is constructed and shown to be an isomorphism (over suitable fields), linking diagrammatic categorification to Lusztig’s and Ariki–Koike frameworks and enabling a canonical basis interpretation via indecomposable projectives. The paper also explores tight monomials, the role of cycles in $\Gamma$, and proposes a program to categorify irreducible representations through quotients $R(\nu;\lambda)$ with functors lifting quantum group actions, suggesting deep connections to geometric realizations and affine Hecke algebras in special cases.

Abstract

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.

Theorems & Definitions (63)

• Theorem 1.1
• Conjecture 1.2
• Proposition 2.3
• proof
• Example 2.4
• Theorem 2.5
• proof
• Corollary 2.6
• Proposition 2.7
• proof