TL;DR
This paper develops a diagrammatic categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras, connecting dg-enhanced KLR algebras with quantum algebra representations.
Contribution
It introduces a dg-enhanced categorification framework for parabolic Verma modules using KLR-like diagrams, extending the usual categorification of integrable modules.
Findings
Constructed a categorification of parabolic Verma modules.
Connected dg-enhancements of cyclotomic KLR quotients to module categorification.
Introduced dg-2-representations for quantum Kac-Moody algebras.
Abstract
We construct a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac--Moody algebras, and in particular of parabolic 2-Verma module.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Topics in Algebra