An introduction to diagrammatic algebra and categorified quantum sl(2)

Abstract

This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of categorified quantum sl2 and highlight some of the new structure that arises in categorified quantum groups. The expert will find a discussion of rescalling isomorphisms for categorified quantum sl2, a proof that cyclotomic quotients of the nilHecke algebra are isomorphic to matrix rings over the cohomology ring of Grassmannians, and an interpretation of `fake bubbles' using symmetric functions.

64 pages. v2 corrects typos

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Cited by in corpus (19)

• Adjunctions and defects in Landau-Ginzburg models
• The odd nilHecke algebra and its diagrammatics
• Deformations of colored sl(N) link homologies via foams
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• An approach to categorification of some small quantum groups II
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• Trace decategorification of categorified quantum sl(2)
• Current algebras and categorified quantum groups
• An odd categorification of quantum sl(2)
• Cyclicity for categorified quantum groups
• Oddification of the cohomology of type A Springer varieties
• The Categorified Heisenberg Algebra I: A Combinatorial Representation
• A diamond lemma for Hecke-type algebras
• A hitchhiker's guide to Khovanov homology
• Logic and linear algebra: an introduction
• Clifford-symmetric polynomials
• Foams, iterated wreath products, field extensions and Sylvester sums
• Trace and categorical sl(n) representations
• Schubert Class and cyclotomic nilHecke algebras