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An Introduction To Diagrammatic Algebra And Categorified Quantum $Sl_2$
by Aaron D. Lauda

Vol. 7 No. 2 (2012) P.165~P.270

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 $\mathfrak{sl}_2$. We derive the definition of categorified quantum $\mathfrak{sl}_2$ and highlight some of the new structure that arises in categorified quantum groups. The expert will find a discussion of rescaling isomorphisms for categorified quantum $\mathfrak{sl}_2$, 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.

KEYWORDS
Categorification, quantum sl(2), cohomology of Grassmannians, diagrammatic algebra, cyclotomic quotient

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 81R50, 17B37, 16W50, 18D05

MILESTONES