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Extended graphical calculus for categorified quantum sl(2)
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A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical
calculus introduced and developed in that paper to include two-morphisms between divided
powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas
for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis
elements. These formulas have integral coefficients and imply that one of the main results
of Lauda’s paper—identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)—also holds when the 2-category is defined over the ring of integers
rather than over a field.
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American Mathematical Societ