Extended graphical calculus for categorified quantum sl(2)

Details Extended graphical calculus for categorified quantum sl(2) Extended graphical calculus for categorified quantum sl(2)

2010-06-15

arxiv.org/abs/1006.2866v1

Mathematics

Quantum Algebra

72 pages, LaTeX2e with xypic and pstricks macros

Scientific paper

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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