**Memoirs of the American Mathematical Society**

2012;
87 pp;
Softcover

MSC: Primary 81;
Secondary 18; 05

Print ISBN: 978-0-8218-8977-0

Product Code: MEMO/219/1029

List Price: $67.00

Individual Member Price: $40.20

**Electronic ISBN: 978-0-8218-9110-0
Product Code: MEMO/219/1029.E**

List Price: $67.00

Individual Member Price: $40.20

# Extended Graphical Calculus for Categorified Quantum sl(2)

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*Mikhail Khovanov; Aaron D. Lauda; Marco Mackaay; Marko Stošić*

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in a paper (arXiv:0803.3652) by Aaron D. Lauda. Here the authors enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. They 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. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).