Extended Graphical Calculus for Categorified Quantum Sl(2): 219 (Memoirs of the American Mathematical Society)

Product Description 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. <br><br>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). About the Author <strong>Mikhail Khovanov</strong> is at Columbia University, New York, USA.<br><br>||<strong>Aaron D. Lauda</strong> is at University of Southern California, USA. <br><br>|<strong>Marco Mackaay</strong> is at Universidade do Algarve, Portugal. <br><br>|<strong>Marko Stosic</strong> is at Instituto Superior Tecnico, Portugal. <br><br