A categorification of quantum sl(2)

Details A categorification of quantum sl(2) A categorification of quantum sl(2)

2008-03-26

arxiv.org/abs/0803.3652v3

Advances in Mathematics, Volume 225, Issue 6, 2010, Pages 3327-3424

Mathematics

Quantum Algebra

89 pages, LaTeX2e with xypic and pstricks macros v3. updates notation to match arXiv:0807.3250 and corrects misprints

Scientific paper

10.1016/j.aim.2010.06.003

We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our graphical calculus. Using iterated flag varieties, a representation of the 2-category is constructed for each positive integer N. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).

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