A categorification of quantum sl(2)

We categorify Lusztig's — a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to the 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 . Graded lifts of various homomorphisms and antihomomorphisms of arise naturally in the context of our graphical calculus. For each positive integer N a representation of is constructed using iterated flag varieties that categorifies the irreducible (N+1)-dimensional representation of .