Constructs an action of the positive Witt algebra on categorified quantum groups for simply-laced Lie algebras, recovering the foam action in type A and inducing the current-algebra action via trace decategorification.
A categorification of quantum sl(2)
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abstract
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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Action of the Witt algebra on categorified quantum groups
Constructs an action of the positive Witt algebra on categorified quantum groups for simply-laced Lie algebras, recovering the foam action in type A and inducing the current-algebra action via trace decategorification.