Categorification of Quantum Groups

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Building off geometric constructions of Beilinson, Lusztig, and MacPherson, the PI in collaboration with Mikhail Khovanov categorified quantum sl(n), as well as the quantum deformation of the universal enveloping algebra of the ``lower-triangular'' subalgebra for an arbitrary Kac-Moody Lie algebra. While these categorifications are completely combinatorial, they utilize various diagrammatic calculi emphasizing a new interplay between topology and algebra. The proposal aims to further develop the theory of categorified quantum groups, study their Hopf structure (comultiplication and antipode), further develop their representation theory, categorify the entire quantum enveloping algebra for other Kac-Moody Lie algebras, and understand the relationship to geometric representation theory. Potential applications of categorified of quantum groups include the study of positivity properties for quantum enveloping algebras, a representation theoretic explanation of Khovanov homology and a categorification of the Witten-Reshetikhin-Turaev quantum 3-manifold invariants.Quantum groups are prevalent throughout mathematics and theoretical physics. These Hopf algebras provide the representation theoretic framework for understanding quantum link invariants such as the Jones polynomial, Kauffman and HOMFLY-PT polynomial, as well as the Witten-Reshetikhin-Turaev quantum 3-manifold invariants. They also relate to statistical mechanics, quantum field theory, and affine Lie algebras. Recent work suggests that the representation theory of quantum groups, and the quantum groups themselves, are shadows of a much richer algebraic structure known as categorified quantum groups. These structures were conjectured to exist by Crane and Frenkel and form the focal point of this proposal.