Abstract. We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan-Lusztig polynomials.A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, Θ-networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.

We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke-Clifford algebra (AHCA),

Abstract. We propose a categorical version of the Boson-Fermion correspondence and its twisted version. One can view it as a relative of the Frenkel-Kac-Segal construction of quantum affine algebras.

The trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra W 1+∞ . This induces an action of W 1+∞ on symmetric functions.

No abstract

No abstract

Abstract. We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel-Thomas) twists and are reminiscent of the Rickard complexes defined by Chuang-Rouquier. Conjecturally, one can relate our complexes to Rickard complexes using categorical vertex operators.

We study the density of the Burau representation from the perspective of a nonsemisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature. We prove that the image of the braid group in the space of possibly indefinite unitary representations is dense. We also argue for the potential applications of non-semisimple TQFTs toward topological quantum computation.