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We develop the categorical context for defining Hermitian non-semisimple TQFTs. We prove that relative Hermitian modular categories give rise to modified Hermitian WRT-TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev-Viro TQFT is applied to define new pseudo-Hermitian topological phases that can be considered as non-semisimple analogs of Levin-Wen models., 39 pages, tikz figures

We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations of the mapping class group in the group of indefinite unitary matrices., 18 pages

We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic $p$ descends to an invariant in the homotopy category finite-dimensional $p$-complexes. A $p$-extended differential on the triply graded homology discovered by Cautis is compatible with the $p$-DG structure. As a consequence we get a categorification of the Jones polynomial evaluated at an odd prime root of unity, 53 pages. V2 contains corrections from referee reports. Comments welcome

We construct a braid group action on a homotopy category of $p$-DG modules of a deformed Webster algebra., Comment: 36 pages. Multiple diagrammatic computations. V2 contains corrections from referee reports. Comments welcome

The trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra W_{1+\infty}. This induces an action of W_{1+\infty} on symmetric functions., Comment: 31 pages, tikz diagrams

We categorify a quantized twisted Heisenberg algebra associated to a finite subgroup of SL ( 2 , C ) .

We describe a collection of differential graded rings that categorify weight spaces of the positive half of the quantized universal enveloping algebra of the Lie superalgebra g l ( m | 1 ) gl(m|1) .

Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson-Seed arXiv:1303.6240 and Gorsky-Hogancamp arXiv:1712.03938 to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in arXiv:1410.7156 to categorified quantum groups (this was the original motivation for this paper)., 29 pages, tikz figures

We formulate a relative, representation theoretic, notion of the algebraic cone construction. This motivates a generalization of the cone corresponding to a preprojective algebra.

We construct a 2-representation categorifying the symmetric Howe representation of $\mathfrak{gl}_m$ using a deformation of an algebra introduced by Webster. As a consequence, we obtain a categorical braid group action taking values in a homotopy category., Comment: 71 pages, tikz diagrams. v2 corrects typos

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., 35 pages

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at $\sigma = \bar\sigma^{-1} = q$. A key step in the proof may be of independent interest: we show that the sum (over $n$) of the Hochschild homologies of the positive affine Hecke algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the $q$-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes., Comment: 49 pages, numerous figures

We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke–Clifford algebra (AHCA), \({\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}\). Our construction includes an analogue of Zelevinsky’s segment representations, a complete combinatorial description of the simple calibrated \({\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}\)-modules, and a classification of the simple integral \({\mathcal{H}_{\mathcal{C}\ell}^{{\rm aff}}(d)}\)-modules. Our main tool is an analogue of the Arakawa–Suzuki functor for the Lie superalgebra \({\mathfrak{q}(n)}\).

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.

We survey some recent results in hopfological algebra and the program of categorification at prime roots of unity. A categorical Jones-Wenzl projector at prime roots of unity is studied, and it is shown that this projector is hopfologically finite in special characteristics, while generically it is infinite., Dedicated to Christian Blanchet on the occasion of his sixtieth birthday

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., 20 pages

We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra used by Khovanov and Seidel to construct a categorical braid group action. We show there is a braid group action in this p-DG setting., Comment: 33 pages, many PSTricks figures, color viewing not essential, v2 contains minor corrections, comments welcome

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, Theta-networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants, \cite{TV}. 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., Several minor revisions throughout the paper. Titel shortened. Expanded section on the categorified colored unknot. The paper gives an important application of the notion of completed Grothendieck groups introduced in arXiv:1105.2715