Related papers: Odd Grassmannian bimodules and derived equivalence…
We show that actions of the odd categorification of sl(2) induce derived superequivalences analogous to those introduced by Chuang and Rouquier. Using Kang, Kashiwara, and Oh's action of the odd 2-category on blocks of the cyclotomic affine…
We define a 2-category that categorifies the covering Kac-Moody algebra for sl(2) introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category…
We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for…
This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We…
In this paper we prove that Brou\'{e}'s abelian defect group conjecture is true for the finite odd-dimensional orthogonal groups $\SO_{2n+1}(q)$ at linear primes with $q$ odd. We first make use of the reduction theorem of…
We develop the rewriting theory for monoidal supercategories and 2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for…
We extend the covering of even and odd Khovanov link homology to tangles, using arc algebras. For this, we develop the theory of quasi-associative algebras and bimodules graded over a category with a 3-cocycle. Furthermore, we show that a…
This is the second paper of a series (started by arXiv:0711.2083) which describes a conjectural analog of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to…
We categorify a coideal subalgebra of the quantum group of $\mathfrak{sl}_{2r+1}$ by introducing a $2$-category \`a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this…
We show that the bigroupoid of separable symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of finitely semi-simple Calabi-Yau categories are equivalent. To this end, we construct a trace on the category of…
We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions:…
This is a companion paper of arXiv:1909.11492 and arXiv:1912.01930. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of twisted $Sp(2n,{\mathbb C}[\![t]\!])$-equivariant…
We equip Ellis and Brundan's version of the odd categorified quantum group for sl(2) with a differential giving it the structure of a graded dg-2-supercategory. The presence of the super grading gives rise to two possible…
We use the theta correspondence to study the equivalence between Godement-Jacquet and Jacquet-Langlands L-functions for $\mathrm{GL}(2)$. We show that the resulting comparison is in fact an exotic symmetric monoidal structure on the…
We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as…
In their 1988 paper "Gluing of perverse sheaves and discrete series representations," D. Kazhdan and G. Laumon constructed an abelian category $\mathcal{A}$ associated to a reductive group $G$ over a finite field with the aim of using it to…
Khovanov-Lauda define a 2-category $\mathcal{U}$ such that the split Grothendieck group $K_0(\mathcal{U})$ is isomorphic to an integral version of the quantized universal enveloping algebra $\mathbf{U}(\mathfrak{sl}_n)$, $n \geq 2$.…
We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was…
We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ of internal $2$-groupoids is a Birkhoff subcategory of the category $ \mathrm{Grpd}^2(\mathscr{C}) $ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite…
Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups $\mathrm{SCG}$ are…