Related papers: On monoidal Koszul duality for the Hecke category

We prove in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain…

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived…

We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$-forms of a…

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting…

We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of $K$-theory. For this, we introduce and study…

We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.

The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a…

We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, $S_2$-property and their Bass numbers

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…

We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example…

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan-Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category…

This article introduces the duplex Hecke algebra, which is an infinite dimensional algebra generated by two Hecke algebras. This concept originates from the degenerate duplex Hecke algebra in the theory of Schur-Weyl duality related to…

We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras…

In the early 1990's Andersen, Jantzen and Soergel introduced a category in order to give a combinatorial model for certain representations of quantum groups at a root of unity and simultaneously of Lie algebras of semisimple algebraic…

For a system of non-homogeneous polynomials it was constructed explicit complex morphism of a dual complex to the Koszul complex into the Koszul complex. If the ideal of these polynomials is 0-dimensional, then this mapping is a homotopic…

We investigate the homotopy type of the Alexander dual of a simplicial complex. In general the homotopy type of K does not determine the homotopy type of its dual K*. Moreover, one can construct for each finitely presented group G, a simply…

We classify simple transitive $2$-representations of certain $2$-sub\-ca\-te\-go\-ri\-es of the $2$-category of Soergel bimodules over the coinvariant algebra in Coxeter types $B_2$ and $I_2(5)$. In the $I_2(5)$ case it turns out that…

The aim of this short note is to present a proof of the existence of an $A_\infty$-quasi-isomorphism between the $A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule $K$, introduced in \cite{CFFR}, and the Koszul complex $\mathrm K(V)$ of…

We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of K\'alm\'an.

This lecture reviews the classification of simple modules of double affine Hecke algebras via the K-theory of Steinberg varieties of affine type