In this thesis, we examine three different versions of "categorification" of the affine Hecke algebra and its periodic module: the first is by equivariant coherent sheaves on the Grothendieck resolution (and related objects), the second is by certain classes on bimodules over polynomial rings, called Soergel bimodules, and the third is by certain categories of constructible sheaves on the affin...

Abstract Answering a question of Hatcher and Vogtmann, we prove that the top homology group free factor complex is not dualizing module for $\operatorname {Aut}(F_{n})$, at least $n = 5$.

Throughout this paper k denotes an algebraically closed field, and all algebras considered here are assumed to be basic, connected, finite-dimensional k-algebras with identity. For an algebra Λ, we denote by modΛ and by modΛ the category of finitedimensional (right) Λ-modules and its stable category, respectively. Recall that for two algebras Λ and Π, an equivalence modΛ → modΠ is called a stab...

We provide an elementary introduction to the (characteristic zero) theory of Letterplace Superalgebras, regarded as bimodules with respect to the superderivation actions of a pair of general linear Lie superalgebras, and discuss some applications.

Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.

We study coactions of concrete Hopf C -bimodules in the framework of (weak) C pseudo-Kac systems, define reduced crossed products and dual coactions, and prove an analogue of Baaj-Skandalis duality.

Bisets can be considered as categories. This note uses this point of view to give a simple proof of a Mackey-like formula expressing the tensor product of two induced bimodules. AMS subject classification (2000) : 16D20, 20C20.

We try to determine the partition function of N = 4 ADE gauge theory on K3 by self-dualizing our previous ADE partition function. The resulting partition function satisfies gap condition.

Fibrations of flux backgrounds by supersymmetric cycles are investigated. For an internal sixmanifold M with static SU(2) structure and mirror M̂ , it is argued that the product M×M̂ is doubly fibered by supersymmetric three-tori, with both sets of fibers transverse to M and M̂ . The mirror map is then realized by T-dualizing the fibers. Mirror-symmetric properties of the fluxes, both geometric an...