We define coadmissible equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces and relate them to admissible locally representations of semisimple $p$-adic Lie groups.

Let H be a finite-dimensional quasibialgebra. We show that H is a quasi-Hopf algebra if and only if the monoidal category of its finite-dimensional left modules is rigid, if and only if a structure theorem for Hopf modules over H holds. We also show that a dual structure theorem for Hopf modules over a coquasibialgebra H holds if and only if the category of finite-dimensional right H-comodules ...

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding semisim-ple simply connected complex algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

By providing a general correspondence between Landau-Ginzburg orbifolds and nonlinear sigma models, we find that the elusive mirror of a rigid manifold is actually a supermanifold. We also discuss when sigma models with super-target spaces are conformally invariant and describe their chiral rings. Both supermanifolds with and without Kähler moduli are considered. This work leads us to conclude ...

We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. show that ICE-closed over the path algebra Dynkin type are in bijection with basic rigid modules precisely some also study natural maps from to terms modules. Finally, we prove number does not depend on orientation quiver, give an explicit...