Koszul duality for extension algebras of standard modules

We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category O, and some quasi-hereditary algebras with Cartan decomposition in the sense of König.     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

Diagonals of separately pointwise Lipschitz mappings

It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X ²? →?Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X?×?Y?→?Z, where X, Y, and Z are metric spaces, is investigated.     

Direct Photocontrol of Peptidomimetics: An Alternative to Oxygen‐Dependent Photodynamic Cancer Therapy

Conventional photodynamic treatment strategies are based on the principle of activating molecular oxygen in situ by light, mediated by a photosensitizer, which leads to the generation of reactive oxygen species and thereby causes cell death. A diarylethene‐derived peptidomimetic is presented that is suitable for photodynamic cancer therapy without any involvement of oxygen. This light‐sensitive molecule is not a mediator but is itself the cytotoxic agent. As a derivative of the cyclic amphiphilic peptide gramicidin S, the peptidomimetic exists in two thermally stable photoforms that are interconvertible by light of different wavelengths. The isomer generated by visible light shows much stronger toxicity against tumor cells than the UV‐generated isomer. First in vivo applications are demonstrated on a tumor animal model to illustrate how the peptidomimetic can be administered in the less toxic form and then activated locally in a solid tumor by visible light.