Electrophotocatalytic Undirected C−H Trifluoromethylations of (Het)Arenes

Electrophotochemistry has enabled arene C−H trifluoromethylation with the Langlois reagent CF₃SO₂Na under mild reaction conditions. The merger of electrosynthesis and photoredox catalysis provided a chemical oxidant-free approach for the generation of the CF₃ radical. The electrophotochemistry was carried out in an operationally simple manner, setting the stage for challenging C−H trifluoromethylations of unactivated arenes and heteroarenes. The robust nature of the electrophotochemical manifold was reflected by a wide scope, including electron-rich and electron-deficient benzenes, as well as naturally occurring heteroarenes. Electrophotochemical C−H trifluoromethylation was further achieved in flow with a modular electro-flow-cell equipped with an in-operando monitoring unit for on-line flow-NMR spectroscopy, providing support for the single electron transfer processes.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

On a conjecture of Vorst

Let k be an infinite perfect field of positive characteristic such that strong resolution of singularities holds over k. We prove that a localization of a d-dimensional commutative k-algebra R of finite type is K _d+1-regular if and only if it is regular. This partially affirms a conjecture of Vorst.     

Hopf algebroids and H-separable extensions

Since an H-separable extension is of depth two, we associate to it dual bialgebroids and over the centralizer as in Kadison-Szlachányi. We show that has an antipode and is a Hopf algebroid. is also Hopf algebroid under the condition that the centralizer is an Azumaya algebra over the center of . For depth two extension , we show that .

     

Boundary and Lens Rigidity of Lorentzian Surfaces

Let be a Lorentzian metric on the plane that agrees with the standard metric outside a compact set and so that there are no conjugate points along any time-like geodesic of . Then and are isometric. Further, if and are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of maximize the distances between their points and and are ``boundary isometric', then there is a conformal diffeomorphism between and and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

     

A jump operator on honest subrecursive degrees

It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let and denote respectively the join and the meet of the degrees and . This paper introduces a jump operator () on the honest elementary degrees and defines canonical degrees and low and high degrees analogous to the corresponding concepts for the Turing degrees. Among others, the following results about the structure of the honest elementary degrees are shown: There exist low degrees, and there exist degrees which are neither low nor high. Every degree above is the jump of some degree, moreover, for every degree above there exist degrees such that . We have and . The jump operator is of course monotonic, i.e. . We prove that every situation compatible with is realized in the structure, e.g. we have incomparable degrees such that and incomparable degrees such that etcetera. We are able to prove all these results without the traditional recursion theoretic constructions. Our proof method relies on the fact that the growth of the functions in a degree is bounded. This technique also yields a very simple proof of an old result, namely that the structure is a lattice. Received October 4, 1995