Dilations of partial representations of Hopf algebras

We introduce the notion of a dilation for a partial representation (i.e. a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (i.e.a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial $H$-modules, a category of (global) $H$-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon $H$, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory.Comment: 25 pages. Corrected several typos, final version to appear in Journal of the London Mathematical Societ

Semi-classical limitations for photon emission in strong external fields

The semi-classical heuristic emission formula of Baier-Katkov [Sov. Phys. JETP \textbf{26}, 854 (1968)] is well-known to describe radiation of an ultrarelativistic electron in strong external fields employing the electron's classical trajectory. To find the limitations of the Baier-Katkov approach, we investigate electron radiation in a strong rotating electric field quantum mechanically using the Wentzel-Kramers-Brillouin approximation. Except for an ultrarelativistic velocity, it is shown that an additional condition is required in order to recover the widely used semi-classical result. A violation of this condition leads to two consequences. First, it gives rise to qualitative discrepancy in harmonic spectra between the two approaches. Second, the quantum harmonic spectra are determined not only by the classical trajectory but also by the dispersion relation of the effective photons of the external field

A comment on BCC crystalization in higher dimensions

The result that near the melting point three-dimensional crystals have an octahedronic structure is generalized to higher flat non compact dimensions