We describe arbitrary multiplicative differential forms on Lie groupoids infinitesimally, i.e., in terms of Lie algebroid data. This description is based on the study of linear differential forms on Lie algebroids and encompasses many known integration results related to Poisson geometry. We also revisit multiplicative multivector fields and their infinitesimal counterparts, drawing a parallel between the two theories.
Abstract. We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their relation via differentiation and integration. We also show how to extend our techniques to describe the more general Lie theory underlying double Lie algebroids and LA-groupoids.
We report here the unexpected observation of significant room-temperature ferromagnetism in a semiconductor doped with nonmagnetic impurities, Cu-doped TiO 2 thin films grown by pulsed laser deposition. The magnetic moment, calculated from the magnetization curves, resulted surprisingly large, about 1.5 B per Cu atom. A large magnetic moment was also obtained from ab initio calculations, but only if an oxygen vacancy in the nearest-neighbor shell of Cu was present. This result suggests that the role of oxygen vacancies is crucial for the appearance of ferromagnetism. The calculations also predict that Cu doping favors the formation of oxygen vacancies.
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