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After introducing the notion of Vinogradov Lie n-algebroids and their canonical Poisson structures, I will show that derived brackets for their algebra of functions give the standard Lie bracket for n=1 and the Courant bracket for n=2. Introducing furthermore basics of double field theory, the case n=2 can be used to understand its gauge structure (in particular the C-bracket) and the strong section condition in a geometric way. If there is still time, expressions for Gualtieri-torsion and curvature for the cases n=1 and n=2 are given.
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