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Ordinary Lie algebra cohomology of a Lie algebra g has a nice reformulation in terms of the Koszul dual differential algebra of the Lie 2-algebra of inner derivations of g. For every transgressive degree n element in g-cohomology there is a short exact sequence of Lie n-algebras. These are characterized by the fact that n-connections taking values in them come from the corresponding Chern-Simons forms and characteristic classes. A straightforward generalization of this construction yields a notion of cohomology, invariant polynomials and transgression elements for arbitrary Lie n-algebras. And in turn, each such element of degree d induces a new Lie max(n, d)-algebra. From the invariant polynomials of a Lie n-algebra one obtains characteristic classes of the corresponding n-bundles

Year: 2007

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