Differential Structures And Cohomology Of An N-Lie Algebroid

The notion of n-Lie algebroids is a generalization of n-Lie algebra and of the tangent bundle of a manifold.We mainly introduce differential structures and cohomology of an n-Lie algebroid.In this paper,we study basic properties of an n-Lie algebroid and give the definition of the exterior algebra of sections of n-Lie algebroids.We also study the theory of Lie derivatives and exterior derivatives in the general setting of an n-Lie algebroid,which is a generalization of Lie derivatives and exterior derivatives of sections of the exterior powers of a tangent bundle and of its dual.We prove that for each smooth section V ∈ A1(M,∧^n-1E)of an n-Lie algebroid,there exists a unique graded endomorphism of degree 0 of both the graded algebra of exterior forms ?(M,∧^n-1E)and A(M,∧^n-1E),called the Lie derivative with respect to V and denoted by La（V）.On the other hand,we see that the ?(M,∧^n-1E)-valued exterior derivative extends as a graded endomorphism of degree 1 of the graded algebra of ?(M,∧^n-1E).In fact,we will see that it is a derivation of degree 1 of ?(M,∧^n-1E).Moreover,it is a coboundary operator.Finally we introduce the representation and cohomology of n-Lie algebroids.