n -Lie conformal algebras and its associated infinite-dimensional n -Lie algebras

In this paper, we introduce a {λ1→n−1}-bracket and a distribution notion of an n-Lie conformal algebra. For any n-Lie conformal algebra R, there exists a series of associated infinite-dimensional linearly compact n-Lie algebras {(Liep R)_}(p≥1). We show that torsionless finite n-Lie conformal algebras R and S are isomorphic if and only if (Liep R)_≃(Liep S)_ as linearly compact n-Lie algebras with ∂ti-action for any p≥1. Moreover, the representation and cohomology theory of n-Lie conformal algebras are established. In particular, the complex of R is isomorphic to a subcomplex of n-Lie algebra (Liep R)_.