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

TL;DR

This paper introduces a new structure called $n$-Lie conformal algebra, explores its associated infinite-dimensional $n$-Lie algebras, and establishes their isomorphism criteria, representation, and cohomology theories.

Contribution

It defines $n$-Lie conformal algebras with a $\{\lambda_{1 o n-1} ight\}$-bracket, constructs associated infinite-dimensional $n$-Lie algebras, and develops their representation and cohomology theories.

Findings

Finite $n$-Lie conformal algebras are isomorphic iff their associated $n$-Lie algebras are isomorphic.

Established the representation theory for $n$-Lie conformal algebras.

Developed the cohomology theory and related complexes for these algebras.

Abstract

In this paper, we introduce a $\{\lambda_{1\to n-1}\}$-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 $\{(\mathscr{L}ie_p\mbox{ }R)_\_\}_{(p\ge1)}$. We show that torsionless finite $n$-Lie conformal algebras $R$ and $S$ are isomorphic if and only if $(\mathscr{L}ie_p\mbox{ }R)_\_\simeq (\mathscr{L}ie_p\mbox{ }S)_\_$ as linearly compact $n$-Lie algebras with $\partial_{t_i}$-action for any $p\ge1$. 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 $(\mathscr{L}ie_p\mbox{ }R)_\_$.