n-Lie bialgebras

TL;DR

This paper explores the structure and classification of n-Lie bialgebras, including their definitions, extensions, and specific cases like simple n-Lie algebras, providing new insights into their algebraic properties.

Contribution

It introduces the concept of n-Lie bialgebras, investigates their structure via constants, and classifies bialgebra structures on simple n-Lie algebras.

Findings

n-Lie bialgebras are characterized by conformal 1-cocycle conditions.

Two-dimensional extensions of finite-dimensional n-Lie bialgebras are constructed.

Only rank zero and rank two bialgebra structures exist on simple n-Lie algebra A_n.

Abstract

The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, \mu, \Delta)$ is an $n$-Lie bialgebra if and only if $\Delta$ is a conformal $1$-cocycle on the $n$-Lie algebra $L$ associated to $L$-modules $(L^{\otimes n}, \rho_s^{\mu})$, $1\leq s\leq n$, and the structure of $n$-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional $n$-Lie bialgebras are studied. For an $m$ dimensional $n$-Lie bialgebra $(L, \mu, \Delta)$, and an $ad_{\mu}$-invariant symmetric bilinear form on $L$, the $m+2$ dimensional $(n+1)$-Lie bialgebra is constructed. In the last section, the bialgebra structure on the finite dimensional simple $n$-Lie algebra $A_n$ is discussed. It is proved that only bialgebra structures on the simple $n$-Lie algebra $A_n$ are rank zero, and rank two.