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Manin triples associated to $n$-Lie bialgebras

Ying Chen, Chuangchuang Kang, Jiafeng Lü, Shizhuo Yu

Abstract

In this paper, we study the Manin triples associated to $n$-Lie bialgebras. We introduce the concept of operad matrices for $n$-Lie bialgebras. In particular, by studying a special case of operad matrices, it leads to the notion of local cocycle $n$-Lie bialgebras. Furthermore, we establish a one-to-one correspondence between the double of $n$-Lie bialgebras and Manin triples of $n$-Lie algebras.

Manin triples associated to $n$-Lie bialgebras

Abstract

In this paper, we study the Manin triples associated to -Lie bialgebras. We introduce the concept of operad matrices for -Lie bialgebras. In particular, by studying a special case of operad matrices, it leads to the notion of local cocycle -Lie bialgebras. Furthermore, we establish a one-to-one correspondence between the double of -Lie bialgebras and Manin triples of -Lie algebras.
Paper Structure (10 sections, 10 theorems, 94 equations)

This paper contains 10 sections, 10 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminary results on $n$-Lie algebras
  3. $n$-Lie bialgebras
  4. $n$-Lie algebra cohomology
  5. $n$-Lie bialgebras
  6. The coadjoint representation of $n$-Lie algebras
  7. The dual of $n$-Lie bialgebras
  8. The double of $n$-Lie bialgebras.
  9. Local cocycle $n$-Lie bialgebras
  10. Manin triples of $n$-Lie algebras and the double of $n$-Lie bialgebras.

Key Result

Proposition 2.3

Let $\mathfrak g$ be an $n$-Lie algebra, then for all $x_1,\cdots,x_{n-1},y_1,\cdots,y_{n}\in \mathfrak g$,

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 35 more