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Nijenhuis BiHom-Lie bialgebras and differential Lie bialgebras

Jiaqi Liu, Lin Gao, Yuanyuan Zhang

Abstract

In this paper, we first introduce the concept of Nijenhuis BiHom-Lie algebras. We then establish the equivalence relations between the Manin triples of Nijenhuis BiHom-Lie algebras, Nijenhuis BiHom-Lie bialgebras, and matched pairs of Nijenhuis BiHom-Lie algebras. Furthermore, we show that such an equivalence also holds for differential Lie bialgebras, together with their associated Manin triples and corresponding matched pairs.

Nijenhuis BiHom-Lie bialgebras and differential Lie bialgebras

Abstract

In this paper, we first introduce the concept of Nijenhuis BiHom-Lie algebras. We then establish the equivalence relations between the Manin triples of Nijenhuis BiHom-Lie algebras, Nijenhuis BiHom-Lie bialgebras, and matched pairs of Nijenhuis BiHom-Lie algebras. Furthermore, we show that such an equivalence also holds for differential Lie bialgebras, together with their associated Manin triples and corresponding matched pairs.
Paper Structure (11 sections, 32 theorems, 108 equations, 1 table)

This paper contains 11 sections, 32 theorems, 108 equations, 1 table.

Table of Contents

  1. Introduction
  2. BiHom-Lie bialgebras
  3. Operators on (generalized) Lie algebras
  4. Manin triples, matched pair and bialgebras structures
  5. Nijenhuis BiHom-Lie algebras and Nijenhuis BiHom-Lie bialgebras
  6. BiHom-Lie algebras and BiHom-Lie coalgebras
  7. Nijenhuis BiHom-Lie algebras and their representations
  8. Manin triples of Nijenhuis BiHom-Lie algebras
  9. Differential Lie algebras and differential Lie bialgebras
  10. Differential Lie algebras and their representations
  11. Differential Lie bialgebras and Manin triples of differential Lie algebras

Key Result

Theorem 1.1

double-B Let $(L,[-,-])$ and $(L^*,[-,-]_{*})$ be two Lie algebras. Then the following conditions are equivalent:

Theorems & Definitions (90)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • ...and 80 more