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Cohomologies, non-abelian extensions and Wells sequences of lambda-weighted Rota-Baxter Lie coalgebras

Qinxiu Sun, Zhen Li

Abstract

In this paper, we investigate cohomologies and non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras. First, we consider Lie comodules and cohomologies of lambda-weighted Rota-Baxter Lie coalgebras. Next, we study non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras and classify the non-abelian extensions in terms of non-abelian cohomology group. Furthermore, we explore extensibility of a pair of automorphisms about a non-abelian extension of lambda-weighted Rota-Baxter Lie coalgebras, and derive the fundamental sequences of Wells in the context of lambda-weighted Rota-Baxter Lie coalgebras. Finally, we discuss the previous results in the case of abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras.

Cohomologies, non-abelian extensions and Wells sequences of lambda-weighted Rota-Baxter Lie coalgebras

Abstract

In this paper, we investigate cohomologies and non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras. First, we consider Lie comodules and cohomologies of lambda-weighted Rota-Baxter Lie coalgebras. Next, we study non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras and classify the non-abelian extensions in terms of non-abelian cohomology group. Furthermore, we explore extensibility of a pair of automorphisms about a non-abelian extension of lambda-weighted Rota-Baxter Lie coalgebras, and derive the fundamental sequences of Wells in the context of lambda-weighted Rota-Baxter Lie coalgebras. Finally, we discuss the previous results in the case of abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras.
Paper Structure (7 sections, 26 theorems, 116 equations)

This paper contains 7 sections, 26 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Comodules of $\lambda$-weighted Rota-Baxter Lie coalgebras
  3. Cohomologies of $\lambda$-weighted Rota-Baxter Lie coalgebras
  4. Non-abelian extensions of $\lambda$-weighted Rota-Baxter Lie coalgebras
  5. Extensibility of a pair of automorphisms
  6. Wells exact sequences for $\lambda$-weighted Rota-Baxter Lie coalgebras
  7. Particular case: abelian extensions of $\lambda$-weighted Rota-Baxter Lie coalgebras

Key Result

Proposition 2.1

Let $(C,\Delta_C,R_C)$ be a $\lambda$-weighted Rota-Baxter Lie coalgebra. Then $(C^*,[\ ,\ ]_{C^*},R_{C}^*)$ is a $\lambda$-weighted Rota-Baxter Lie algebra.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 50 more