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Deformations and extensions of modified $λ$-differential $3$-Lie Algebras

Wen Teng, Hui Zhang

TL;DR

The work develops a cohomology framework for modified lambda-differential 3-Lie algebras with representations to control deformations and extensions. It defines adjoint and dual representations and builds a cochain complex whose cohomology groups, particularly the second group, govern deformation theory and extension classifications. Infinitesimal deformations correspond to 2-cocycles in the adjoint cohomology, with equivalence classes captured by the second cohomology; Nijenhuis and O-operators are introduced to produce trivial deformations and relate to semidirect products. The paper further provides a cohomology-based classification of abelian and T*-extensions and demonstrates how metrised structures arise from symmetry constraints on the cocycles, enabling construction of metrised extended algebras.

Abstract

In this paper, we introduce the representation of modified $λ$-differential $3$-Lie algebras and define the cohomology of modified $λ$-differential $3$-Lie algebras with coefficients in a representation. As applications of the proposed cohomology theory, we study linear deformations, abelian extensions and $T^*$-extensions of modified $λ$-differential $3$-Lie algebras.

Deformations and extensions of modified $λ$-differential $3$-Lie Algebras

TL;DR

The work develops a cohomology framework for modified lambda-differential 3-Lie algebras with representations to control deformations and extensions. It defines adjoint and dual representations and builds a cochain complex whose cohomology groups, particularly the second group, govern deformation theory and extension classifications. Infinitesimal deformations correspond to 2-cocycles in the adjoint cohomology, with equivalence classes captured by the second cohomology; Nijenhuis and O-operators are introduced to produce trivial deformations and relate to semidirect products. The paper further provides a cohomology-based classification of abelian and T*-extensions and demonstrates how metrised structures arise from symmetry constraints on the cocycles, enabling construction of metrised extended algebras.

Abstract

In this paper, we introduce the representation of modified -differential -Lie algebras and define the cohomology of modified -differential -Lie algebras with coefficients in a representation. As applications of the proposed cohomology theory, we study linear deformations, abelian extensions and -extensions of modified -differential -Lie algebras.
Paper Structure (5 sections, 15 theorems, 55 equations)

This paper contains 5 sections, 15 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Representations and cohomologies of modified $\lambda$-differential $3$-Lie algebras
  3. Deformations of modified $\lambda$-differential 3-Lie algebras
  4. Abelian extensions of modified $\lambda$-differential 3-Lie algebras
  5. $T^*$-extensions of modified $\lambda$-differential 3-Lie algebras

Key Result

Proposition 2.3

Let $(\mathfrak{A}, [-, -, -],\mathrm{d})$ be a modified $\lambda$-differential 3-Lie algebra. Then, $(\wedge^2 \mathfrak{A}, [-,-]_\mathcal{F}, \mathrm{d}_{\mathcal{F}})$ is a Leibniz algebra with a derivation, where $\mathrm{d}_{\mathcal{F}}(a_1\wedge a_2)=\mathrm{d}(a_1)\wedge a_2+a_1\wedge\mathr

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 40 more