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Cohomologies of modified $λ$-differential Lie triple systems and applications

Wen Teng, Fengshan Long, Yu Zhang

TL;DR

This work develops a cohomology framework for modified $\lambda$-differential Lie triple systems, integrating Lie triple systems with a weight-$\lambda$ differential operator. It defines representations, adjoint and dual structures, and a semidirect product to study modules, and constructs a Yamaguti-based cohomology $\mathcal{H}_{\mathrm{mDLts^\lambda}}^*$ using a cochain map $\Phi$ and the coboundary $\delta$. The theory then governs deformations and abelian extensions through the third cohomology $\mathcal{H}_{\mathrm{mDLts^\lambda}}^3$, with infinitesimals as $3$-cocycles and rigidity/extension classification arising from vanishing or nontrivial cohomology. Overall, the paper generalizes Lie triple systems with derivations to the modified $\lambda$-differential setting, providing a unified approach to deformation and extension problems with substantial algebraic and geometric implications.

Abstract

In this paper, we introduce the concept and representation of modified $λ$-differential Lie triple systems. Next, we define the cohomology of modified $λ$-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $λ$-differential Lie triple systems.

Cohomologies of modified $λ$-differential Lie triple systems and applications

TL;DR

This work develops a cohomology framework for modified -differential Lie triple systems, integrating Lie triple systems with a weight- differential operator. It defines representations, adjoint and dual structures, and a semidirect product to study modules, and constructs a Yamaguti-based cohomology using a cochain map and the coboundary . The theory then governs deformations and abelian extensions through the third cohomology , with infinitesimals as -cocycles and rigidity/extension classification arising from vanishing or nontrivial cohomology. Overall, the paper generalizes Lie triple systems with derivations to the modified -differential setting, providing a unified approach to deformation and extension problems with substantial algebraic and geometric implications.

Abstract

In this paper, we introduce the concept and representation of modified -differential Lie triple systems. Next, we define the cohomology of modified -differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified -differential Lie triple systems.
Paper Structure (5 sections, 13 theorems, 53 equations)

This paper contains 5 sections, 13 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Representations of modified $\lambda$-differential Lie triple systems
  3. Cohomology of modified $\lambda$-differential Lie triple systems
  4. Deformations of modified $\lambda$-differential Lie triple systems
  5. Abelian extensions of modified $\lambda$-differential Lie triple systems

Key Result

Proposition 2.6

Let $(\mathfrak{L}, [\cdot, \cdot, \cdot ])$ be a Lie triple system. Then, a linear operator $d:\mathfrak{L}\rightarrow \mathfrak{L}$ is a modified $\lambda$-differential operator if and only if $d+\frac{\lambda}{2} \mathrm{id}_\mathfrak{L}$ is a derivation on $\mathfrak{L}$.

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • ...and 37 more