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Omni-representations of Leibniz algebras

Zhangju Liu, Yunhe Sheng

Abstract

In this paper, first we introduce the notion of an omni-representation of a Leibniz algebra $\g$ on a vector space $V$ as a Leibniz algebra homomorphism from $\g$ to the omni-Lie algebra $\gl(V)\oplus V$. Then we introduce the omni-cohomology theory associated to omni-representations and establish the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.

Omni-representations of Leibniz algebras

Abstract

In this paper, first we introduce the notion of an omni-representation of a Leibniz algebra on a vector space as a Leibniz algebra homomorphism from to the omni-Lie algebra . Then we introduce the omni-cohomology theory associated to omni-representations and establish the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.
Paper Structure (4 sections, 11 theorems, 41 equations)

This paper contains 4 sections, 11 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Representations of Leibniz algebras
  3. Omni-representations of Leibniz algebras
  4. Omni-cohomologies of Leibniz algebras

Key Result

Theorem 2.2

(BalavoineFialowski) With the above notations, $(C^*(\mathfrak g,\mathfrak g),[\cdot,\cdot]_{\mathsf{B}})$ is a graded Lie algebra.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 13 more