Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Levi-Civita products of Leibniz algebras with nondegenerate skew-symmetric 2-cocycles

Quan Zhao, Guilai Liu

TL;DR

This work investigates Levi-Civita products on a Leibniz algebra equipped with a nondegenerate skew-symmetric $2$-cocycle, showing these products organize as an anti-pre-Leibniz algebra. It provides a precise criterion for when a compatible anti-pre-Leibniz structure exists—namely the existence of an invertible anti-$\mathcal{O}$-operator—and develops dual-representation and doubled-space constructions with a natural $2$-cocycle. The paper then recharacterizes Novikov dialgebras as admissible anti-pre-Leibniz algebras arising from transformed pre-Leibniz algebras and situates these within the broader landscape of GD dialgebras and affinization. Overall, it unifies operator, representation, and dialgebra perspectives to advance the theory of Leibniz-type algebras and their nondegenerate bilinear forms.

Abstract

This paper studies the associated Levi-Civita products of a Leibniz algebra with a nondegenerate skew-symmetric $2$-cocycle. Such products form into the notion of an anti-pre-Leibniz algebra, which is characterized as a Leibniz-admissible algebra which renders a representation of the sub-adjacent Leibniz algebra through the negative multiplication operators. Such a characterization serves as the converse side of the role that pre-Liebniz algebras play in the splitting theory of Leibniz algebras, which justifies the name of anti-pre-Leibniz algebras. There is a compatible anti-pre-Leibniz algebra structure on a Leibniz algebra if and only if there is an invertible anti-$\mathcal{O}$-operator of the Leibniz algebra. Another important role that anti-pre-Leibniz algebras play is that they give a new characterization of Novikov dialgebras, that is, a Novikov dialgebra is interpreted as a transformed pre-Leibniz algebra which gives rise to an anti-pre-Leibniz algebra structure through specific combinations of multiplications. The properties of Novikov dialgebras are also further investigated.

The Levi-Civita products of Leibniz algebras with nondegenerate skew-symmetric 2-cocycles

TL;DR

This work investigates Levi-Civita products on a Leibniz algebra equipped with a nondegenerate skew-symmetric -cocycle, showing these products organize as an anti-pre-Leibniz algebra. It provides a precise criterion for when a compatible anti-pre-Leibniz structure exists—namely the existence of an invertible anti--operator—and develops dual-representation and doubled-space constructions with a natural -cocycle. The paper then recharacterizes Novikov dialgebras as admissible anti-pre-Leibniz algebras arising from transformed pre-Leibniz algebras and situates these within the broader landscape of GD dialgebras and affinization. Overall, it unifies operator, representation, and dialgebra perspectives to advance the theory of Leibniz-type algebras and their nondegenerate bilinear forms.

Abstract

This paper studies the associated Levi-Civita products of a Leibniz algebra with a nondegenerate skew-symmetric -cocycle. Such products form into the notion of an anti-pre-Leibniz algebra, which is characterized as a Leibniz-admissible algebra which renders a representation of the sub-adjacent Leibniz algebra through the negative multiplication operators. Such a characterization serves as the converse side of the role that pre-Liebniz algebras play in the splitting theory of Leibniz algebras, which justifies the name of anti-pre-Leibniz algebras. There is a compatible anti-pre-Leibniz algebra structure on a Leibniz algebra if and only if there is an invertible anti--operator of the Leibniz algebra. Another important role that anti-pre-Leibniz algebras play is that they give a new characterization of Novikov dialgebras, that is, a Novikov dialgebra is interpreted as a transformed pre-Leibniz algebra which gives rise to an anti-pre-Leibniz algebra structure through specific combinations of multiplications. The properties of Novikov dialgebras are also further investigated.

Paper Structure

This paper contains 3 sections, 22 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Anti-pre-Leibniz algebras and nondegenerate skew-symmetric 2-cocycles on Leibniz algebras
  3. Admissible Novikov dialgebras as a subclass of anti-pre-Leibniz algebras

Key Result

Proposition 2.2

Let $(A,\circ_{A})$ be a Leibniz algebra, $V$ be a vector space and $l_{\circ_{A}},r_{\circ_{A}}:A\rightarrow \mathrm{End}(V)$ be linear maps. Then the triple $(l_{\circ_{A}},r_{\circ_{A}},V)$ is a representation of the $(A,\circ_{A})$ if and only if there is a Leibniz algebra structure on the direc We denote the Leibniz algebra $(A\oplus V,\circ_{d})$ by $A\ltimes_{l_{\circ_{A}},r_{\circ_{A}}}V$.

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Definition 2.8
  • ...and 45 more