# Enhanced Leibniz Algebras: Structure Theorem and Induced Lie 2-Algebra

## TL;DR

This paper introduces enhanced Leibniz algebras, proves a structure theorem relating them to underlying Leibniz algebras and cohomology classes, and constructs a functor to Lie 2-algebras, with implications for higher gauge theories.

## Contribution

It provides a structure theorem for enhanced Leibniz algebras and establishes a functor to Lie 2-algebras, linking algebraic structures to higher gauge theory applications.

## Key findings

- Enhanced Leibniz algebras are determined by a Leibniz algebra, an abelian ideal, and a cohomology class.
- Positive quadratic enhanced Leibniz algebras require the underlying algebra to be a hemisemidirect product.
- A functor from enhanced Leibniz algebras to Lie 2-algebras is constructed.

## Abstract

An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, a bilinear form on $\mathbb{V}$ with values in another vector space $\mathbb{W}$, and a map $t \colon \mathbb{W} \to \mathbb{V}$, satisfying altogether four compatibility relations. Our structure theorem asserts that an enhanced Leibniz algebra is uniquely determined by the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, an appropriate abelian ideal ${\mathfrak i}$ inside it, as well as a cohomology 2-class $[\Delta]$ which only effects the $\mathbb{W}$-valued product. Positive quadratic enhanced Leibniz algebras, as needed for the definition of a Yang-Mills type action functional, turn out to be rather restrictive on the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \dot ])$: $\mathbb{V}$ has to be the hemisemidirect product of a positive quadratic Lie algebra ${\mathfrak g}$ with a ${\mathfrak g}$-module ${\mathfrak i}$, $\mathbb{V} \cong {\mathfrak g}\ltimes{\mathfrak i}$, with ${\mathfrak i}$ the above-mentioned ideal in this case. The second main result of this article is the construction of a functor from the category of such enhanced Leibniz algebras to the category of (semi-strict) Lie 2-algebras or, equivalentely, of two-term $L_\infty$-algebras.