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

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 $t:\mathbb{W}\to \mathbb{V}$, 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 $\mathfrak{i}$, $\mathbb{V}\cong \mathfrak{g}⋉\mathfrak{i}$, 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.