Corpus ID: 212737155

$L_\infty$-algebra extensions of Leibniz algebras

  title={\$L\_\infty\$-algebra extensions of Leibniz algebras},
  author={Sylvain Lavau and Jim Stasheff},
  journal={arXiv: Mathematical Physics},
Leibniz algebras have been increasingly used in gauging procedures in supergravity. Their relationship with $L_\infty$-algebras and tensor hierarchies have been explored in the physics literature. This paper is devoted to showing that a Leibniz algebra $V$ gives rise to a non-positively graded $L_\infty$-algebra. We call such an $L_\infty$-algebra an '$L_\infty$-extension of the Leibniz algebra $V$' and show that this construction is functorial. We will also use the opportunity of building this… Expand
2 Citations
Differential Graded Lie Algebras and Leibniz Algebra Cohomology
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras andExpand
The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples
In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then weExpand


The Embedding Tensor, Leibniz–Loday Algebras, and Their Higher Gauge Theories
We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebrasExpand
$$L_{\infty }$$L∞ Algebras for Extended Geometry from Borcherds Superalgebras
We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in aExpand
Leibniz Gauge Theories and Infinity Structures
‘infinity-enhanced Leibniz algebras’ are defined that guarantee the existence of consistent tensor hierarchies to arbitrary level and can be used to define topological field theories for which all curvatures vanish. Expand
Infinity-enhancing of Leibniz algebras.
We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies, and differential graded Lie algebras, which have been already usedExpand
Strongly homotopy Lie algebras
The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32,Expand
Algèbres de Leibnitz : définitions, propriétés
— Leibnitz algebras are algebras whose product, denoted [., . ], satisfies a certain form of Jacobi's identity, without any symmetry assumption. So, ail Lie algebras are Leibniz. A (co)homologicalExpand
Universal enveloping algebras of Leibniz algebras and (co)homology
The homology of Lie algebras is closely related to the cyclic homology of associative algebras [LQ]. In [L] the first author constructed a "noncommutative" analog of Lie algebra homology which is,Expand
The tensor hierarchy algebra
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respectExpand
$L_{\infty}$ Algebras and Field Theory
We review and develop the general properties of L∞ algebras focusing on the gauge structure of the associated field theories. Motivated by the L∞ homotopy Lie algebra of closed string field theoryExpand
Manin Triples for Lie Bialgebroids
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket doesExpand