Related papers: Strongly homotopy Lie bialgebras and Lie quasi-bia…

We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras,…

In this paper,we will show how to kill the obstructions to Lie algebra deformations via a method which essentially embeds a Lie algebra into Strong homotopy Lie algebra or $L_{\infty}$ algebra. All such obstructions have been transfered to…

The aim of this article is to give a quantization of some coisotropic subalgebras in complex semisimple Lie bialgebras. The coisotropic subalgebras that will be quantized are those given by Zambon in his paper "`A Construction for…

We study Maurer-Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We…

We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an…

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary…

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media -…

We introduce a new type of algebra, which is called a Lie-Leibniz algebra. This concept is an abstraction of derived bracket construction. It will be proved that the operad of Lie-Leibniz algebras is Koszul. The strong homotopy version of…

In the present paper we shall investigate the Lie bialgebra structures on the Lie algebra $\widetilde{\frak{sl}_2(C_q[x,y])}$, which are shown to be triangular coboundary.

It is shown that any compact K\"ahler manifold $M$ gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the…

Lie $n$-algebras are the $L_\infty$ analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite type Lie $n$-algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie…

A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic…

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent \Delta operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized…

Recently, relative Rota-Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota-Baxter Leibniz algebras (rRB Leibniz algebras) as the object of…

We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with a reductive decomposition, we extend the description of the moduli space of…

The purpose of this memoir is to study pre-Lie algebras up to homotopy with divided powers, and to use this algebraic structure for the study of mapping spaces in the category of operads. We define a new notion of algebra called…

In this paper, we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras. Explicitly, we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan…

We introduce a notion of a para-K\"{a}hler strict Lie 2-algebra, which can be viewed as a categorification of a para-K\"{a}hler Lie algebra. In order to study para-K\"{a}hler strict Lie 2-algebra in terms of strict pre-Lie 2-algebras, we…

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative)…

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients…