Related papers: LIE n-RACKS

In this work we state a result that relates the cohomology groups of a Lie algebra $\mathfrak{g}$ and a current Lie algebra $\mathfrak{g} \otimes \mathcal{S}$, by means of a short exact sequence -- similar to the universal coefficients…

Each choice of a K\"ahler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such K\"ahler classes is given, then we prove that the corresponding…

We first recall two equivalent definitions of Lie $2$-algebras, categorification of Lie algebras and $2$-term $L_\infty$-algebras. Then we present four different kinds of Lie $2$-algebras from $2$-plectic manifolds, Courant algebroids,…

In this paper, we introduce the notion of modular class of a Lie algebroid $A$ equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that…

Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by…

A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$,…

Two types of higher order Lie $\ell$-ple systems are introduced in this paper. They are defined by brackets with $\ell > 3$ arguments satisfying certain conditions, and generalize the well known Lie triple systems. One of the…

We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real…

We introduce and investigate a natural family of metrics on connected components of a rack. The metrics are closely related to certain bi-invariant metrics on the group of inner automorphisms of the rack. We also introduce a bounded…

Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…

We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on…

We establish vanishing results for the first cohomology group of nilpotent groups and Lie rings when the submodule of invariants is trivial. Our results are obtained within a model-theoretic setting, namely for structures that are definable…

The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of K\"ahler differentials. Here we generalize it for simplicial, graded and anticommutative…

The aim of this paper is to extend the notion of bialgebra for Leibniz algebras (and Lie algebras) to $3$-Leibniz algebras (and $3$-Lie algebras) by use of the cohomology complex of $3$-Leibniz algebras. Also, some theorems about Leibniz…

In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for…

We discuss groups corresponding to Kohno Lie algebra of infinitesimal braids and actions of such groups. We construct homomorphisms of Lie braid groups to the group of symplectomorphisms of the space of point configurations in $R^3$ and to…

In this paper, we introduce twisted relative Rota-Baxter operators on a Leibniz algebra as a generalization of twisted Poisson structures. We define the cohomology of a twisted relative Rota-Baxter operator $K$ as the Loday-Pirashvili…

We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground…

Poisson-NIjenhuis structures for an arbitrary Lie agebroid are defined and studied by means of tangent lifts of tensor fields.

This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear…