Related papers: From pre-Lie rings back to braces
We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which $G$ acts, then the…
In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$ is irreducible. This conjecture remains open, but it…
In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable…
We give a simple characterization of Lie elements in free pre-Lie algebras as elements of the kernel of a map between spaces of trees. We explain how this result is related to natural operations on the Chevalley-Eilenberg complex of a Lie…
In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.
In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study…
A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without…
Thin Lie algebras are Lie algebras L, graded over the positive integers, with all homogeneous components of dimension at most two, and satisfying a more stringent but natural narrowness condition modeled on an analogous one for pro-p…
We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra…
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical $R$-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie…
A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras…
It is shown that over an arbitrary field there exists a nil algebra $R$ whose adjoint group $R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997 [5] and answers related questions from [3, 44]. The case of an…
In this paper, we introduce an element $\phi$-$\delta$-primary to another element in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between…
Novikov algebras provide a simple but powerful algebraic axiomatization of important features of classical diferential calculus. We study their structure properties, modeling their relationships with commutative algebras with a derivation,…
We give an explicit description of the Lie algebra of derivations for a class of infinite dimensional algebras which are given by \'etale descent. The algebras under consideration are twisted forms of central algebras over rings, and…
We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel'd homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms,…
The general theory of the radicals of Lie algebras are established. Baer radicals of untwisted affine Lie algebras are found.
We define a family of multigraded operads $O_\lambda$ depending on a scalar parameter, such that forgetting the multigraduation gives back the pre-Lie operad when the parameter $\lambda$ is equal to one, and the NAP operad governing…
In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and…
On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the…