Related papers: Free post-groups, post-groups from group actions, …
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with…
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and…
We determine the isomorphism classes of the first family of infinite dimensional simple Lie algebras recently introduced by Xu. The structure space of these algebras is given explicitly. The derivations of these algebras are also…
The expansion method of Lie algebras by a semigroup or S-expansion is generalized to act directly on the group manifold, and not only at the level of its Lie algebra. The consistency of this generalization with the dual formulation of the…
Recently the notion of post-Hopf algebra was introduced, with the universal enveloping algebra of a post-Lie algebra as the fundamental example. A novel property is that any cocommutative post-Hopf algebra gives rise to a sub-adjacent Hopf…
We describe the structure of ``K-approximate subgroups'' of torsion-free nilpotent groups, paying particular attention to Lie groups. Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which independently address related…
We describe, up to degree equal to the rank, the Lie algebra associated with the automorphism group of a free group. We compute in particular the ranks of its homogeneous components, and their structure as modules over the linear group.…
In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UP-semigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right…
We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence…
We investigate the algebraic structure underlying Voiculescu's S-transform in the setting of operator-valued free probability. We show that its twisted factorisation property gives rise to post-groups, crossed morphisms, as well as pre- and…
We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary…
This paper defines the algebraic structure of tracial post-Lie-Rinehart algebras and describes the free object in this category. Post-Lie-Rinehart algebras is a generalisation of pre-Lie-Rinehart algebras, and of post-Lie algebroids.
Hom-Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom-Lie algebras studied further. In the theory of groups,…
In this paper, we characterize the graded post-Lie algebra structures and a class of shifting post-Lie algebra structures on the Witt algebra. We obtain some new Lie algebras and give a class of their modules. As an application, the…
We study post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for nilpotent Lie algebras. First we show that if $\mathfrak{g}$ is nilpotent such that $H^0(\mathfrak{g},\mathfrak{n})=0$, then also $\mathfrak{n}$ must be nilpotent, of…
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…
We study the existence of post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent,…
For a large class of finite-dimensional Lie superalgebras (including the classical simple ones) a Lie supergroup associated to the algebra is defined by fixing the Hopf superalgebra of functions on the supergroup. Then it is shown that on…
We construct the universal enveloping preassociative and postassociative algebra for a pre-Lie and a postLie algebra respectively. We show that the pairs $(\mathrm{preLie},\mathrm{preAs})$ and $(\mathrm{postLie},\mathrm{postAs})$ are…
We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…