post-Lie Algebra Research Articles

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

Derivations and biderivations of the non-square linear Lie algebra gl(m×n, 𝔽)

Let m, n be integers with m > n . Denote M m , n by the set of all m × n matrices over the field F of characteristic zero. Let I be an n × m matrix with ( i , i ) -position 1 for any 1 ≤ i ≤ n , and 0 in other position. Define a bracket [ A , B ] = AIB − BIA , where A , B ∈ M m , n . Then M m , n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl ( m × n , F ) . In this paper, all derivations and biderivations of gl ( m × n , F ) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl ( m × n , F ) are given.

Post-Lie algebra structures for perfect Lie algebras

We study the existence of post-Lie algebra structures on pairs of Lie algebras ( g , n ) , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect non-semisimple, and n is s l 3 ( C ) . We also show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect and n is reductive with a 1-dimensional center.

Algebraic aspects of connections: From torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials

Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work, we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a D -algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations among the post-Lie Magnus expansion, the Grossman–Larson product, and the K -map, \alpha -map, and \beta -map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker–Campbell–Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds.

Post-Hopf algebras, relative Rota–Baxter operators and solutions to the Yang–Baxter equation

In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman–Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang–Baxter equation. Then, we introduce the notion of relative Rota–Baxter operator on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota–Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota–Baxter operator also induces a post-Hopf algebra. Finally, we show that relative Rota–Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota–Baxter operators give solutions to the Yang–Baxter equation in certain cocommutative Hopf algebras.

Biderivations of a Lie algebra of Block type and applications

Let be a Lie algebra of Block type with basis and relations . In the present paper, the biderivations of are explicitly described. In particular, it is shown that any biderivation of is inner. As applications, the linear commuting maps and commutative post-Lie algebra structures on this algebra are given.

Homotopy Rota–Baxter operators and post-Lie algebras

Rota–Baxter operators and the more general \mathcal{O} -operators, together with their interconnected pre-Lie and post-Lie algebras, are important algebraic structures, with Rota–Baxter operators and pre-Lie algebras instrumental in the Connes–Kreimer approach to renormalization of quantum field theory. This paper introduces the notions of a homotopy Rota–Baxter operator and a homotopy \mathcal{O} -operator on a symmetric graded Lie algebra. Their characterization by Maurer–Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy \mathcal{O} -operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer–Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.

Post-Lie algebras in Regularity Structures

Abstract In this work, we construct the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. We show that this can be done using either of the two combinatorial structures that have been proposed in the context of singular SPDEs: decorated trees and multi-indices. Our construction is inspired from multi-indices where the Hopf algebra was obtained as the universal envelope of a Lie algebra, and it has been proved that one can find a basis that is symmetric with respect to certain elements. We show that this Lie algebra comes from an underlying post-Lie structure.

Certain lie algebraic structures on Riemannian manifolds with semi-symmetric non-metric connection

As a natural consequence of the Levi-Civita connection on a Riemannian manifold, there is a Lie algebra structure on a Riemannian manifold. Lie Algebras and Lie Groups are the mathematical structure of continuous symmetries in physics. In this paper, semi-symmetric non-metric connection is considered instead of Levi-Civita connection of Riemann manifold, and accordingly the existence of algebraic structures is investigated. First, it is shown that there is not always a Lie algebra structure on a Riemannian manifold with a semi-symmetric non-metric connection. Then, necessary and sufficient conditions for Lie admissible algebra, pre-Lie algebra and post Lie algebra on a Riemann manifold with semi-symmetric non-metric connection are obtained depending on geometric terms. In addition, the cases of the Riemannian manifold with such algebraic structures according to the semi-symmetric non-metric connection being Einstein manifold and being flat manifold have been also investigated.

Rigidity results for Lie algebras admitting a post-Lie algebra structure

We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. We show that if [Formula: see text] is semisimple and [Formula: see text] is arbitrary, then we have rigidity in the sense that [Formula: see text] and [Formula: see text] must be isomorphic. The proof uses a result on the decomposition of a Lie algebra [Formula: see text] as the direct vector space sum of two semisimple subalgebras. We show that [Formula: see text] must be semisimple and hence isomorphic to the direct Lie algebra sum [Formula: see text]. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras [Formula: see text]. We prove additional existence results for pairs [Formula: see text], where [Formula: see text] is complete, and for pairs, where [Formula: see text] is reductive with [Formula: see text]-dimensional center and [Formula: see text] is solvable or nilpotent.

Derivations and biderivations of the n-th Schrödinger algebra

The n-th Schrödinger algebra is the semi-direct of the Lie algebra with the n-th Heisenberg Lie algebra . In this paper, all derivations and biderivations of the n-th Schrödinger algebra are determined. As applications, all linear commuting maps and commutative post-Lie algebra structures on are obtained.

Derivations and Biderivations of the Schrödinger algebra in (n + 1)-dimensional space-time

In this paper, all derivations of the Schrödinger algebra in -dimensional space-time of the Schrödinger Lie group are determined. As applications, all biderivations are obtained. Finally, as applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on this algebra are given.

Simply transitive NIL-affine actions of solvable Lie groups

Abstract Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻. So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action ρ : G → Aff ⁡ ( H ) \rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ : g → aff ⁡ ( h ) \varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.

Crystallographic actions on Lie groups and post-Lie algebra structures

Abstract This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.

Crossed morphisms, integration of post-Lie algebras and the post-Lie Magnus expansion

This letter is divided in two parts. In the first one it will be shown that the datum of a post-Lie product is equivalent to the one of an invertible crossed morphism between two Lie algebras. Moreover it will be argued that the integration of such a crossed morphism yields the post-Lie Magnus expansion associated to the original post-Lie algebra. The second part is devoted to present two combinatorial methods to compute the coefficients of this remarkable formal series. Both methods are based on special tubings on planar trees.

N-Derivations of the Extended Schrödinger–Virasoro Lie Algebra

Let [Formula: see text] be the extended Schrödinger–Virasoro Lie algebra and [Formula: see text] an integer. A map [Formula: see text] is called an [Formula: see text]-derivation if it is a derivation in one variable while other variables fixed. We investigate [Formula: see text]-derivations of the extended Schrödinger–Virasoro Lie algebra [Formula: see text]. The main result when [Formula: see text] is then applied to characterize the linear commuting maps and the commutative post-Lie algebra structures on [Formula: see text].

Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra

<p style='text-indent:20px;'>In this paper, we characterize the graded post-Lie algebra structures on the Schrödinger-Virasoro Lie algebra. Furthermore, as an application, we obtain the all homogeneous Rota-Baxter operator of weight <inline-formula><tex-math id="M1">$ 1 $</tex-math></inline-formula> on the Schrödinger-Virasoro Lie algebra.</p>

Biderivations and linear commuting maps on the restricted contact Lie algebras K(n; )

Let K be the restricted contact Lie algebras K(n, ) over an algebraically closed field F of characteristic p > 3. We prove that each skew-symmetric biderivation of K is inner and show that commuting maps on K are scalar multiplication maps. Moreover, it is showed that the commuting automorphisms and dertivations of K are proved to be the identity mappings and zero mappings, respectively. Meanwhile, the commutative post-Lie algebra structures on K are given.

Monomial Bases for Free Post-Lie Algebras

Many attempts have been made to describe bases for the post-Lie algebra, using many types of Lie bases. Here, we try to describe special kind of post-Lie bases using those bases described in the pre-Lie (respectively Lie) contexts, founded before in [8, 9]. In addition, we try to understand the effect of the second operation of the post-Lie structure on the form of these bases, using some cases of the generating set, and translate it in terms of rooted trees.

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.