Crystallographic actions on Lie groups and post-Lie algebra structures

Dietrich Burde

Displaying similar documents to “Crystallographic actions on Lie groups and post-Lie algebra structures”

Integrating central extensions of Lie algebras via Lie 2-groups

Christoph Wockel, Chenchang Zhu (2016)

Journal of the European Mathematical Society

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The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of $π_{2}$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated...

Quantization of semisimple real Lie groups

Kenny De Commer (2024)

Archivum Mathematicum

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We provide a novel construction of quantized universal enveloping $*$ -algebras of real semisimple Lie algebras, based on Letzter’s theory of quantum symmetric pairs. We show that these structures can be ‘integrated’, leading to a quantization of the group C $^{*}$ -algebra of an arbitrary semisimple algebraic real Lie group.

Algorithmic computations of Lie algebras cohomologies

Šilhan, Josef

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From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant’s result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package $L i E$ offers the data structures and corresponding procedures for...

The variety of dual mock-Lie algebras

Luisa M. Camacho, Ivan Kaygorodov, Viktor Lopatkin, Mohamed A. Salim (2020)

Communications in Mathematics

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We classify all complex $7$ - and $8$ -dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex $9$ -dimensional dual mock-Lie algebras.

Weighted diffeomorphism groups of Banach spaces and weighted mapping groups

Boris Walter

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In this work, we construct and study certain classes of infinite-dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group $D i f f_{} (X)$ of diffeomorphisms, for each Banach space X and each set of weights on X containing the constant weights. We also construct certain types of “weighted mapping groups”. These are Lie groups modelled on weighted function spaces of the form $_{}^{k} (U, L (G))$ , where G is a given (finite- or infinite-dimensional) Lie group. Both the...

When unit groups of continuous inverse algebras are regular Lie groups

Helge Glöckner, Karl-Hermann Neeb (2012)

Studia Mathematica

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It is a basic fact in infinite-dimensional Lie theory that the unit group $A^{\times}$ of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group $A^{\times}$ is regular in Milnor’s sense. Notably, $A^{\times}$ is regular if A is Mackey-complete and locally m-convex.

Universal central extension of direct limits of Hom-Lie algebras

Valiollah Khalili (2019)

Czechoslovak Mathematical Journal

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We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $(ℒ_{i}, α_{ℒ_{i}})$ is (isomorphic to) the direct limit of universal central extensions of $(ℒ_{i}, α_{ℒ_{i}})$ . As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras ${({sl}_{k} (å), α_{k})}_{k \in I}$ and describe the universal central extension of its direct limit.

The diffeomorphism group of a non-compact orbifold

A. Schmeding

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We endow the diffeomorphism group $D i f f_{O r b} (Q,)$ of a paracompact (reduced) orbifold with the structure of an infinite-dimensional Lie group modeled on the space of compactly supported sections of the tangent orbibundle. For a second countable orbifold, we prove that $D i f f_{O r b} (Q,)$ is C⁰-regular, and thus regular in the sense of Milnor. Furthermore, an explicit characterization of the Lie algebra associated to $D i f f_{O r b} (Q,)$ is given.

Knit products of graded Lie algebras and groups

Michor, Peter W.

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Let $A = ⨁_{k} A_{k}$ and $B = ⨁_{k} B_{k}$ be graded Lie algebras whose grading is in $𝒵$ or $𝒵_{2}$ , but only one of them. Suppose that $(α, β)$ is a derivatively knitted pair of representations for $(A, B)$ , i.e. $α$ and $β$ satisfy equations which look “derivatively knitted"; then $A \oplus B : = ⨁_{k, l} (A_{k} \oplus B_{l})$ , endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A \oplus_{(α, β)} B$ . This graded Lie algebra is called the knit product of $A$ and $B$ . The author investigates the general situation for any graded Lie subalgebras $A$ and...

Extensions of hom-Lie algebras in terms of cohomology

Abdoreza R. Armakan, Mohammed Reza Farhangdoost (2017)

Czechoslovak Mathematical Journal

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We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra $𝔤$ by another hom-Lie algebra $𝔥$ and discuss the case where $𝔥$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological...

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

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In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano...