scholarly journals COMMUTATORS OF SKEW-SYMMETRIC MATRICES

2005 ◽  
Vol 15 (03) ◽  
pp. 793-801 ◽  
Author(s):  
ANTHONY M. BLOCH ◽  
ARIEH ISERLES
Keyword(s):  
Optimal Control ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Integration ◽  
Symmetric Matrices ◽  
Frobenius Norm

In this paper we develop a theory for analysing the "radius" of the Lie algebra of a matrix Lie group, which is a measure of the size of its commutators. Complete details are given for the Lie algebra 𝔰𝔬(n) of skew symmetric matrices where we prove [Formula: see text], X, Y ∈ 𝔰𝔬(n), for the Frobenius norm. We indicate how these ideas might be extended to other matrix Lie algebras. We discuss why these ideas are of interest in applications such as geometric integration and optimal control.

COMPUTING THE LAW OF A FAMILY OF SOLVABLE LIE ALGEBRAS

2009 ◽  
Vol 19 (03) ◽  
pp. 337-345 ◽  
Author(s):  
JUAN C. BENJUMEA ◽  
JUAN NÚÑEZ ◽  
ÁNGEL F. TENORIO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Computational Data ◽  
Solvable Lie Algebras ◽  
Final Output ◽  
The Matrix ◽  
The Law

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that basis. These brackets constitute the final output of the algorithm, whose unique input is the matrix order n. Besides, its complexity is proved to be polynomial and some complementary computational data relative to its implementation are also shown.

scholarly journals Simply transitive NIL-affine actions of solvable Lie groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jonas Deré ◽  
Marcos Origlia
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Main Tool ◽  
Full Description ◽  
Nilpotent Lie Group ◽  
Affine Transformations ◽  
Solvable Lie Algebra ◽  
Solvable Lie Group

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.

scholarly journals Pro-Lie groups which are infinite-dimensional Lie groups

2009 ◽  
Vol 146 (2) ◽  
pp. 351-378 ◽  
Author(s):  
K. H. HOFMANN ◽  
K.-H. NEEB
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Exponential Function ◽  
Lie Group ◽  
Smooth Manifold ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Lie Groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

XIX.—Metrisable Lie Groups and Algebras

1957 ◽  
Vol 64 (3) ◽  
pp. 290-304 ◽  
Author(s):  
S.-T. Tsou ◽  
A. G. Walker
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Necessary Conditions ◽  
Existence Theorems ◽  
Riemannian Metric ◽  
Abelian Algebra ◽  
Lie Groups And Algebras ◽  
Complex Extension

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

scholarly journals On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

2020 ◽  
Vol 4 (2) ◽  
pp. 107
Author(s):  
Edi Kurniadi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Plancherel Measure ◽  
Nilpotent Lie Algebra ◽  
Direct Sum ◽  
Unitary Dual ◽  
Dimension One ◽  
The Given ◽  
Frobenius Lie Algebra

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit  how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group  and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

Maximal Abelian subalgebras of complex Euclidean Lie algebras

1994 ◽  
Vol 72 (7-8) ◽  
pp. 389-404 ◽  
Author(s):  
E. G. Kalnins ◽  
P. Winternitz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Conjugacy Classes ◽  
Abelian Subalgebras ◽  
Affine Lie Algebra ◽  
Orthogonal Lie Algebra ◽  
Nilpotent Subalgebras

Maximal Abelian subalgebras (MASAs) of the complex Euclidean Lie algebra [Formula: see text] are classified into conjugacy classes under the action of the Lie group [Formula: see text] Use is made of an earlier classification of MASAs of the orthogonal Lie algebra [Formula: see text] These are then extended to nonsplitting MASAs of [Formula: see text] in which maximal Abelian nilpotent subalgebras of [Formula: see text] are coupled with translations in a nontrivial manner. The methods presented are applicable to the classification of MASAs of any affine Lie algebra.

AFFINE STRUCTURES ON NILMANIFOLDS

1996 ◽  
Vol 07 (05) ◽  
pp. 599-616 ◽  
Author(s):  
DIETRICH BURDE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Variety ◽  
Lie Group ◽  
Affine Structure ◽  
Nilpotent Lie Algebra ◽  
Minimal Dimension ◽  
Affine Structures ◽  
Filiform Algebra ◽  
Filiform Lie Algebras

We investigate the existence of affine structures on nilmanifolds Γ\G in the case where the Lie algebra g of the Lie group G is filiform nilpotent of dimension less or equal to 11. Here we obtain examples of nilmanifolds without any affine structure in dimensions 10, 11. These are new counterexamples to the Milnor conjecture. So far examples in dimension 11 were known where the proof is complicated, see [5] and [4]. Using certain 2-cocycles we realize the filiform Lie algebras as deformation algebras from a standard graded filiform algebra. Thus we study the affine algebraic variety of complex filiform nilpotent Lie algebra structures of a given dimension ≤11. This approach simplifies the calculations, and the counterexamples in dimension 10 are less complicated than the known ones. We also obtain results for the minimal dimension µ(g) of a faithful g-module for these filiform Lie algebras g.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

scholarly journals Electrical network and Lie theory

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Yi Su
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Electrical Network ◽  
Type B ◽  
Electrical Networks ◽  
Combinatorial Properties ◽  
Type D ◽  
International Audience ◽  
Type C

International audience Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group $EL_{2n}$ whose positive part $(EL_{2n})_{\geq 0}$ naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical networks. The Lie algebra $el_{2n}$ is semisimple and isomorphic to the symplectic algebra. In the end of their paper, they suggest a generalization of electrical Lie algebras to all finite Dynkin types. We give the structure of the type $B$ electrical Lie algebra $e_{b_{2n}}$. The nonnegative part $(E_{B_{2n}})_{\geq 0}$ of the corresponding Lie group conjecturally acts on a class of "mirror symmetric circular planar electrical networks". This class of networks has interesting combinatorial properties. Finally, we mention some partial results for type $C$ and $D$ electrical Lie algebras, where an analogous story needs to be developed. Curtis, Ingerman et Morrow ont étudié l’espace des réseaux électriques circulaires plans et ont classifié toutes les matrices de réponses possibles pour ces réseaux. Lam et Pylyavskyy ont trouvé un groupe de Lie $EL_{2n}$ dont la partie positive $(EL_{2n})_{\geq 0}$ agit naturellement sur le réseau électrique circulaire plan par une description combinatoire, où l’action est inspirée par la transformation étoile vers triangle des réseaux électriques. L’algèbre de Lie $el_{2n}$ est semi-simple et isomorphe à l’algèbre symplectique. A la fin de leur article, ils proposent une généralisation des algèbres de Lie électriques pour tous les types de Dynkin finis. Nous donnons la structure de l’algèbre de Lie électrique $e_{b_{2n}}$ du type $B$. La partie positive $(E_{B_{2n}})_{\geq 0}$ du groupe de Lie correspondant agit conjecturalement sur une famille de ”miroirs réseaux électriques circulaires symétriques plans”. Cette famille de réseaux a des propriétés combinatoires intéressantes. Nous donnons enfin quelques résultats partiels de l’algèbre de Lie électrique du type $C$ et du type $D$, où une étude analogue doit être développée.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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