scholarly journals Derivation double Lie algebras

2016 ◽  
Vol 15 (06) ◽  
pp. 1650114
Author(s):  
Dietrich Burde
Keyword(s):  
Recent Work ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Classical Formula ◽  
Simple Derivation ◽  
Affine Actions

We study classical [Formula: see text]-matrices [Formula: see text] for Lie algebras [Formula: see text] such that [Formula: see text] is also a derivation of [Formula: see text]. This yields derivation double Lie algebras [Formula: see text]. The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

scholarly journals Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1354 ◽  
Author(s):  
Hassan Almusawa ◽  
Ryad Ghanam ◽  
Gerard Thompson
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Solvable Lie Groups ◽  
Related System ◽  
Geodesic Equations ◽  
Solvable Lie Algebras ◽  
Symmetry Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

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 Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 119-148
Author(s):  
Thomas Ernst
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Homogeneous Spaces ◽  
Iwasawa Decomposition ◽  
Base Field ◽  
Killing Form ◽  
Exact Sequences

Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples S U q ( 1 , 1 ) S O q ( 2 ) {{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and S O q ( 3 ) S O q ( 2 ) {{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).

scholarly journals Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds

2019 ◽  
Vol 31 (3) ◽  
pp. 563-578
Author(s):  
Marcos Origlia
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Type I ◽  
Symplectic Structures ◽  
Locally Conformal

Abstract We study Lie algebras of type I, that is, a Lie algebra {\mathfrak{g}} where all the eigenvalues of the operator {\operatorname{ad}_{X}} are imaginary for all {X\in\mathfrak{g}} . We prove that the Morse–Novikov cohomology of a Lie algebra of type I is trivial for any closed 1-form. We focus on locally conformal symplectic structures (LCS) on Lie algebras of type I. In particular, we show that for a Lie algebra of type I any LCS structure is of the first kind. We also exhibit lattices for some 6-dimensional Lie groups of type I admitting left invariant LCS structures in order to produce compact solvmanifolds equipped with an invariant LCS structure.

Almost contact Hom-Lie algebras and Sasakian Hom-Lie algebras

2020 ◽  
pp. 2250005
Author(s):  
E. Peyghan ◽  
L. Nourmohammadifar
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Matrix ◽  
Contact Structures ◽  
Sasakian Structure ◽  
Sasakian Structures ◽  
Matrix Formula ◽  
Skew Symmetric Matrix

In this paper, we consider Hom-Lie groups and introduce left invariant almost contact structures on them (almost contact Hom-Lie algebras). On such Hom-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe [Formula: see text]-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition, it is shown that any Sasakian Hom-Lie algebra is a [Formula: see text]-contact Hom-Lie algebra. Finally, we present examples of Sasakian Hom-Lie algebras and in particular, we show that the skew symmetric matrix [Formula: see text] carries a Sasakian structure.

scholarly journals Affine actions on Lie groups and post-Lie algebra structures

2012 ◽  
Vol 437 (5) ◽  
pp. 1250-1263 ◽  
Author(s):  
Dietrich Burde ◽  
Karel Dekimpe ◽  
Kim Vercammen
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Affine Actions

scholarly journals Homogeneous Lie algebras

1971 ◽  
Vol 4 (3) ◽  
pp. 349-353 ◽  
Author(s):  
S. Świerczkowski
Keyword(s):  
Automorphism Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
One Dimensional ◽  
Connected Set ◽  
Real Matrices

It is shown that the automorphism group of a real Lie algebra operates transitively on the set of its one-dimensional subspaces iff the Lie algebra is abelian, or isomorphic to the algebra of skew-symmetric 3 × 3 real matrices. This allows to conclude that R, S0(2), S0(3) and Spin(3) are the only connected Lie groups such that: (1) the conjugates of every connected set containing e cover a neighbourhood of e, (2) every point sufficiently close to e lies on exactly one 1-parameter subgroup.

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