scholarly journals LINEARIZATION OF POISSON–LIE STRUCTURES ON THE 2D EUCLIDEAN AND (1 + 1) POINCARÉ GROUPS

2021 ◽  
Vol 7 (2) ◽  
pp. 33
Author(s):  
Bousselham Ganbouri ◽  
Mohamed Wadia Mansouri
Keyword(s):  
Lie Algebras ◽  
Explicit Form ◽  
Lie Bialgebra ◽  
Linearization Problem

The paper deals with linearization problem of Poisson-Lie structures on the  \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.

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 Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

2019 ◽  
Vol 16 (07) ◽  
pp. 1950097
Author(s):  
Ghorbanali Haghighatdoost ◽  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Baxter Equation ◽  
Complex Structures ◽  
Lie Bialgebra ◽  
Text Structures ◽  
Generalized Complex Structures ◽  
Generalized Complex

We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all [Formula: see text]-[Formula: see text] structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between [Formula: see text]-[Formula: see text] structures and the generalized complex structures on the Lie algebras [Formula: see text] and also the solutions of modified Yang–Baxter equation (MYBE) on the double of Lie bialgebra [Formula: see text]. The results are applied to some relevant examples.

Lie bialgebras of not-finitely graded Lie algebras related to generalized Virasoro algebras

2014 ◽  
Vol 25 (05) ◽  
pp. 1450049 ◽  
Author(s):  
Qiufan Chen ◽  
Jianzhi Han ◽  
Yucai Su
Keyword(s):  
Lie Algebras ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Graded Lie Algebras

In this paper, the Lie bialgebra structures on a class of not-finitely graded Lie algebras related to generalized Virasoro algebras Ŵ are considered. We prove that all Lie bialgebras on Ŵ are coboundary triangular.

scholarly journals On the Fiber Preserving Transformations for the Fifth-Order Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. Suksern ◽  
W. Pinyo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Explicit Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linearization Problem ◽  
Necessary And Sufficient ◽  
Fifth Order

This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of fiber preserving transformations. The necessary and sufficient conditions for linearization are obtained. The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented.

Lie Bialgebra Structures on Lie Algebras of Block Type

2009 ◽  
Vol 16 (04) ◽  
pp. 677-690 ◽  
Author(s):  
Yongsheng Cheng ◽  
Guang'ai Song ◽  
Bin Xin
Keyword(s):  
Lie Algebras ◽  
Block Type ◽  
Lie Bialgebra ◽  
Lie Bialgebras

This paper considers Lie bialgebra structures on Lie algebras of Block type and proves that all such Lie bialgebras are triangular coboundary.

Lie Bialgebra Structures on Lie Algebras of Generalized Weyl Type

2008 ◽  
Vol 36 (4) ◽  
pp. 1537-1549 ◽  
Author(s):  
Xiaoqing Yue ◽  
Yucai Su
Keyword(s):  
Lie Algebras ◽  
Lie Bialgebra

scholarly journals Structure of the Enveloping Algebras

10.14311/926 ◽  
2007 ◽  
Vol 47 (2-3) ◽  
Author(s):  
Č. Burdík ◽  
O. Navrátil ◽  
S. Pošta
Keyword(s):  
Lie Algebras ◽  
Explicit Form ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Raising Operators ◽  
General Method

The adjoint representations of several small dimensional Lie algebras  on their universal enveloping algebras  are explicitly decomposed. It is shown that commutants of raising operators are generated as polynomials in several basic elements. The explicit form of these elements is given and the general method for obtaining  these elements is described. 

scholarly journals CLASSIFICATION OF SIX-DIMENSIONAL REAL DRINFELD DOUBLES

2002 ◽  
Vol 17 (28) ◽  
pp. 4043-4067 ◽  
Author(s):  
LIBOR ŠNOBL ◽  
LADISLAV HLAVATÝ
Keyword(s):  
Lie Algebras ◽  
Explicit Form ◽  
Complete List ◽  
Invariant Form ◽  
Manin Triples

Starting from the classification of real Manin triples we look for those that are isomorphic as six-dimensional Drinfeld doubles i.e. Lie algebras with the ad-invariant form used for construction of the Manin triples. We use several invariants of the Lie algebras to distinguish the nonisomorphic structures and give the explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of six-dimensional real Drinfeld doubles. It consists of 22 classes of nonisomorphic Drinfeld doubles.

Quantization of Lie Algebras of Generalized Weyl Type

2009 ◽  
Vol 16 (03) ◽  
pp. 437-448 ◽  
Author(s):  
Xiaoqing Yue ◽  
Qifen Jiang ◽  
Bin Xin
Keyword(s):  
Lie Algebras ◽  
Lie Bialgebra

Recently, in [25], Lie bialgebra structures on Lie algebras of generalized Weyl type were considered, which are shown to be triangular coboundary. In this paper, we quantize these algebras with their Lie bialgebra structures.

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