scholarly journals Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1249
Author(s):  
Ivan Gutierrez-Sagredo ◽  
Francisco Jose Herranz
Keyword(s):  
Lie Algebras ◽  
De Sitter ◽  
Lie Bialgebras ◽  
R Matrix ◽  
First Order ◽  
Double Structure ◽  
Drinfel’D Double ◽  
Real Algebra ◽  
Noncommutative Spaces ◽  
Drinfel'd Doubles

The Cayley–Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for so(5) together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter so(4,1) and anti-de Sitter so(3,2) as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

scholarly journals Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

2017 ◽  
Vol 2017 ◽  
pp. 1-19 ◽  
Author(s):  
Ángel Ballesteros ◽  
N. Rossano Bruno ◽  
Francisco J. Herranz
Keyword(s):  
Quantum Gravity ◽  
Cosmological Constant ◽  
Lie Algebras ◽  
Deformation Parameter ◽  
Planck Scale ◽  
Quantum Deformation ◽  
De Sitter ◽  
Planck Length ◽  
Unified Approach ◽  
Drinfel’D Double

Theκ-deformation of the (2 + 1)D anti-de Sitter, Poincaré, and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson–Lie structure underlying theκ-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson–Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize theκ-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged.

scholarly journals UNDECIDABILITY OF THE FIRST ORDER THEORIES OF FREE NONCOMMUTATIVE LIE ALGEBRAS

2018 ◽  
Vol 83 (3) ◽  
pp. 1204-1216 ◽  
Author(s):  
OLGA KHARLAMPOVICH ◽  
ALEXEI MYASNIKOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Model Theory ◽  
Characteristic Zero ◽  
Elementary Theory ◽  
Independence Property ◽  
First Order ◽  
Free Lie Algebras ◽  
Image Position ◽  
Do So

AbstractLet R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

scholarly journals Manifold invariants affect dynamics in anti de Sitter gravity

2013 ◽  
Vol 91 (6) ◽  
pp. 465-466 ◽  
Author(s):  
Tomáš Liko
Keyword(s):  
Cosmological Constant ◽  
Configuration Space ◽  
De Sitter ◽  
First Order ◽  
Diffeomorphism Invariance ◽  
Negative Cosmological Constant ◽  
Weight Factors ◽  
Fixed Weight ◽  
Affect Dynamics

The first-order Holst action with negative cosmological constant is rendered finite by requiring functional differentiability on the configuration space of tetrads and connections. The surface terms that arise in the action for anti de Sitter gravity are equivalent to the Euler and Pontryagin densities with fixed weight factors; these terms modify the Noether charges that arise from diffeomorphism invariance of the action.

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 On 3+1 Anti-de Sitter and de Sitter Lie Bialgebras with Dimensionful Deformation Parameters

2004 ◽  
Vol 54 (11) ◽  
pp. 1321-1327 ◽  
Author(s):  
Francisco J. Herranz ◽  
Angel Ballesteros ◽  
N. Rossano Bruno
Keyword(s):  
De Sitter ◽  
Lie Bialgebras ◽  
Deformation Parameters

scholarly journals Gauge dependence in QED amplitudes in expanding de Sitter space

2016 ◽  
Vol 31 (10) ◽  
pp. 1650050 ◽  
Author(s):  
Nistor Nicolaevici
Keyword(s):  
External Field ◽  
Gauge Invariance ◽  
De Sitter Space ◽  
External Fields ◽  
De Sitter ◽  
First Order ◽  
Gauge Dependence ◽  
Invariance Violation ◽  
The Face ◽  
First Order Transition

We consider first-order transition amplitudes in external fields in QED in the expanding de Sitter space and point out that they are gauge dependent quantities. We examine the gauge variations of the amplitudes assuming a decoupling of the interaction at large times, which allows to conclude that the source of the problem lies in the fact that the frequencies of the modes in the infinite future become independent of the comoving momenta. We show that a possibility to assure the gauge invariance of the external field amplitudes is to restrict to potentials which vanish sufficiently fast at infinite times, and briefly discuss a number of options in the face of the possible gauge invariance violation in the full interacting theory.

scholarly journals ON LIE INDUCTION AND THE EXCEPTIONAL SERIES

2005 ◽  
Vol 04 (06) ◽  
pp. 707-737 ◽  
Author(s):  
JAN E. GRABOWSKI
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Point Of View ◽  
Lie Bialgebras ◽  
Simple Complex ◽  
Single Node ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
Induction Process ◽  
And Algebra

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

Sign in / Sign up

Export Citation Format

Share Document