κ-（A）dS量子代数：（3 + 1）维的Poisson结构与双叉积

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"这篇论文详细探讨了κ-（A）dS量子代数在（3+1）维空间中的实现，利用量子对偶原理来构建其Poisson类似物。研究涉及Poisson-Lie群上的结构，以及如何将宇宙常数Λ作为一个压缩参数来考虑。当Λ趋向于0时，κ-庞加莱代数的双叉积基础得以显现。此外，还讨论了κ-（A）dS变形的扭曲形式，这些形式与Drinfel'd双结构相关联。" κ-（A）dS量子代数是物理学中的一种非平凡代数结构，特别是在相对论性量子场论和反德西特（AdS）空间的研究中扮演重要角色。在这个（3+1）维的框架下，量子对偶原理被用来建立κ-（A）dS代数的Poisson结构。这是一种经典对量子代数的模拟，它允许研究者在经典力学的框架下理解和分析量子系统的某些特性。 Poisson-Lie群是一个Lie群，其上的Poisson结构使得它可以与Lie代数的对偶结构相互作用。在本文中，κ-（A）dS量子代数的Poisson-Lie结构是在对偶可解Lie群上定义的。这一构造允许研究人员深入理解κ-（A）dS代数的动力学特性，同时还能得到变形的卡西米尔函数，这些函数在代数的表示理论中是至关重要的。宇宙常数Λ是广义相对论中的一个关键参数，它与宇宙的全局几何和膨胀有关。在这里，Λ被用作Poisson-Lie群的压缩参数，意味着κ-（A）dS量子代数的性质会随着Λ的变化而变化。当Λ趋近于0时，这个代数的结构简化，对应于κ-庞加莱代数，这是一个在平坦时空中的重要代数结构。 Drinfel'd双结构的概念是量子群理论中的一个重要工具，它提供了构造非abelian量子群的一种方式。在κ-（A）dS量子代数的扭曲形式中，这种双结构的体现揭示了代数的更深层次的对称性和结构关系。论文的发表表明，κ-（A）dS量子代数的这些研究不仅有助于深化我们对量子代数和Poisson-Lie群的理解，而且可能对AdS/CFT（反德西特/共形场论）对偶等现代物理理论有实际应用，特别是对于理解高维空间中的引力和量子效应。关键词包括：反德西特空间、宇宙常数、量子群、Poisson-Lie群、Lie双代数和量子对偶原理。这些关键词揭示了论文研究的核心领域和方法，强调了理论物理学中的关键概念。

Physics Letters B 766 (2017) 205–211

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

The κ-(A)dS quantum algebra in (3+1) dimensions

Ángel Ballesteros

a,∗

, Francisco J. Herranz

, Fabio Musso

, Pedro Naranjo

Departamento de Física, Universidad de Burgos, E-09001 Burgos, Spain

Istituto Comprensivo “Leonardo da Vinci”, Via della Grande Muraglia 37, I-0014 Rome, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received

9 December 2016

Accepted

11 January 2017

Available

online 16 January 2017

Editor:

M. Cveti

Keywords:

Anti-de

Sitter

Cosmological

constant

Quantum

groups

Poisson–Lie

groups

Lie

bialgebras

Quantum

duality principle

The quantum duality principle is used to obtain explicitly the Poisson analogue of the κ-(A)dS quantum

algebra in (3+1) dimensions as the corresponding Poisson–Lie structure on the dual solvable Lie

group. The construction is fully performed in a kinematical basis and deformed Casimir functions are

also explicitly obtained. The cosmological constant  is included as a Poisson–Lie group contraction

parameter, and the limit  → 0leads to the well-known κ-Poincaré algebra in the bicrossproduct basis.

A twisted version with Drinfel’d double structure of this κ-(A)dS deformation is sketched.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Since Classical Gravity is essentially a theory describing the

geometry of spacetime, it seems natural to consider that a suit-

able

deﬁnition of a “quantum” spacetime geometry emerging at

the Planck energy regime could be a reasonable feature of Quan-

tum

Gravity. Indeed, speciﬁc mathematical frameworks for such

“quantum geometry” have to be proposed. In particular, “quantum

spacetime” is frequently introduced as a noncommutative alge-

bra

whose noncommutativity is governed by a parameter related

to the Planck scale, thus leading to minimum length frameworks

through generalized spacetime uncertainty relations (see, for in-

stance, [1–5] and

references therein).

this context, quantum groups [6–8] provide a consistent ap-

proach

to noncommutative spacetimes, since the latter are ob-

tained

as noncommutative algebras that are covariant under the

action of quantum kinematical groups. For instance, the well-

known

κ -Minkowski spacetime [9–12] was obtained as a byprod-

uct

of the κ-Poincaré quantum algebra, which was introduced

in [13] (see also [14–16]) by making use of quantum group

contraction techniques [17,18]. One of the main features of the

κ-Poincaré quantum algebra (which is the Hopf algebra dual to the

quantum Poincaré group, and is deﬁned as a deformation of the

Corresponding author.

E-mail

addresses: angelb@ubu.es (Á. Ballesteros), fjherranz@ubu.es (F.J. Herranz),

fmusso@ubu.es (F. Musso), pnaranjo@ubu.es (P. Naranjo).

Poincaré algebra in terms of the dimensionful parameter κ) con-

sists

in its associated deformed second-order Casimir, which leads

to a modiﬁed energy-momentum dispersion relation. From the

phenomenological side, this type of deformed dispersion relations

have been proposed as possible experimentally testable footprints

of quantum gravity effects in very different contexts (see [19–21]

and

references therein).

Moreover,

if the interplay between quantum spacetime and

gravity at cosmological distances is to be modeled, then the cur-

vature

of spacetime cannot be neglected and models with non-

vanishing

cosmological constant have to be considered [22–25].

Thus, the relevant kinematical groups (and spacetimes) would be

the (anti-)de Sitter ones, hereafter (A)dS, and the construction

of quantum (A)dS groups should be faced. In (1+1) and (2+1)

dimensions, the corresponding κ-deformations have been con-

structed [26,27] (see

also [28–30] for classiﬁcation approaches).

In fact, it is worth stressing that the κ-(A)dS deformation intro-

duced

in [27] was proposed in [31] as the algebra of symmetries

for (2+1) quantum gravity (see also [32]), and compatibility con-

ditions

imposed by the Chern–Simons approach to (2+1) gravity

have been recently used [33] in order to identify certain privi-

leged

(A)dS quantum deformations [34] (among them, the twisted

κ-(A)dS algebra [35–38]).

Concerning

(3+1) dimensions, we recall that in the papers

[13–16] the κ-Poincaré algebra was obtained as a contraction of

the Drinfel’d–Jimbo quantum deformation [6,39] of the so(3, 2)

and so(4, 1) Lie algebras, by starting from the latter written in the

http://dx.doi.org/10.1016/j.physletb.2017.01.020

0370-2693/

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