In this paper, we study quantum group deformations of the infinite-dimensional symmetry algebra of asymptotically AdS spacetimes in three dimensions. Building on previous results in the finite-dimensional subalgebras we classify all possible Lie bial
gebra structures and for selected examples, we explicitly construct the related Hopf algebras. Using cohomological arguments we show that this construction can always be performed by a so-called twist deformation. The resulting structures can be compared to the well-known $kappa$-Poincare Hopf algebras constructed on the finite-dimensional Poincare or (anti) de Sitter algebra. The dual $kappa$ Minkowski spacetime is supposed to describe a specific non-commutative geometry. Importantly, we find that some incarnations of the $kappa$-Poincare can not be extended consistently to the infinite-dimensional algebras. Furthermore, certain deformations can have potential physical applications if subalgebras are considered. The presence of the full symmetry algebra might have observable consequences that could be used to rule out these deformations.
BMS symmetry is a symmetry of asymptotically flat spacetimes in the vicinity of the null boundary of spacetime and it is expected to play a fundamental role in physics. It is interesting therefore to investigate the structures and properties of quant
um deformations of these symmetries, which are expected to shed some light on symmetries of quantum spacetime. In this paper we discuss the structure of the algebra of extended BMS symmetries in 3 and 4 spacetime dimensions, realizing that these algebras contain an infinite number of distinct Poincare subalgebras, a fact that has previously been noted in the 3-dimensional case only. Then we use these subalgebras to construct an infinite number of different Hopf algebras being quantum deformations of the BMS algebras. We also discuss different types of twist-deformations and the dual Hopf algebras, which could be interpreted as noncommutative, extended quantum spacetimes.
We present the quantum $kappa$-deformation of BMS symmetry, by generalizing the lightlike $kappa$-Poincare Hopf algebra. On the technical level, our analysis relies on the fact that the lightlike $kappa$-deformation of Poincare algebra is given by a
twist and the lightlike deformation of any algebra containing Poincare as a subalgebra can be done with the help of the same twisting element. We briefly comment on the physical relevance of the obtained $kappa$-BMS Hopf algebra as a possible asymptotic symmetry of quantum gravity.
According to the work of Laitinen, Morimoto, Oliver and Pawal{}owski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and f
or $G=A_5times C_n$ for $n=3,5,7$, we present a strategy of excluding of smooth effective one fixed point $G$-actions on low-dimensional spheres.
We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $mathfrak{o}(4;mathbb{C})cong mathfrak{o}(3;mathbb{C})oplus mathfrak{o}(3;mathbb{C})$, describing quantum rotational symmetry of f
our-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $mathfrak{o}(4;mathbb{C})$: Euclidean $mathfrak{o}(4)$, Lorentz $mathfrak{o}(3,1)$, Kleinian $mathfrak{o}(2,2)$ and quaternionic $mathfrak{o}^{star}(4)$. For $mathfrak{o}(3,1)$ we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra $mathfrak{o}(4;mathbb{C})$ we present new results.
In our previous paper we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex $D=4$ Euclidean Lie symmetry $mathfrak{o}(4;mathbb{C})$. The result was presented in the form of a list consisting of three
three-parameter, one two-parameter and one one-parameter nonisomorphic classical $r$-matrices which provide directions of the nonequivalent quantizations of $mathfrak{o}(4;mathbb{C})$. Applying reality conditions to the complex $mathfrak{o}(4;mathbb{C})$ $r$-matrices we obtained the nonisomorphic classical $r$-matrices for all possible real forms of $mathfrak{o}(4;mathbb{C})$: Euclidean $mathfrak{o}(4)$, Lorentz $mathfrak{o}(3,1)$, Kleinian $mathfrak{o}(2,2)$ and quaternionic $mathfrak{o}^{star}(4)$ Lie algebras. In the case of $mathfrak{o}(4)$ and $mathfrak{o}(3,1)$ real symmetries these $r$-matrices give the full classifications of the inequivalent quasitriangular quantum deformations, however for $mathfrak{o}(2,2)$ and $mathfrak{o}^{star}(4)$ the classifications are not full. In this paper we complete these classifications by adding three new three-parameter $mathfrak{o}(2,2)$-real $r$-matrices and one new three-parameter $mathfrak{o}^{star}(4)$-real $r$-matrix. All nonisomorphic classical $r$-matrices for all real forms of $mathfrak{o}(4;mathbb{C})$ are presented in the explicite form what is convenient for providing the quantizations. We will mention also some applications of our results to the deformations of space-time symmetries and string $sigma$-models.
We employ new calculational technique and present complete list of classical $r$-matrices for $D=4$ complex homogeneous orthogonal Lie algebra $mathfrak{o}(4;mathbb{C})$, the rotational symmetry of four-dimensional complex space-time. Further applyin
g reality conditions we obtain the classical $r$-matrices for all possible real forms of $mathfrak{o}(4;mathbb{C})$: Euclidean $mathfrak{o}(4)$, Lorentz $mathfrak{o}(3,1)$, Kleinian $mathfrak{o}(2,2)$ and quaternionic $mathfrak{o}^{star}(4)$ Lie algebras. For $mathfrak{o}(3,1)$ we get known four classical $D=4$ Lorentz $r$-matrices, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we provide new results and mention some applications.
We demonstrate a polarization-managed 8-dimensional modulation format that is time domain coded to reduce inter-channel nonlinearity. Simulation results show a 2.33 dB improvement in maximum net system margin (NSM) relative to polarization multiplexe
d (PM)-BPSK, and a 1.0 dB improvement relative to time interleaved return to zero (RZ)-PM-BPSK, for a five channel fill propagating on 20x80 km spans of 90% compensated ELEAF. In contrast to the other modulations considered, the new 8-dimentional (8D) format has negligible sensitivity to the polarization states of the neighboring channels. Laboratory results from High-density WDM (HD-WDM) propagation experiments on a 5000 km dispersion-managed link show a 1 dB improvement in net system margin relative to PM-BPSK.
This article presents an extended model of gravity obtained by gauging the AdS-Mawell algebra. It involves additional fields that shift the spin connection, leading effectively to theory of two independent connections. Extension of algebraic structur
e by another tetrad gives rise to the model described by a pair of Einstein equations.
We present the class of deformations of simple Euclidean superalgebra, which describe the supersymmetrization of some Lie algebraic noncommutativity of D=4 Euclidean space-time. The presented deformations are generated by the supertwists. We provide
new explicit formulae for a chosen twisted D=4 Euclidean Hopf superalgebra and describe the corresponding quantum covariant deformation of chiral Euclidean superspace.
