Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting

Abstract

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 applying 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.

13 pages; typos corrected. v3 matches version published in PLB

References in corpus (15)

• Double Field Theory
• Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory
• Deformed integrable $\sigma$-models, classical $R$-matrices and classical exchange algebra on Drinfel'd doubles
• Yang-Baxter sigma models based on the CYBE
• Towards a two-parameter q-deformation of AdS_3 x S^3 x M^4 superstrings
• Integrability of classical strings dual for noncommutative gauge theories
• Supergravity backgrounds for deformations of AdS_n x S^n supercoset string models
• A Jordanian deformation of AdS space in type IIB supergravity
• String Theory as a Higher Spin Theory
• Quantum deformations of the flat space superstring
• On classical Yang-Baxter based deformations of the AdS_5 x S^5 superstring
• Type IIB supergravity solution for the T-dual of the eta-deformed AdS_5 x S^5 superstring
• Twisted Quantum Deformations of Lorentz and Poincaré algebras
• Towards (3+1) gravity through Drinfel'd doubles with cosmological constant
• Real and pseudoreal forms of D=4 complex Euclidean (super)algebras and super-Poincare / super-Euclidean r-matrices

Cited by in corpus (19)

• Poisson-Lie T-duals of the bi-Yang-Baxter models
• Symmetries of quantum space-time in 3 dimensions
• The kappa-(A)dS quantum algebra in (3+1) dimensions
• Noncommutative spaces of worldlines
• Curved momentum spaces from quantum groups with cosmological constant
• Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum
• AdS Poisson homogeneous spaces and Drinfel'd doubles
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• The Poincaré group as a Drinfel'd double
• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Quantizations of D=3 Lorentz symmetry
• $\kappa$-deformed BMS symmetry
• Quantum Klein Space and Superspace
• Drinfel'd double structures for Poincaré and Euclidean groups
• Quantum groups and noncommutative spacetimes with cosmological constant
• Fuzzy worldlines with $\kappa$-Poincaré symmetries
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• 3-dimensional $\Lambda$-BMS Symmetry and its Deformations