Classification of 6-dimensional real Drinfeld doubles

Abstract

Starting from the classification of real Manin triples done in a previous paper we look for those that are isomorphic as 6-dimensional 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 non-isomorphic structures and give explicit form of maps between Manin triples that are decompositions of isomorphic Drinfeld doubles. The result is a complete list of 6-dimensional real Drinfeld doubles. It consists of 22 classes of non-isomorphic Drinfeld doubles.

27 pages, corrected minor mistakes and typos, added references

References in corpus (4)

• Dual Non-Abelian Duality and the Drinfeld Double
• Poisson-Lie T-Duality beyond the classical level and the renormalization group
• Poisson-Lie T-Duality and Bianchi Type Algebras
• Classification of Poisson-Lie T-dual models with two-dimensional targets

Cited by in corpus (46)

• Non-geometric backgrounds in string theory
• Poisson-Lie T-plurality
• Poisson-Lie U-duality in Exceptional Field Theory
• U-duality extension of Drinfel'd double
• Type II DFT solutions from Poisson-Lie T-duality/plurality
• Quantum equivalence in Poisson-Lie T-duality
• Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle
• Comments on $\eta$-deformed principal chiral model from 4D Chern-Simons theory
• Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
• Exploring Exceptional Drinfeld Geometries
• Drinfel'd doubles for (2+1)-gravity
• Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models II: Nondiagonal metrics and dilaton puzzle
• Non-abelian U-duality at work
• Generalized Parallelizable Spaces from Exceptional Field Theory
• On non-abelian U-duality of 11D backgrounds
• Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum
• Worldsheet boundary conditions in Poisson-Lie T-duality
• Towards (3+1) gravity through Drinfel'd doubles with cosmological constant
• AdS Poisson homogeneous spaces and Drinfel'd doubles
• Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations
• Classification of four and six dimensional Drinfel'd superdoubles
• Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models
• Dressing cosets revisited
• Quantization of Drinfel'd doubles
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• The Poincaré group as a Drinfel'd double
• The Drinfeld double gl(n) \oplus t_n
• Jacobi-Lie T-plurality
• Classical r-matrices of two and three dimensional Lie super-bialgebras and their Poisson-Lie supergroups
• Poisson-Lie T-plurality as canonical transformation
• Poisson-Lie transformations and Generalized Supergravity Equations
• Classical solutions of sigma models in curved backgrounds by the Poisson-Lie T-plurality
• T-dualization of Gödel string cosmologies via Poisson-Lie T-duality approach
• Flat coordinates and dilaton fields for three--dimensional conformal sigma models
• Poisson-Lie T-plurality revisited. Is T-duality unique?
• Poisson-Lie identities and dualities of Bianchi cosmologies
• Effective Actions for $\sigma$-Models of Poisson-Lie Type
• Integrability structures in string theory
• Classification of real three-dimensional Poisson-Lie groups
• One-loop renormalizability of all 2d dimensional Poisson-Lie sigma-models
• T-folds as Poisson-Lie plurals
• Non-Abelian T-duality of $Ad{S}_{d\le 3}$ families by Poisson-Lie T-duality
• Poisson-Lie T-plurality for dressing cosets
• Classification of standard Manin triples in dimension 4+4
• Non-Abelian target space duals of Thurston geometries