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O(4,4) dualities and Manin triples

Angelina Kurenkova, Edvard T. Musaev

TL;DR

This paper develops an $O(4,4)$-invariant framework to classify 8-dimensional Manin triples $( rak{g}, ilde{ rak{g}}, abla)$ that encode Poisson--Lie T-dualities between 4D group-manifold supergravity backgrounds. Leveraging Mubarakzyanov’s classification of real 4D Lie algebras, it solves the Drinfeld double consistency conditions to enumerate duals for each $ rak{g}_{4,n}$, identifying dualities and a Poisson--Lie triality, and then refines the classification by marking whether a given double can arise from a Yang–Baxter deformation of $( rak{g}, 4 rak{g}_1)$. Dualities are organized via invariants such as Killing-form eigenvalues and the abelian/nonabelian character of the maximal nilpotent ideal, with a Mathematica implementation used to generate explicit duals. The results provide a structured catalog of Poisson–Lie dualities and set the stage for explicit geometric realizations and higher-dimensional extensions, offering practical criteria for constructing PL backgrounds via bi-vector deformations.

Abstract

We provide a coarse classification of all 8-dimensional Manin triples, that describe Poisson--Lie T-dualities between 4-dimensional group manifold solutions to supergravity equations. We find several such dualities and one Poisson--Lie triality. For each class we refine the classification by marking whether the corresponding Drinfeld double can be obtained by a Yang--Baxter deformation of the initial algebra paired by the four-dimensional Abelian algebra.

O(4,4) dualities and Manin triples

TL;DR

This paper develops an -invariant framework to classify 8-dimensional Manin triples that encode Poisson--Lie T-dualities between 4D group-manifold supergravity backgrounds. Leveraging Mubarakzyanov’s classification of real 4D Lie algebras, it solves the Drinfeld double consistency conditions to enumerate duals for each , identifying dualities and a Poisson--Lie triality, and then refines the classification by marking whether a given double can arise from a Yang–Baxter deformation of . Dualities are organized via invariants such as Killing-form eigenvalues and the abelian/nonabelian character of the maximal nilpotent ideal, with a Mathematica implementation used to generate explicit duals. The results provide a structured catalog of Poisson–Lie dualities and set the stage for explicit geometric realizations and higher-dimensional extensions, offering practical criteria for constructing PL backgrounds via bi-vector deformations.

Abstract

We provide a coarse classification of all 8-dimensional Manin triples, that describe Poisson--Lie T-dualities between 4-dimensional group manifold solutions to supergravity equations. We find several such dualities and one Poisson--Lie triality. For each class we refine the classification by marking whether the corresponding Drinfeld double can be obtained by a Yang--Baxter deformation of the initial algebra paired by the four-dimensional Abelian algebra.

Paper Structure

This paper contains 5 sections, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Detailed comments on the classification
  4. Conclusions
  5. List of classes

Figures (2)

  • Figure 1: Schematic depiction of four-dimensional Lie algebras reflecting moduli space of each class. Wave line means that the parameter cannot take values there, arrow means that the parameter can go to infinity. Algebras with no free parameters are depicted by a dot.
  • Figure 2: Duality relations between 4d real Lie algebras: algebras on sides of an arrow can be part of a Manin triple. Red lines denote part of the algebras module space in a given class to be dualised. Algebras are not taken in the standard form.