Classification of Poisson–lie T-Dual Models With Two-Dimensional Targets

Abstract

Four-dimensional Manin triples and Drinfeld doubles are classified and corresponding two-dimensional Poisson-Lie T-dual sigma models on them are constructed. The simplest example of a Drinfeld double allowing decomposition into two nontrivially different Manin triples is presented.

Fabrication of Catalyst-Functionalized Three-Dimensional Micromesh Structures

“…This is much too simple a model to be physically interesting, but it serves as a tractable toy model to illustrate the main features of the duality transformation. The dual sigma models on this double have been explicitly worked out in [18,33]; here we analyse their worldsheet boundary conditions.…”

“…Note that the explicit expressions for E(g) and e E(e g) differ from those of [18,33], because they use right-translation on G, e G while we use left-translation, and also we have used slightly different definitions of the fields.…”

Classification of Six-Dimensional Real Drinfeld Doubles

“…In the next subsection we analyze the subsets of non-isomorphic Manin triples characterized by invariants described in the Section 3 and displayed in the Table 1. 4.1 Manin triples with the Killing form of signature (3,3,0) In this case the signature of the Killing form itself fixes the Lie algebra D of the Drinfeld double uniquely. It is the well-known so(3, 1) which is simple as a real Lie algebra and its complexification is semisimple; it decomposes into two copies of sl(2, C).…”

“…The classification of the two-dimensional Drinfeld doubles is trivial and the four-dimensional Drinfeld doubles can be found e.g. in the paper [3] together with the corresponding two-dimensional T-dual models. Examples of sixdimensional Drinfeld doubles and three-dimensional dual models were given e.g.…”

Quantization of Drinfel'd doubles

“…That result is tantamount to the (first order) classification of quantum deformations of 2D and 3D real Lie algebras. Complementarily, the classification of 4D and 6D Drinfel'd doubles has been performed in [22] and [24], respectively, by following a "direct" approach. Throughout the paper we will preserve the notation and labeling for the Lie bialgebras and Drinfel'd doubles given in [23], although both classifications are fully equivalent (up to the fact that [23] does not consider 6D classical doubles coming from a trivial Lie bialgebra with vanishing cocommutator δ or, equivalently, with abelian dual g * ).…”