Extended Drinfel’d algebras and non-Abelian duality

Abstract

Abstract The Drinfel’d algebra provides a method to construct generalized parallelizable spaces and this allows us to study an extended $T$-duality, known as the Poisson–Lie $T$-duality. Recently, in order to find a generalized $U$-duality, an extended Drinfel’d algebra (ExDA), called the exceptional Drinfel’d algebra (EDA), was proposed and a natural extension of Abelian $U$-duality was studied both in the context of supergravity and membrane theory. In this paper, we clarify the general structure of ExDAs and show that an ExDA always gives a generalized parallelizable space, which may be regarded as a group manifold with generalized Nambu–Lie structures. We then discuss the non-Abelian duality that is based on a general ExDA. For a coboundary ExDA, this non-Abelian duality reduces to a generalized Yang–Baxter deformation and we find a general formula for the twist matrix. In order to study the non-Abelian $U$-duality, we particularly focus on the $E_{n(n)}$ EDA for $n\leq 8$ and study various aspects, both in terms of M-theory and type IIB theory.