[Paper Review] Extended Drinfel'd algebras and non-Abelian duality

Yuho Sakatani|arXiv (Cornell University)|Sep 9, 2020

Advanced Topics in Algebra81 references4 citations

TL;DR

This paper introduces the Extended Drinfel'd Algebra (ExDA) as a unified framework for generalized parallelizable spaces underlying non-Abelian duality, generalizing Drinfel'd algebras to exceptional groups like $E_{n(n)}$. It proves that every ExDA yields a generalized Nambu–Lie structure and enables generalized Yang–Baxter deformations, with explicit constructions of $E_{n(n)}$ ExDAs for $n \leq 8$ in both M-theory and type IIB pictures, unifying $U$-duality and Poisson–Lie duality in exceptional field theory.

ABSTRACT

A Drinfel'd algebra gives the systematic construction of 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 the usual 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 also discuss generalized Yang-Baxter deformations that are based on coboundary ExDAs. As important examples, we consider the $E_{n(n)}$ EDA for $n\leq 8$ and study various aspects, both in terms of M-theory and type IIB theory.

Motivation & Objective

• To generalize Drinfel'd algebras to exceptional groups, enabling a unified framework for $U$-duality analogues of Poisson–Lie duality.
• To establish that every ExDA gives rise to a generalized parallelizable space with Nambu–Lie algebraic structure.
• To construct explicit $E_{n(n)}$ ExDAs for $n \leq 8$, valid in both M-theory and type IIB duality pictures.
• To demonstrate that coboundary ExDAs support generalized Yang–Baxter deformations.
• To unify the description of $U$-duality in supergravity and membrane theory via a single algebraic structure.

Proposed method

• Define the ExDA via a generalized Lie algebra with a maximally isotropic subalgebra and a section condition on generalized frame fields $E_A^I$.
• Derive Leibniz identities and fundamental cocycle conditions that ensure consistency of the algebraic structure.
• Introduce generalized frame fields $E_A^I$ satisfying $\hat{\mathcal{L}}_{E_A} E_B^I = -X_{AB}{}^C E_C^I$, linking to generalized diffeomorphisms.
• Construct explicit matrix representations of the $Y$-tensor and $\chi$-matrices for $E_{n(n)}$ algebras in both M-theory and type IIB pictures.
• Use the generalized classical Yang–Baxter equation to define coboundary ExDAs and study their associated deformations.
• Apply the formalism to $E_{n(n)}$ for $n \leq 8$, showing consistent algebraic and geometric structures across both duality pictures.

Experimental results

Research questions

• RQ1 Can a unified algebraic framework be constructed to generalize Poisson–Lie duality to $U$-duality via exceptional groups?
• RQ2 Does every ExDA naturally give rise to a generalized parallelizable space with Nambu–Lie algebraic structure?
• RQ3 How do $E_{n(n)}$ ExDAs behave in both M-theory and type IIB duality pictures, and what are the differences in subalgebra dimensions?
• RQ4 Can generalized Yang–Baxter deformations be consistently defined on coboundary ExDAs?
• RQ5 What is the explicit structure of $E_{n(n)}$ ExDAs for $n \leq 8$, and how do they realize $U$-duality covariance?

Key findings

• Every ExDA admits a generalized parallelizable structure, meaning it supports a generalized frame field $E_A^I$ satisfying the generalized Lie derivative condition $\hat{\mathcal{L}}_{E_A} E_B^I = -X_{AB}{}^C E_C^I$.
• The ExDA framework naturally incorporates Nambu–Lie algebraic structures, generalizing the notion of group manifolds to higher-graded algebras.
• Coboundary ExDAs allow for generalized Yang–Baxter deformations, extending the known construction from Drinfel'd algebras to exceptional cases.
• Explicit $E_{n(n)}$ ExDAs are constructed for $n \leq 8$, with consistent matrix representations of the $Y$-tensor and $\chi$-matrices in both M-theory and type IIB pictures.
• In the M-theory picture, the maximally isotropic subalgebra $\mathfrak{g}$ has dimension $n$, while in the type IIB picture, it has dimension $n-1$, despite the same total algebra dimension.
• The $E_{n(n)}$ ExDAs for $n \leq 8$ satisfy the section condition and preserve $U$-duality covariance, enabling applications in supergravity and membrane theory.