U-duality extension of Drinfel'd double

TL;DR

This paper introduces an extended algebraic framework for U-duality that generalizes the Drinfel'd double, enabling systematic construction of generalized frame fields with potential applications in non-Abelian U-duality and deformations.

Contribution

It proposes a new family of algebras extending the Drinfel'd double, facilitating the construction of generalized frame fields with a Nambu-Poisson twist.

Findings

Constructed a family of algebras $\\mathcal{E}_n$ extending the Drinfel'd double.

Developed a method to systematically build generalized frame fields using these algebras.

Discussed potential applications to non-Abelian U-duality and Yang-Baxter deformations.

Abstract

A family of algebras $\mathcal{E}_n$ that extends the Lie algebra of the Drinfel'd double is proposed. This allows us to systematically construct the generalized frame fields $E_A{}^I$ which realize the proposed algebra by means of the generalized Lie derivative, i.e., $\hat{\mathcal{L}}_{E_A}E_B{}^I = - \mathcal{F}_{AB}{}^C\,E_C{}^I$. By construction, the generalized frame fields include a twist by a Nambu-Poisson tensor. A possible application to the non-Abelian extension of U-duality and a generalization of the Yang-Baxter deformation are also discussed.