U -duality extension of Drinfel’d double

Abstract

<jats:title>Abstract</jats:title> <jats:p>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., $\hat{\pounds}_{E_A}E_B{}^I =-\mathcal{F}_{AB}{}^C\,E_C{}^I$. 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.</jats:p>

Tri-vector deformations in d = 11 supergravity

“…Recently using a generalisation of Poisson-Lie T-duality to the U-duality setup and to M-theory brane dynamics [36,52], a proposal for the sufficient algebraic constraint for ρ αβγ has been made in [36]. Both non-abelian deformed solutions described in the present work are in the non-unimodular class, meaning ∂ m Ω mnk ≠ 0, therefore the corresponding ρ αβγ cannot satisfy the equations of [36] as the latter suppose unimodularity. It is then natural to expect that the algebraic constraints for the tri-vector components ρ αβγ , such as (4.18), are manifestations of the M-theory generalisation of the CYBE with non-unimodularity properly taken into account.…”

“…Note that while in the d = 10 case both unimodular and non-unimodular deformations are required to satisfy the same classical Yang-Baxter equation, this seems not to be the case for M-theory. Moreover, the condition of vanishing R-flux, which is equivalent to the CYBE in d = 10, appears to be only a part of the equations of [36].…”

“…Depending on the labeling scheme, the tetrahedron equation may be referred to as Zamolodchikov or Frenkel-Moore equation. Deriving a representation independent form of semi-classical limit of the tetrahedron equation and comparing the results to that of [36] is an open problem.…”

“…Supergravity formulations that are natural to look at in this context are Double [26] and Exceptional [27] Field Theories (DFT and ExFT, respectively). Specifically designed to render supergravities in various dimensions covariant under T-and U-duality groups at the expense of extending the spacetime dimension, they are useful in describing Yang-Baxter deformations [15,[28][29][30][31] and Poisson-Lie T-duality [32][33][34][35][36][37]. The proof of [23] that (1.1), (1.2) is a supergravity symmetry relied upon DFT techniques, in particular the β-supergravity formalism [38][39][40].…”

Poisson-Lie U-duality in exceptional field theory

“…Note added: While finalising this manuscript, the paper [44] appeared which proposes a Uduality extension of Drinfeld Doubles and has some overlap with our sections 3 and 4 in the case where our I a = τ a5 .…”