E6(6) exceptional Drinfel’d algebras

Abstract

The exceptional Drinfel’d algebra (EDA) is a Leibniz algebra introduced to provide an algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence, it provides an M-theoretic analogue of the way a Drinfel’d double encodes generalised T-dualities of strings. In this note we detail the construction of the EDA in the case where the regular U-duality group is E6(6). We show how the EDA can be realised geometrically as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle for a six-dimensional group manifold G, endowed with a Nambu-Lie structure. When the EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples including some which embed Drinfel’d doubles and others that are not of this type.

Jacobi-Lie T-plurality

“…In [20][21][22], the Poisson-Lie T -duality/T -plurality has been proven to be a symmetry of DFT. As a natural extension, non-Abelian U-duality associated with EDA has been discussed in [4][5][6][7][8][9][10]16], and several examples of the non-Abelian U-duality have been found in [11]. Here, we show that the non-Abelian duality based on a DD + , i.e., the Jacobi-Lie T -plurality, is a symmetry of the DFT equations of motion.…”

“…When α = 0 , we fail to find an O(3, 3) transformation which maps this fluxes into any DD + . 7 To be a little more specific, let us consider the case ξ 0 = 0 . By considering the eigenvalues of the Killing form, the only possible Drinfel'd doubles that may be related to orbit 6 are DD14-DD17.…”

“…Recently the Poisson-Lie T -duality [1,2] or T -plurality [3] and their U-duality extensions [4][5][6][7][8][9][10][11] have been studied and developed by using the duality-covariant formulations, such as double field theory (DFT) [12][13][14][15] and its U-duality extensions. The Poisson-Lie T -duality is based on a Lie algebra called the Drinfel'd double while the U-duality variant is based on the exceptional Drinfel'd algebra (EDA) [4-7, 9, 16], which is an extension of the Drinfel'd double.…”

Currents, charges and algebras in exceptional generalised geometry

“…The type of finite-dimensional algebras, as for example the recently discussed exceptional Drinfeld algebras [51,52,[56][57][58][59][60][61][62][63], are contained in the current algebra as the algebra of the zero modes z A = − Z A in case F is constant: 6 The SL( 5)-vielbeine and their invariance conditions, used here, are:…”

“…Success in the strive for an E-model like formulation has, by now, only been achieved for the E 3(3) -theory and partly the E 4(4) -theory [32,42,43,[50][51][52]. Generalising [51], such a formulation might -similar to Poisson-Lie T -duality as canonical transformation of classical string theory -help to understand to which extent the recently discussed Poisson-Lie U -duality, transformations of exceptional Drinfeld algebras or other interesting flux configurations [51][52][53][54][55][56][57][58][59][60][61][62][63], can be understood as some kind of classical duality transformation of world-volume theories in exceptional field theory.…”