Non-Abelian U-duality for membranes

The T-duality of string theory can be extended to the Poisson-Lie T-duality when the target space has a generalized isometry group given by a Drinfel'd double. In M-theory, T-duality is understood as a subgroup of U-duality, but the non-Abelian extension of U-duality is still a mystery. In this paper we study membrane theory on a curved background with a generalized isometry group given by the epsilon(n) algebra. This provides a natural setup to study non-Abelian U-duality because the epsilon(n) algebra has been proposed as a U-duality extension of the Drinfel'd double. We show that the standard treatment of Abelian U-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian U-duality still exists in the non-Abelian extension.