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 $\mathcal{E}_n$ algebra. This provides a natural setup to study non-Abelian U-duality because the $\mathcal{E}_n$ 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.

中文翻译：

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膜的非阿贝尔 U 对偶性

当目标空间具有由 Drinfel'd 对偶给出的广义等距群时，弦理论的 T 对偶可以扩展到 Poisson-Lie T 对偶。在M-理论中，T-对偶被理解为U-对偶的一个子群，但U-对偶的非阿贝尔扩展仍然是一个谜。在本文中，我们研究了曲面背景下的膜理论，其中包含 $\mathcal{E}_n$ 代数给出的广义等距群。这为研究非阿贝尔 U 对偶提供了一个自然的设置，因为 $\mathcal{E}_n$ 代数已被提议作为 Drinfel'd 双的 U 对偶扩展。我们表明，阿贝尔 U 对偶的标准处理可以扩展到非阿贝尔设置。然而，阿贝尔 U 对偶的一个著名问题仍然存在于非阿贝尔扩展中。