Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.

中文翻译：

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连续统 Kac-Moody 代数的量化

最近作者和 O. Schiffmann 介绍了连续统 Kac-Moody 代数。这些是由连续根系统支配的李代数，可以将其实现为 Borcherds-Kac-Moody 代数的不可数共界。在本文中，我们证明了任何连续统 Kac-Moody 代数都具有典型的非退化不变双线性形式。正负 Borel 子代数相对于这种配对形成 Manin 三元组，在连续统 Kac-Moody 代数上引入拓扑准三角李双代数结构。然后我们构造了一个显式量化，我们将其称为连续量子群，并且我们表明后者同样被实现为 Drinfeld-Jimbo 量子群的不可数共限。