  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.
