Continuum Kac–Moody algebras have been recently introduced by the authors and O. Schiffmann in [ASS18]. 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 g 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, which allows to define on g a topological quasi–triangular Lie bialgebra structure. We then construct an explicit quantization of g, 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.
|Number of pages||35|
|Journal||Pure and applied mathematics quarterly|
|Publication status||Accepted/In press - 8 Oct 2019|