Quantization of continuum Kac–Moody algebras

Continuum Kac–Moody algebras have been recently introduced by the authors and O. Schiffmann in [2]. 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 $\mathfrak{g}$ 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 $\mathfrak{g}$ a topological quasi–triangular Lie bialgebra structure. We then construct an explicit quantization of $\mathfrak{g}$, 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.

Keywords

quantum groups, infinite-dimensional Lie bialgebras

2010 Mathematics Subject Classification

Primary 17B65. Secondary 17B67, 81R50.

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The first-named author was partially supported by the ERC Grant 637618.

The second-named author was partially supported by World Premier International Research Center Initiative (WPI), MEXT, Japan, by JSPS KAKENHI Grant number JP17H06598 and by JSPS KAKENHI Grant number JP18K13402.

Received 4 March 2019

Accepted 8 October 2019

Published 11 November 2020