Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 3

Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

Quantization of continuum Kac–Moody algebras

Pages: 439 – 493

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a5


Andrea Appel (Dipartimento di Scienze Matematiche, Fisiche e Naturali, Università di Parma, Italy)

Francesco Sala (Dipartimento di Matematica, Università di Pisa, Italy)


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.


quantum groups, infinite-dimensional Lie bialgebras

2010 Mathematics Subject Classification

Primary 17B65. Secondary 17B67, 81R50.

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