# Quantization of continuum Kac-Moody algebras

## TL;DR

This paper introduces a quantization framework for continuum Kac-Moody algebras, establishing their bilinear forms, Lie bialgebra structures, and realizing them as colimits of quantum groups, advancing the understanding of infinite-dimensional algebraic structures.

## Contribution

It constructs a continuum quantum group as a quantization of continuum Kac-Moody algebras, extending quantum group theory to uncountable colimits.

## Key findings

- Existence of a non-degenerate invariant bilinear form
- Positive and negative Borel subalgebras form a Manin triple
- Continuum quantum groups are colimits of Drinfeld-Jimbo quantum groups

## Abstract

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.