Braided-Lie bialgebras associated to Kac-Moody algebras

TL;DR

This paper constructs new infinite-dimensional braided-Lie bialgebras from Kac-Moody algebra inclusions, providing explicit examples and analyzing their structures, especially for affine and simple Lie algebras like sl_3.

Contribution

It extends the theory of braided-Lie bialgebras to include those associated with Kac-Moody algebra inclusions, offering explicit constructions and examples.

Findings

New braided-Lie bialgebras associated to Kac-Moody inclusions

Explicit braided cobracket for sl_3

Identification of current algebra as braided-Lie bialgebra

Abstract

Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra $sl_3$.