Uniqueness of Coxeter structures on Kac-Moody algebras

TL;DR

This paper proves the uniqueness of the braided quasi-Coxeter structure on integrable category O representations of symmetrisable Kac-Moody algebras, extending previous results and connecting to monodromy of the Casimir connection.

Contribution

It establishes the unique existence of a certain algebraic structure on category O representations of Kac-Moody algebras, generalizing prior semisimple cases.

Findings

Proves the uniqueness of the braided quasi-Coxeter structure up to equivalence.

Extends results from semisimple to symmetrisable Kac-Moody algebras.

Connects the structure to monodromy of the Casimir connection.

Abstract

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g.