Uniqueness of Coxeter categories for Kac-Moody algebras

Andrea Appel, Valerio Toledano Laredo

Research output: Contribution to journalArticlepeer-review


Abstract. Let g be a symmetrisable Kac–Moody algebra, and Ung the corresponding quantum group. We showed in [1, 2] that the braided Coxeter structure on integrable, category O representations of Ung which underlies the R–matrix actions arising from the Levi subalgebras of Ung and the quantum Weyl group action of the generalised braid group Bg 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 [3] to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of
Ung. 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.
Original languageEnglish
Pages (from-to)1-104
Number of pages81
JournalAdvances in Mathematics
Early online date21 Feb 2019
Publication statusPublished - 30 Apr 2019


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