2012/12/30 by Andrea Appel, Appel, Andrea, Valerio Toledano-Laredo +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Braid group #Category Theory (math.CT) #Coxeter group #FOS: Mathematics #Functor #Hecke algebra #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Tensor product #Universal enveloping algebra #math.CT #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.1212.6720
63 pages. Exposition in sections 1 and 4 improved. Material added: definition of a split pair of Lie bialgebras (sect. 5.2-5.5), 1-jet of the relative twist (5.20), PROP description of the Verma modules L_-,N*_+ (7.6), restriction to Levi subalgebras (8.4), D-structures on Kac-Moody algebras (9.1)
openalex publication_date 2012/12/30 · arxiv created 2013/05/10 · arxiv updated 2013/05/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/08/05
Let g be a symmetrizable Kac-Moody algebra and Uh(g) its quantized\nenveloping algebra. The quantum Weyl group operators of Uh(g) and the\nuniversal R-matrices of its Levi subalgebras endow Uh(g) with a natural\nquasi-Coxeter quasitriangular quasibialgebra structure which underlies the\naction of the braid group of g and Artin's braid groups on the tensor product\nof integrable, category O modules. We show that this structure can be\ntransferred to the universal enveloping algebra Ug[[h]]. The proof relies on a\nmodification of the Etingof-Kazhdan quantization functor, and yields an\nisomorphism between (appropriate completions of) Uh(g) and Ug[[h]] preserving\na given chain of Levi subalgebras. We carry it out in the more general context\nof chains of Manin triples, and obtain in particular a relative version of the\nEtingof-Kazhdan functor with input a split pair of Lie bialgebras. Along the\nway, we develop the notion of quasi-Coxeter categories, which are to\ngeneralized braid groups what braided tensor categories are to Artin's braid\ngroups. This leads to their succint description as a 2-functor from a\n2-category whose morphisms are De Concini-Procesi associahedra. These results\nwill be used in the sequel to this paper to give a monodromic description of\nthe quantum Weyl group operators of an affine Kac-Moody algebra, extending the\none obtained by the second author for a semisimple Lie algebra.\n