Cohomological construction of relative twists

Abstract: Let be a complex, semi-simple Lie algebra, a Cartan subalgebra and D a subdiagram of the Dynkin diagram of . Let be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions and of the pentagon equation for and respectively. Motivated by the theory of quasi-Coxeter quasitriangular quasibialgebras [V. Toledano Laredo, Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups, math.QA/0506529], we study in this paper the existence of a relative twist, that is an element such that the twist of Φ by F is . Adapting the method of Donin and Shnider [J. Donin, S. Shnider, Cohomological construction of quantized universal enveloping algebras, Trans. Amer. Math. Soc. 349 (1997) 1611–1632], who treated the case of an empty D, so that and , we give a cohomological construction of such an F under the assumption that is the image of Φ under the generalised Harish-Chandra homomorphism . We also show that F is unique up to a gauge transformation if is of corank 1 or F satisfies where is an involution acting as −1 on . [Copyright &y& Elsevier]