Cohomological construction of relative twists

Let g be a complex, semi-simple Lie algebra, h⊂g a Cartan subalgebra and D a subdiagram of the Dynkin diagram of g. Let gD⊂lD⊆g be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions Φ∈g(Ug⊗3〚ℏ〛) and of the pentagon equation for g and gD 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 F∈lD(Ug⊗2〚ℏ〛) such that the twist of Φ by F is ΦD. 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 lD=h and ΦD=1⊗3, we give a cohomological construction of such an F under the assumption that ΦD is the image of Φ under the generalised Harish-Chandra homomorphism . We also show that F is unique up to a gauge transformation if lD is of corank 1 or F satisfies FΘ=F21 where Θ∈Aut(g) is an involution acting as −1 on h.