Cohomological construction of relative twists

Abstract

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 g D ⊂ l D ⊆ g be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions Φ ∈ ( U g ⊗ 3 〚 ℏ 〛 ) g and Φ D ∈ ( U g D ⊗ 3 〚 ℏ 〛 ) g D of the pentagon equation for g and g D 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 ∈ ( U g ⊗ 2 〚 ℏ 〛 ) l D such that the twist of Φ by F is Φ D . 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 l D = h and Φ D = 1 ⊗ 3 , we give a cohomological construction of such an F under the assumption that Φ D is the image of Φ under the generalised Harish-Chandra homomorphism ( U g ⊗ 3 ) l D → ( U g D ⊗ 3 ) g D . We also show that F is unique up to a gauge transformation if l D is of corank 1 or F satisfies F Θ = F 21 where Θ ∈ Aut ( g ) is an involution acting as −1 on h .