Mathematics – Quantum Algebra
Scientific paper
2005-06-26
Advances in Mathematics 210 (2007), 375-403.
Mathematics
Quantum Algebra
minor touch-ups, to appear in Advances in Math
Scientific paper
Let g be a complex, semi-simple Lie algebra, h a Cartan subalgebra of g and D a subdiagram of the Dynkin diagram of g. Let g_D and l_D be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions Phi, Phi_D of the pentagon equation for g and g_D respectively. Motivated by the theory of quasi-Coxeter quasitriangular quasibialgebras \cite{TL3}, we study in this paper the existence of a relative twist, that is an element F invariant under l_D such that the twist of Phi by F is Phi_D. Adapting the method of Donin and Shnider, who treated the case of an empty D, so that l_D=h and Phi_D=1, we give a cohomological construction of such an F under the assumption that Phi_D is the image of Phi under the generalised Harish-Chandra homomorphism. We also show that F is unique up to a gauge transformation if l_D is of corank 1 or F satisfies F^\Theta= F^{21} where \Theta is an involution of g acting as -1 on h.
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