On Quasi-Hopf superalgebras

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In this work we investigate several important aspects of the structure theory of the recently introduced quasi-Hopf superalgebras (QHSAs), which play a fundamental role in knot theory and integrable systems. In particular we introduce the opposite structure and prove in detail (for the graded case) Drinfeld's result that the coproduct Δ' ≡ (S⊗ S)· T· Δ· S⁻¹ induced on a QHSA is obtained from the coproduct Δ by twisting. The corresponding ``Drinfeld twist'' FD is explicitly constructed, as well as its inverse, and we investigate the complete QHSA associated with Δ'. We give a universal proof that the coassociator Φ'=(S⊗ S⊗ S)Φ₃₂₁ and canonical elements α' = S(β), β' = S(α) correspond to twisting the original coassociator Φ= Φ₁₂₃ and canonical elements α,β with the Drinfeld twist FD. Moreover in the quasi-triangular case, it is shown algebraically that the R-matrix R' = (S⊗ S)R corresponds to twisting the original R-matrix R with FD. This has important consequences in knot theory, which will be investigated elsewhere.