On Quasi-Hopf superalgebras

Abstract

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 ${\Delta }^{\prime }\equiv (S\otimes S)\cdot T\cdot \Delta \cdot S^{-1}$ induced on a QHSA is obtained from the coproduct $\Delta$ by twisting. The corresponding ``Drinfeld twist'' $F_D$ is explicitly constructed, as well as its inverse, and we investigate the complete QHSA associated with ${\Delta }^{\prime }$. We give a universal proof that the coassociator ${\Phi }^{\prime }=(S\otimes S\otimes S){\Phi }_{321}$ and canonical elements ${\alpha }^{\prime }=S(\beta ),$ ${\beta }^{\prime }=S(\alpha )$ correspond to twisting the original coassociator $\Phi ={\Phi }_{123}$ and canonical elements $\alpha ,\beta$ with the Drinfeld twist $F_D$. Moreover in the quasi-triangular case, it is shown algebraically that the R-matrix $R^{\prime }=(S\otimes S)R$ corresponds to twisting the original R-matrix $R$ with $F_D$. This has important consequences in knot theory, which will be investigated elsewhere.

Latex file, 34 pages; typo corrections (in some formulae), minor changes and one reference added