Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras (Q1612175)

Let \(A\) be a complex algebra and \(\sigma\) an automorphism of \(A\otimes A\) with some braiding conditions. Define the quantum shuffle multiplication \(\phi_\sigma\colon A^{\otimes p}\otimes A^{\otimes q}\to A^{\otimes(p+q)}\) by \[ a\otimes a'\to\sum_{w\in S_{p,p+q}}(-1)^{|w|}T_\sigma(w)(a\otimes a'), \] where \(T_\sigma\) is the representation of the braided group \(B_{p+q}\) on \(A^{\otimes(p+q)}\). It is shown that \(\phi_\sigma=\mu^{\otimes*}\Theta\), where \(\mu\) is a multiplication in \(A\) and \(\Theta\colon T(A)\otimes T(A)\to T(A\otimes_\sigma A)\), where \(A\otimes_\sigma A\) is a braid of the algebra \(A\otimes A\) by \(\sigma\). Moreover, \(1\otimes\Theta\) is a special morphism of complexes. It is shown that the Hochschild-Serre identity is the dual statement of this result. An extension of the identity mentioned to Hopf algebras and a construction of quasi-Hopf algebras associated to finite dimensional cocommutative Hopf algebras is presented.