Quantum symmetric algebras as braided hopf algebras

We review some of our results in [FG2] about the quantum symmetric algebra S-c(V), associated with a braided K-vector space (V, c). Using one of our formulae in the K-algebra K[B-n], obtained in [FG1], we give another proof of the fact that S-c(V) is a twisted bialgebra. Then we study S-c(V) and T-c(V), when V is a Yetter-Drinfeld module over a Hopf algebra H and c is the braiding induced from its Yetter-Drinfeld module structure. We prove that the induced braidings on the Yetter-Drinfeld modules T(V) and S(V) are equal to the twist F-c used by Rosso to define T-c(V) and S-c(V); this leads us to prove that T-c(V) and S-c(V) are Hopf algebras in the category of Yetter-Drinfeld modules over H, and that S-c(V) is the Nichols algebra S(V) associated with V, as considered in [A - S1].