Quantum symmetric algebras II

M. Rosso has generalized G. Lusztigʹs construction of the Drinfeld–Jimbo quantum group [G. Lusztig, in: Progr. Math., Vol. 110, Birkhäuser, Boston, 1993], by associating to any (finite-dimensional) braided vector space (V,τ) two “twisted” bialgebras: namely the “quantum shuffle algebra” T(V), and a subalgebra S(V) of T(V), called the “quantum symmetric algebra.” When τ is Lusztigʹs braiding, S(V) is isomorphic to the “plus part” U+ of a certain quantum group Uq(g). Rosso [Invent. Math. 133 (1998) 399–416] generalizes this by using a braiding determined by a family of parameters tij, and our paper [D. Flores, J.A. Green, Algebr. Represent. Theory 4 (2001) 55–76] was concerned with this case. In the present work we study twisted bialgebras in more general context, and get results on Rossoʹs S(V) for arbitrary braidings. It is useful to consider a pair (T(U),T(V)) of twisted bialgebras which are adjoint under a given non-degenerate bilinear form. We give explicit formulae for the comultiplication of T(V), and also for its antipode. Under suitable conditions, S(V) is self-adjoint: this generalizes the part of Lusztigʹs construction which allows the “Drinfeld double” construction.