Comments on Bosonisation and Biproducts

Recently there appeared an interesting paper by Fischman and Montgomery[FM] in which some abstract constructions due to the author for Hopf algebras in braided categories [M6][M7][M8][M9] were applied to obtain a Schur’s double centraliser theorem for color Lie algebras. Unfortunately, in the introduction and preliminaries of their paper, some incorrect statements are made which could cause confusion and which we would like to correct here in this note. The most serious is the identification HM = HM which we cover in Section 8. The main results about color-Lie algebras, etc., in [FM] are, fortunately, not affected. Finally, we add some related remarks about a recent preprint of Pareigis [P2]. Most of these comments were made to Susan Montgomery at the LMS meeting on Noncommutative Rings, Durham, UK in July 1992, where we were shown the first draft of [FM] and pointed out [M6][M7][M8][M9]. We work over a ground field k and use the usual notations S, ǫ for the antipode and counit, and the notation ∆h = ∑ h(1) ⊗h(2) for the coproduct applied to an element h ∈ H[S]. We use the symbols >⊳, etc., for smash products, >◭ for smash coproducts and >⊳· when both are made simultaneously.