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Braided Lie Algebras, Double Biproducts and A Schur's Double Centralizer Theorem

Let $H$ be a quasitriangular Hopf algebra and $B$ a coquasitriangular Hopf algebra. We construct a braided Lie algebra in ${}_H^B{\cal L}$(see [16]) by sketching this procedure in the framework of a braided Lie algebra concerning an algebra in any symmetric braided monoidal category ${\cal C}$, and study the structure of braided Lie algebras in ${}_H^B{\cal L}$, and in particular the braided Lie structure of an algebra $A$ in ${}_H^B{\cal L}$. Next, we study braided enveloping algebra of a braided Lie algebra in ${}_H^B{\cal L}$ by imitating the standard algebraic constructions. One of our results gives a positive answer to a question of [3, p.42]. Finally, we introduce a double biproduct which transforms braided Hopf algebras in ${}_H^B{\cal L}$ to usual Hopf algebra, and study its coalgebra structures. As application, using double biproduct and the above we unify and extend both Cohen-Fischman-Westreich-Schur's double centralizer theorem [5] and Fischman-Montgomery-Schur's double centralizer theorem [9] to the one for the generalized Long module category ${}_H^B{\cal L}$.