The quantum double and related constructions (Q1295625)

The author studies several categorical constructions, going up in the hierarchy ``monoidal categories'', ``braided categories'', ``balanced categories'' and ``cotortile categories''. These constructions are illustrated in the case of categories of modules over a bialgebra, giving, via a Tannakian reconstruction-type argument, various versions of quantum doubles. Results of \textit{V. G. Drinfel'd} [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)], \textit{S. Majid} [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 197-206 (1991; Zbl 0762.18005)], \textit{N. Yu. Reshetikhin} and \textit{V. G. Turaev} [Commun. Math. Phys. 127, No. 1, 1-26 (1990; Zbl 0768.57003)], \textit{C. Kassel} and \textit{V. Turaev} [Acta Math. 175, No. 1, 1-48 (1995; Zbl 0833.18005)], are recovered and generalized.