More on the Quantum Double
Ross Street – 31 July 1996
This was a report on my recent preprint "The Quantum Double and Related Constructions". Three monoidal constructions were discussed:
1) a braided monoidal category Z(V) from any monoidal category V
2) a balanced monoidal category V^Z from any braided monoidal category V
3) a tortile monoidal category N(V) from any balanced monoidal category V
The first two constructions already appear in our paper A.Joyal & R.Street, Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991) 43-51. The last two are actually right adjoints to the forgetfuls. The Representation Theorem from Section 7 of our paper A.Joyal & R.Street, An Introduction to Tannaka duality and quantum groups; Lecture Notes in Mathematics 1488 (Springer-Verlag, 1991) 411-492 was invoked to obtain three corresponding constructions for bialgebras:
1) a cobraided bialgebra D_1(B) from any bialgebra B
2) a cobalanced bialgebra D_2(B) from any cobraided bialgebra B
3) a cotortile Hopf algebra D_3(B) from any cobalanced bialgebra B
Cotortile Hopf algebras provide a good abstract notion of quantum group.
Composing the three monoidal constructions we obtain a tortile monoidal category D(V) for each monoidal category V. This agrees with the double described in C.Kassel & V.Turaev, Double Construction for Monoidal Categories, Acta Math. 175 (1995) 1-48 when V is left autonomous. Composing the three bialgebra constructions, we obtain a quantum group D(B) for any bialgebra B. This agrees with the Reshetikhin-Turaev ribbon algebra of the Drinfield double in the case where B is a finite-dimensional Hopf algebra
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