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Bicrossed product of Hopf algebra

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In quantum group and Hopf algebra, the bicrossed product is a process to create new Hopf algebras from the given ones. It's motivated by the Zappa–Szép product of groups. It was first discussed by M. Takeuchi in 1981,[1] and now a general tool for construction of Drinfeld quantum double.[2][3]

Bicrossed product

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Consider two bialgebras and , if there exist linear maps turning a module coalgebra over , and turning into a right module coalgebra over . We call them a pair of matched bialgebras, if we set and , the following conditions are satisfied

for all and . Here the Sweedler's notation of coproduct of Hopf algebra is used.

For matched pair of Hopf algebras and , there exists a unique Hopf algebra over , the resulting Hopf algebra is called bicrossed product of and and denoted by ,

  • The unit is given by ;
  • The multiplication is given by ;
  • The counit is ;
  • The coproduct is ;
  • The antipode is .

Drinfeld quantum double

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For a given Hopf algebra , its dual space has a canonical Hopf algebra structure and and are matched pairs. In this case, the bicrossed product of them is called Drinfeld quantum double .

References

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  1. ^ Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621
  2. ^ Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, vol. 155, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0783-2, ISBN 9780387943701
  3. ^ Majid, Shahn (1995), Foundations of quantum group theory, Cambridge University Press, doi:10.1017/CBO9780511613104, ISBN 9780511613104