Squared Hopf algebras and reconstruction theorems

Details Squared Hopf algebras and reconstruction theorems Squared Hopf algebras and reconstruction theorems

1996-05-22

arxiv.org/abs/q-alg/9605035v1

Proc. Workshop ``Quantum Groups and Quantum Spaces'' (Warszawa), Banach Center Publ., no. 40, Inst. Math. Polish Acad. Sci. (1

Mathematics

Quantum Algebra

Latex2e, 31 pages, to appear in the Proceedings of Banach Center Minisemester on Quantum Groups, November 1995

Scientific paper

Given an abelian k-linear rigid monoidal category V, where k is a perfect field, we define squared coalgebras as objects of cocompleted V tensor V (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If V is the category of k-vector spaces, squared (co)algebras coincide with conventional ones. If V is braided, a braided Hopf algebra can be obtained from a squared one. Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras in V and corresponding fibre functors to V (which is not the case with other definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to the problem of defining quantum groups in braided categories.

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